Displaying 1-10 of 22 results found.
Number of positive integers that are a divisor of some prime index of n.
+10
26
0, 1, 2, 1, 2, 2, 3, 1, 2, 2, 2, 2, 4, 3, 3, 1, 2, 2, 4, 2, 3, 2, 3, 2, 2, 4, 2, 3, 4, 3, 2, 1, 3, 2, 4, 2, 6, 4, 4, 2, 2, 3, 4, 2, 3, 3, 4, 2, 3, 2, 3, 4, 5, 2, 3, 3, 4, 4, 2, 3, 6, 2, 3, 1, 4, 3, 2, 2, 4, 4, 6, 2, 4, 6, 3, 4, 4, 4, 4, 2, 2, 2, 2, 3, 3, 4, 4
COMMENTS
A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798. This sequence contains all nonnegative integers. In particular, a(prime(n)!) = n.
EXAMPLE
2045 has prime indices {3,80} with combined divisors {1,2,3,4,5,8,10,16,20,40,80}, so a(2045) = 11. In fact, 2045 is the least number with this property.
MATHEMATICA
Table[Length[Union@@Divisors/@PrimePi/@First/@If[n==1, {}, FactorInteger[n]]], {n, 100}]
PROG
(PARI) a(n) = my(list=List(), f=factor(n)); for (i=1, #f~, fordiv(primepi(f[i, 1]), d, listput(list, d))); #Set(list); \\ Michel Marcus, May 02 2024
CROSSREFS
Positions of ones are A000079 except for 1. a(prime(n)!) = a(prime( A005179(n))) = n. Counting prime factors instead of divisors gives A303975. Position of first appearance of n is A371131(n), sorted version A371181. A001221 counts distinct prime factors.
A003963 gives product of prime indices.
A355731 counts choices of a divisor of each prime index, firsts A355732.
A355741 counts choices of a prime factor of each prime index.
Numbers k such that there are fewer divisors of prime indices of k than there are prime indices of k.
+10
19
4, 8, 12, 16, 18, 20, 24, 27, 32, 36, 40, 44, 48, 50, 54, 56, 60, 64, 68, 72, 80, 81, 84, 88, 90, 96, 100, 108, 112, 120, 124, 125, 126, 128, 132, 135, 136, 144, 150, 160, 162, 164, 168, 176, 180, 184, 189, 192, 196, 198, 200, 204, 208, 216, 220, 224, 225, 236, 240, 242, 243, 248, 250, 252, 256
COMMENTS
No multiple of a term is a term of A368110.
EXAMPLE
a(5) = 18 is a term because the prime indices of 18 = 2 * 3^2 are 1,2,2, and there are 3 of these but only 2 divisors of prime indices, namely 1 and 2.
MAPLE
filter:= proc(n) uses numtheory; local F, D, t;
F:= map(t -> [pi(t[1]), t[2]], ifactors(n)[2]);
D:= `union`(seq(divisors(t[1]), t = F));
nops(D) < add(t[2], t = F)
end proc:
select(filter, [$1..300]);
MATHEMATICA
filter[n_] := Module[{F, d},
F = {PrimePi[#[[1]]], #[[2]]}& /@ FactorInteger[n];
d = Union[Flatten[Divisors /@ F[[All, 1]]]];
Length[d] < Total[F[[All, 2]]]];
CROSSREFS
For submultisets instead of parts on the RHS we get A371167. Partitions of this type are counted by A371171. A001221 counts distinct prