Displaying 1-10 of 17 results found.
Number of partitions of n with positive rank.
+10
31
0, 1, 1, 2, 3, 5, 6, 10, 13, 19, 25, 35, 45, 62, 80, 106, 136, 178, 225, 291, 366, 466, 583, 735, 912, 1140, 1407, 1743, 2140, 2634, 3214, 3932, 4776, 5807, 7022, 8495, 10225, 12313, 14762, 17696, 21136, 25236, 30030, 35722, 42367, 50216, 59368, 70138, 82665
COMMENTS
The rank of a partition is the largest summand minus the number of summands.
Also number of partitions of n with negative rank. - Omar E. Pol, Mar 05 2012 Number of partitions p of n such that max(max(p), number of parts of p) is not a part of p. - Clark Kimberling, Feb 28 2014 The sequence enumerates the semigroup of partitions of positive rank for each number n. The semigroup is a subsemigroup of the monoid of partitions of nonnegative rank under the binary operation "*": Let A be the positive rank partition (a1,...,ak) where ak > k, and let B=(b1,...bj) with bj > j. Then let A*B be the partition (a1b1,...,a1bj,...,akb1,...,akbj), which has akbj > kj, thus having positive rank. For example, the partition (2,3,4) of 9 has rank 1, and its product with itself is (4,6,6,8,8,9,12,12,16) of 81, which has rank 7. A similar situation holds for partitions of negative rank--they are a subsemigroup of the monoid of nonpositive rank partitions. - Richard Locke Peterson, Jul 15 2018
FORMULA
a(n) = p(n-2) - p(n-7) + p(n-15) - ... - (-1)^k*p(n-(3*k^2+k)/2) + ..., where p() is A000041(). - Vladeta Jovovic, Aug 04 2004 G.f.: Product_{k>=1} (1/(1-q^k)) * Sum_{k>=1} ( (-1)^k * (-q^(3*k^2/2+k/2))) (conjectured). - Thomas Baruchel, May 12 2018 G.f.: Sum_{k>=1} x^k * Product_{j=1..k} (1-x^(k+j-2)/(1-x^j). - Seiichi Manyama, Jan 25 2022
EXAMPLE
a(20) = p(18) - p(13) + p(5) = 385 - 101 + 7 = 291.
The a(2) = 1 through a(9) = 13 partitions of positive rank:
(2) (3) (4) (5) (6) (7) (8) (9)
(31) (32) (33) (43) (44) (54)
(41) (42) (52) (53) (63)
(51) (61) (62) (72)
(411) (421) (71) (81)
(511) (422) (432)
(431) (441)
(521) (522)
(611) (531)
(5111) (621)
(711)
(5211)
(6111)
(End)
MAPLE
a := 0 ;
for p in combinat[partition](n) do
r := max(op(p))-nops(p) ;
if r > 0 then
a := a+1 ;
end if;
end do:
a ;
end proc:
MATHEMATICA
Table[Count[IntegerPartitions[n], q_ /; First[q] > Length[q]], {n, 24}] (* Clark Kimberling, Feb 12 2014 *) Table[Count[IntegerPartitions[n], p_ /; ! MemberQ[p, Max[Max[p], Length[p]]]], {n, 20}] (* Clark Kimberling, Feb 28 2014 *) P = PartitionsP;
a[n_] := (P[n] - Sum[-(-1)^k (P[n - (3k^2 - k)/2] - P[n - (3k^2 + k)/2]), {k, 1, Floor[(1 + Sqrt[1 + 24n])/6]}])/2;
PROG
(PARI) my(N=66, x='x+O('x^N)); concat(0, Vec(sum(k=1, N, x^k*prod(j=1, k, (1-x^(k+j-2))/(1-x^j))))) \\ Seiichi Manyama, Jan 25 2022
CROSSREFS
Note: A-numbers of ranking sequences are in parentheses below.
These partitions are ranked by ( A340787). A072233 counts partitions by sum and length.
A168659 counts partitions whose length is a multiple of the greatest part.
A200750 counts partitions whose length and greatest part are coprime.
- Rank -
A257541 gives the rank of the partition with Heinz number n.
- Balance -
A340599 counts alt-balanced factorizations.
A340653 counts balanced factorizations.
Heinz numbers of integer partitions of odd positive rank.
+10
29
3, 7, 10, 13, 15, 19, 22, 25, 28, 29, 33, 34, 37, 42, 43, 46, 51, 52, 53, 55, 61, 62, 63, 69, 70, 71, 76, 77, 78, 79, 82, 85, 88, 89, 93, 94, 98, 101, 105, 107, 113, 114, 115, 116, 117, 118, 119, 121, 123, 130, 131, 132, 134, 136, 139, 141, 146, 147, 148, 151
COMMENTS
The Dyson rank of a nonempty partition is its maximum part minus its number of parts. The rank of an empty partition is 0.
The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.
EXAMPLE
The sequence of partitions with their Heinz numbers begins:
3: (2) 46: (9,1) 82: (13,1)
7: (4) 51: (7,2) 85: (7,3)
10: (3,1) 52: (6,1,1) 88: (5,1,1,1)
13: (6) 53: (16) 89: (24)
15: (3,2) 55: (5,3) 93: (11,2)
19: (8) 61: (18) 94: (15,1)
22: (5,1) 62: (11,1) 98: (4,4,1)
25: (3,3) 63: (4,2,2) 101: (26)
28: (4,1,1) 69: (9,2) 105: (4,3,2)
29: (10) 70: (4,3,1) 107: (28)
33: (5,2) 71: (20) 113: (30)
34: (7,1) 76: (8,1,1) 114: (8,2,1)
37: (12) 77: (5,4) 115: (9,3)
42: (4,2,1) 78: (6,2,1) 116: (10,1,1)
43: (14) 79: (22) 117: (6,2,2)
MATHEMATICA
rk[n_]:=PrimePi[FactorInteger[n][[-1, 1]]]-PrimeOmega[n];
Select[Range[100], OddQ[rk[#]]&&rk[#]>0&]
CROSSREFS
Note: Heinz numbers are given in parentheses below.
These partitions are counted by A101707. A001222 gives number of prime indices.
- Rank -
A257541 gives the rank of the partition with Heinz number n.
A340653 counts balanced factorizations.
- Odd -
A339890 counts factorizations of odd length.
A340101 counts factorizations into odd factors.
A340102 counts odd-length factorizations into odd factors.
Cf. A001221, A006141, A056239, A112798, A168659, A200750, A316413, A325134, A340601, A340602, A340608, A340609, A340610.
Number of integer partitions of n of even rank.
+10
27
1, 1, 0, 3, 1, 5, 3, 11, 8, 18, 16, 34, 33, 57, 59, 98, 105, 159, 179, 262, 297, 414, 478, 653, 761, 1008, 1184, 1544, 1818, 2327, 2750, 3480, 4113, 5137, 6078, 7527, 8899, 10917, 12897, 15715, 18538, 22431, 26430, 31805, 37403, 44766, 52556, 62620, 73379
COMMENTS
The Dyson rank of a nonempty partition is its maximum part minus its number of parts. For this sequence, the rank of an empty partition is 0.
FORMULA
G.f.: 1 + Sum_{i, j>0} q^(i*j) * ( (1+(-1)^(i+j))/2 + Sum_{k>0} q^k * q_binomial(k,i-2) * (1+(-1)^(i+j+k))/2 ). - John Tyler Rascoe, Apr 15 2024
EXAMPLE
The a(1) = 1 through a(9) = 18 partitions (empty column indicated by dot):
(1) . (3) (22) (5) (42) (7) (44) (9)
(21) (41) (321) (43) (62) (63)
(111) (311) (2211) (61) (332) (81)
(2111) (322) (521) (333)
(11111) (331) (2222) (522)
(511) (4211) (531)
(2221) (32111) (711)
(4111) (221111) (4221)
(31111) (4311)
(211111) (6111)
(1111111) (32211)
(33111)
(51111)
(222111)
(411111)
(3111111)
(21111111)
(111111111)
MAPLE
b:= proc(n, i, r) option remember; `if`(n=0, 1-max(0, r),
`if`(i<1, 0, b(n, i-1, r) +b(n-i, min(n-i, i), 1-
`if`(r<0, irem(i, 2), r))))
end:
a:= n-> b(n$2, -1):
MATHEMATICA
Table[If[n==0, 1, Length[Select[IntegerPartitions[n], EvenQ[Max[#]-Length[#]]&]]], {n, 0, 30}]
(* Second program: *)
b[n_, i_, r_] := b[n, i, r] = If[n == 0, 1 - Max[0, r], If[i < 1, 0, b[n, i - 1, r] + b[n - i, Min[n - i, i], 1 - If[r < 0, Mod[i, 2], r]]]];
a[n_] := b[n, n, -1];
PROG
(PARI)
p_q(k) = {prod(j=1, k, 1-q^j); }
GB_q(N, M)= {if(N>=0 && M>=0, p_q(N+M)/(p_q(M)*p_q(N)), 0 ); }
A_q(N) = {my(q='q+O('q^N), g=1+sum(i=1, N, sum(j=1, N/i, q^(i*j) * ( ((1/2)*(1+(-1)^(i+j))) + sum(k=1, N-(i*j), ((q^k)*GB_q(k, i-2)) * ((1/2)*(1+(-1)^(i+j+k)))))))); Vec(g)}
CROSSREFS
Note: Heinz numbers are given in parentheses below.
The Heinz numbers of these partitions are A340602. - Rank -
A072233 counts partitions by sum and length.
A257541 gives the rank of the partition with Heinz number n.
A340653 counts factorizations of rank 0.
- Even -
A024430 counts set partitions of even length.
A034008 counts compositions of even length.
A052841 counts ordered set partitions of even length.
A339846 counts factorizations of even length.
Number of partitions of n having positive odd rank (the rank of a partition is the largest part minus the number of parts).
+10
22
0, 0, 1, 0, 2, 1, 4, 2, 7, 6, 13, 11, 22, 22, 38, 39, 63, 69, 103, 114, 165, 189, 262, 301, 407, 475, 626, 733, 950, 1119, 1427, 1681, 2118, 2503, 3116, 3678, 4539, 5360, 6559, 7735, 9400, 11076, 13372, 15728, 18886, 22184, 26501, 31067, 36947, 43242, 51210, 59818, 70576, 82291, 96750
REFERENCES
George E. Andrews, The Theory of Partitions, Addison-Wesley, Reading, Mass., 1976.
FORMULA
G.f.: Sum((-1)^(k+1)*x^((3*k^2+k)/2)/(1+x^k), k=1..infinity)/Product(1-x^k, k=1..infinity). - Vladeta Jovovic, Dec 20 2004
EXAMPLE
a(7)=2 because the only partitions of 7 with positive odd rank are 421 (rank=1) and 52 (rank=3).
Also the number of integer partitions of n into an even number of parts, the greatest of which is odd. For example, the a(2) = 1 through a(10) = 13 partitions (empty column indicated by dot) are:
11 . 31 32 33 52 53 54 55
1111 51 3211 71 72 73
3111 3221 3222 91
111111 3311 3321 3322
5111 5211 3331
311111 321111 5221
11111111 5311
7111
322111
331111
511111
31111111
1111111111
Also the number of integer partitions of n into an odd number of parts, the greatest of which is even. For example, the a(2) = 1 through a(10) = 13 partitions (empty column indicated by dot, A = 10) are:
2 . 4 221 6 421 8 432 A
211 222 22111 422 441 433
411 431 621 442
21111 611 22221 622
22211 42111 631
41111 2211111 811
2111111 22222
42211
43111
61111
2221111
4111111
211111111
(End)
MAPLE
b:= proc(n, i, r) option remember; `if`(n=0, max(0, r),
`if`(i<1, 0, b(n, i-1, r) +b(n-i, min(n-i, i), 1-
`if`(r<0, irem(i, 2), r))))
end:
a:= n-> b(n$2, -1)/2:
MATHEMATICA
Table[Length[Select[IntegerPartitions[n], EvenQ[Length[#]]&&OddQ[Max[#]]&]], {n, 0, 30}] (* Gus Wiseman, Feb 10 2021 *) b[n_, i_, r_] := b[n, i, r] = If[n == 0, Max[0, r],
If[i < 1, 0, b[n, i - 1, r] + b[n - i, Min[n - i, i], 1 -
If[r < 0, Mod[i, 2], r]]]];
a[n_] := b[n, n, -1]/2;
CROSSREFS
Note: A-numbers of ranking sequences are in parentheses below.
The even- but not necessarily positive-rank version is A340601 ( A340602). The Heinz numbers of these partitions are ( A340604). - Rank -
A257541 gives the rank of the partition with Heinz number n.
- Odd -
A026804 counts partitions whose least part is odd.
A339890 counts factorizations of odd length.
Coefficients of the 3rd-order mock theta function f(q).
(Formerly M0433 N0164)
+10
20
1, 1, -2, 3, -3, 3, -5, 7, -6, 6, -10, 12, -11, 13, -17, 20, -21, 21, -27, 34, -33, 36, -46, 51, -53, 58, -68, 78, -82, 89, -104, 118, -123, 131, -154, 171, -179, 197, -221, 245, -262, 279, -314, 349, -369, 398, -446, 486, -515, 557, -614, 671, -715, 767, -845, 920, -977, 1046, -1148, 1244
COMMENTS
a(n) = number of partitions of n with even rank minus number with odd rank. The rank of a partition is its largest part minus the number of parts.
REFERENCES
G. E. Andrews, The Theory of Partitions, Addison-Wesley, 1976, p. 82, Examples 4 and 5.
Srinivasa Ramanujan, Collected Papers, Chelsea, New York, 1962, pp. 354-355
Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, pp. 17, 31.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
FORMULA
G.f.: 1 + Sum_{n>=1} (q^(n^2) / Product_{i=1..n} (1 + q^i)^2).
G.f.: (1 + 4 * Sum_{n>=1} (-1)^n * q^(n*(3*n+1)/2) / (1 + q^n)) / Product_{i>=1} (1 - q^i).
a(n) ~ -(-1)^n * exp(Pi*sqrt(n/6)) / (2*sqrt(n)) [Ramanujan]. - Vaclav Kotesovec, Jun 10 2019 G.f.: 1 - Sum_{n >= 1} (-1)^n*x^n/Product_{k = 1..n} 1 + x^k. See Fine, equation 26.22, p. 55. - Peter Bala, Feb 04 2021 G.f.: 1 + (1/Product_{k>=1} (1-x^k)) * Sum_{k>=1} (-1)^(k-1) * x^(k*(3*k-1)/2) * (1-x^k)^2 / (1+x^k). (End)
EXAMPLE
G.f. = 1 + q - 2*q^2 + 3*q^3 - 3*q^4 + 3*q^5 - 5*q^6 + 7*q^7 - 6*q^8 + 6*q^9 + ...
MAPLE
a:= m-> coeff(series((1+4*add((-1)^n*q^(n*(3*n+1)/2)/
(1+q^n), n=1..m))/mul(1-q^i, i=1..m), q, m+1), q, m):
seq(a(n), n=0..120);
MATHEMATICA
CoefficientList[Series[(1+4Sum[(-1)^n q^(n(3n+1)/2)/(1+q^n), {n, 1, 10}])/Sum[(-1)^n q^(n(3n+1)/2), {n, -8, 8}], {q, 0, 100}], q] (* N. J. A. Sloane *) sgn[P_ (* a partition *)] :=
Signature[
PermutationList[
Cycles[Flatten[
SplitBy[Range[Total[P]], (Function[{x}, x > #1] &) /@
Accumulate[P]], Length[P] - 1]]]]
conjugate[P_List(* a partition *)] :=
Module[{s = Select[P, #1 > 0 &], i, row, r}, row = Length[s];
Table[r = row; While[s[[row]] <= i, row--]; r, {i, First[s]}]]
Total[Function[{x}, sgn[x] sgn[conjugate[x]]] /@
IntegerPartitions[#]] & /@ Range[20]
a[ n_] := If[ n < 0, 0, SeriesCoefficient[ Sum[ x^k^2 / Product[ 1 + x^j, {j, k}]^2, {k, 0, Sqrt@n}], {x, 0, n}]]; (* Michael Somos, Jun 30 2015 *) rnk[prts_]:=Max[prts]-Length[prts]; mtf[n_]:=Module[{pn=IntegerPartitions[n]}, Total[If[ EvenQ[ rnk[#]], 1, -1]&/@pn]]; Join[{1}, Array[mtf, 60]] (* Harvey P. Dale, Sep 13 2024 *)
PROG
(PARI) {a(n) = if( n<0, 0, polcoeff( sum(k=1, sqrtint(n), x^k^2 / prod(i=1, k, 1 + x^i, 1 + x * O(x^(n - k^2)))^2, 1), n))}; /* Michael Somos, Sep 02 2007 */ (PARI) my(N=60, x='x+O('x^N)); Vec(1+1/prod(k=1, N, 1-x^k)*sum(k=1, N, (-1)^(k-1)*x^(k*(3*k-1)/2)*(1-x^k)^2/(1+x^k))) \\ Seiichi Manyama, May 23 2023
Numbers that can be factored into factors > 1, the least of which is odd.
+10
19
3, 5, 7, 9, 11, 12, 13, 15, 17, 18, 19, 21, 23, 24, 25, 27, 29, 30, 31, 33, 35, 36, 37, 39, 40, 41, 42, 43, 45, 47, 48, 49, 50, 51, 53, 54, 55, 56, 57, 59, 60, 61, 63, 65, 66, 67, 69, 70, 71, 72, 73, 75, 77, 78, 79, 80, 81, 83, 84, 85, 87, 89, 90, 91, 93, 95
COMMENTS
These are numbers that are odd or have an odd divisor 1 < d <= n/d.
EXAMPLE
The sequence of terms together with their prime indices begins:
3: {2} 27: {2,2,2} 48: {1,1,1,1,2}
5: {3} 29: {10} 49: {4,4}
7: {4} 30: {1,2,3} 50: {1,3,3}
9: {2,2} 31: {11} 51: {2,7}
11: {5} 33: {2,5} 53: {16}
12: {1,1,2} 35: {3,4} 54: {1,2,2,2}
13: {6} 36: {1,1,2,2} 55: {3,5}
15: {2,3} 37: {12} 56: {1,1,1,4}
17: {7} 39: {2,6} 57: {2,8}
18: {1,2,2} 40: {1,1,1,3} 59: {17}
19: {8} 41: {13} 60: {1,1,2,3}
21: {2,4} 42: {1,2,4} 61: {18}
23: {9} 43: {14} 63: {2,2,4}
24: {1,1,1,2} 45: {2,2,3} 65: {3,6}
25: {3,3} 47: {15} 66: {1,2,5}
For example, 72 is in the sequence because it has three suitable factorizations: (3*3*8), (3*4*6), (3*24).
MATHEMATICA
Select[Range[100], Function[n, n>1&&(OddQ[n]||Select[Rest[Divisors[n]], OddQ[#]&&#<=n/#&]!={})]]
CROSSREFS
The version looking at greatest factor is A057716. These factorization are counted by A340832. A033676 selects the maximum inferior divisor.
A055396 selects the least prime index.
- Factorizations -
A045778 counts strict factorizations.
A316439 counts factorizations by product and length.
A339890 counts factorizations of odd length.
A340653 counts balanced factorizations.
- Odd -
A000009 counts partitions into odd parts.
A024429 counts set partitions of odd length.
A026424 lists numbers with odd Omega.
A066208 lists Heinz numbers of partitions into odd parts.
A174726 counts ordered factorizations of odd length.
A332304 counts strict compositions of odd length.
A340692 counts partitions of odd rank.
Cf. A026804, A027193, A050320, A244991, A340101, A340102, A340596, A340597, A340607, A340654, A340655, A340852.
Heinz numbers of integer partitions of even rank.
+10
18
1, 2, 5, 6, 8, 9, 11, 14, 17, 20, 21, 23, 24, 26, 30, 31, 32, 35, 36, 38, 39, 41, 44, 45, 47, 49, 50, 54, 56, 57, 58, 59, 65, 66, 67, 68, 73, 74, 75, 80, 81, 83, 84, 86, 87, 91, 92, 95, 96, 97, 99, 102, 103, 104, 106, 109, 110, 111, 120, 122, 124, 125, 126, 127
COMMENTS
The Dyson rank of a nonempty partition is its maximum part minus its length. The rank of an empty partition is 0.
The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.
EXAMPLE
The sequence of partitions with their Heinz numbers begins:
1: () 31: (11) 58: (10,1)
2: (1) 32: (1,1,1,1,1) 59: (17)
5: (3) 35: (4,3) 65: (6,3)
6: (2,1) 36: (2,2,1,1) 66: (5,2,1)
8: (1,1,1) 38: (8,1) 67: (19)
9: (2,2) 39: (6,2) 68: (7,1,1)
11: (5) 41: (13) 73: (21)
14: (4,1) 44: (5,1,1) 74: (12,1)
17: (7) 45: (3,2,2) 75: (3,3,2)
20: (3,1,1) 47: (15) 80: (3,1,1,1,1)
21: (4,2) 49: (4,4) 81: (2,2,2,2)
23: (9) 50: (3,3,1) 83: (23)
24: (2,1,1,1) 54: (2,2,2,1) 84: (4,2,1,1)
26: (6,1) 56: (4,1,1,1) 86: (14,1)
30: (3,2,1) 57: (8,2) 87: (10,2)
MATHEMATICA
Select[Range[100], EvenQ[PrimePi[FactorInteger[#][[-1, 1]]]-PrimeOmega[#]]&]
CROSSREFS
Taking only maximum part gives A061395. These partitions are counted by A340601. The case of positive rank is A340605. - Rank -
A257541 gives the rank of the partition with Heinz number n.
A324516 counts partitions with rank = maximum minus minimum part ( A324515).
A340653 counts factorizations of rank 0.
- Even -
A024430 counts set partitions of even length.
A034008 counts compositions of even length.
A052841 counts ordered set partitions of even length.
A339846 counts factorizations of even length.
Cf. A000041, A006141, A056239, A072233, A112798, A168659, A325134, A326836, A326845, A340386, A340387.
Number of factorizations of n into factors > 1 with odd least factor.
+10
15
0, 0, 1, 0, 1, 0, 1, 0, 2, 0, 1, 1, 1, 0, 2, 0, 1, 1, 1, 0, 2, 0, 1, 1, 2, 0, 3, 0, 1, 2, 1, 0, 2, 0, 2, 2, 1, 0, 2, 1, 1, 1, 1, 0, 4, 0, 1, 2, 2, 1, 2, 0, 1, 2, 2, 1, 2, 0, 1, 3, 1, 0, 4, 0, 2, 1, 1, 0, 2, 2, 1, 3, 1, 0, 4, 0, 2, 1, 1, 1, 5, 0, 1, 3, 2, 0, 2, 0, 1, 5, 2, 0, 2, 0, 2, 2, 1, 1, 4, 1, 1, 1, 1, 0, 5, 0, 1, 6
EXAMPLE
The a(n) factorizations for n = 45, 108, 135, 180, 252:
(45) (3*36) (135) (3*60) (3*84)
(5*9) (9*12) (3*45) (5*36) (7*36)
(3*15) (3*4*9) (5*27) (9*20) (9*28)
(3*3*5) (3*6*6) (9*15) (5*6*6) (3*3*28)
(3*3*12) (3*5*9) (3*3*20) (3*4*21)
(3*3*3*4) (3*3*15) (3*4*15) (3*6*14)
(3*3*3*5) (3*5*12) (3*7*12)
(3*6*10) (3*3*4*7)
(3*3*4*5)
MATHEMATICA
facs[n_]:=If[n<=1, {{}}, Join@@Table[Map[Prepend[#, d]&, Select[facs[n/d], Min@@#>=d&]], {d, Rest[Divisors[n]]}]];
Table[Length[Select[facs[n], OddQ@*Min]], {n, 100}]
PROG
(PARI) A340832(n, m=n, fc=1) = if(1==n, (m%2)&&!fc, my(s=0); fordiv(n, d, if((d>1)&&(d<=m), s += A340832(n/d, d, 0*fc))); (s)); \\ Antti Karttunen, Dec 13 2021
CROSSREFS
Positions of nonzero terms are A340855. The version for partitions is A026804. Odd-length factorizations are counted by A339890. The version looking at greatest factor is A340831. - Factorizations -
A045778 counts strict factorizations.
A316439 counts factorizations by product and length.
A340101 counts factorizations into odd factors, odd-length case A340102.
A340607 counts factorizations with odd length and greatest factor.
A340653 counts balanced factorizations.
- Odd -
A000009 counts partitions into odd parts.
A026424 lists numbers with odd Omega.
A027193 counts partitions of odd length.
A066208 lists numbers with odd-indexed prime factors.
A174726 counts ordered factorizations of odd length.
A244991 lists numbers whose greatest prime index is odd.
A340692 counts partitions of odd rank.
Heinz numbers of integer partitions of negative rank.
+10
14
4, 8, 12, 16, 18, 24, 27, 32, 36, 40, 48, 54, 60, 64, 72, 80, 81, 90, 96, 100, 108, 112, 120, 128, 135, 144, 150, 160, 162, 168, 180, 192, 200, 216, 224, 225, 240, 243, 250, 252, 256, 270, 280, 288, 300, 320, 324, 336, 352, 360, 375, 378, 384, 392, 400, 405
COMMENTS
The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), giving a bijective correspondence between positive integers and integer partitions.
The Dyson rank of a nonempty partition is its maximum part minus its length. The rank of an empty partition is undefined.
EXAMPLE
The sequence of partitions together with their Heinz numbers begins:
4: (1,1) 80: (3,1,1,1,1)
8: (1,1,1) 81: (2,2,2,2)
12: (2,1,1) 90: (3,2,2,1)
16: (1,1,1,1) 96: (2,1,1,1,1,1)
18: (2,2,1) 100: (3,3,1,1)
24: (2,1,1,1) 108: (2,2,2,1,1)
27: (2,2,2) 112: (4,1,1,1,1)
32: (1,1,1,1,1) 120: (3,2,1,1,1)
36: (2,2,1,1) 128: (1,1,1,1,1,1,1)
40: (3,1,1,1) 135: (3,2,2,2)
48: (2,1,1,1,1) 144: (2,2,1,1,1,1)
54: (2,2,2,1) 150: (3,3,2,1)
60: (3,2,1,1) 160: (3,1,1,1,1,1)
64: (1,1,1,1,1,1) 162: (2,2,2,2,1)
72: (2,2,1,1,1) 168: (4,2,1,1,1)
MATHEMATICA
Select[Range[2, 100], PrimePi[FactorInteger[#][[-1, 1]]]<PrimeOmega[#]&]
CROSSREFS
Note: A-numbers of Heinz-number sequences are in parentheses below.
These partitions are counted by A064173. A061395 selects the maximum prime index.
A072233 counts partitions by sum and length.
A168659 counts partitions whose length is divisible by maximum.
A200750 counts partitions whose length and maximum are relatively prime.
- Rank -
A257541 gives the rank of the partition with Heinz number n.
Cf. A003114, A006141, A039900, A056239, A096401, A112798, A117409, A316413, A325134, A326845, A340604, A340605, A340828.
Heinz numbers of integer partitions of even positive rank.
+10
13
5, 11, 14, 17, 21, 23, 26, 31, 35, 38, 39, 41, 44, 47, 49, 57, 58, 59, 65, 66, 67, 68, 73, 74, 83, 86, 87, 91, 92, 95, 97, 99, 102, 103, 104, 106, 109, 110, 111, 122, 124, 127, 129, 133, 137, 138, 142, 143, 145, 149, 152, 153, 154, 156, 157, 158, 159, 164, 165
COMMENTS
The Dyson rank of a nonempty partition is its maximum part minus its number of parts. The rank of an empty partition is 0.
The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.
EXAMPLE
The sequence of partitions with their Heinz numbers begins:
5: (3) 57: (8,2) 97: (25)
11: (5) 58: (10,1) 99: (5,2,2)
14: (4,1) 59: (17) 102: (7,2,1)
17: (7) 65: (6,3) 103: (27)
21: (4,2) 66: (5,2,1) 104: (6,1,1,1)
23: (9) 67: (19) 106: (16,1)
26: (6,1) 68: (7,1,1) 109: (29)
31: (11) 73: (21) 110: (5,3,1)
35: (4,3) 74: (12,1) 111: (12,2)
38: (8,1) 83: (23) 122: (18,1)
39: (6,2) 86: (14,1) 124: (11,1,1)
41: (13) 87: (10,2) 127: (31)
44: (5,1,1) 91: (6,4) 129: (14,2)
47: (15) 92: (9,1,1) 133: (8,4)
49: (4,4) 95: (8,3) 137: (33)
MATHEMATICA
rk[n_]:=PrimePi[FactorInteger[n][[-1, 1]]]-PrimeOmega[n];
Select[Range[100], EvenQ[rk[#]]&&rk[#]>0&]
CROSSREFS
Note: Heinz numbers are given in parentheses below.
These partitions are counted by A101708. A072233 counts partitions by sum and length.
- Rank -
A257541 gives the rank of the partition with Heinz number n.
- Even -
A339846 counts factorizations of even length.
Cf. A006141, A024430, A056239, A112798, A340387, A340598, A340600, A340608, A340609, A340610, A340653.
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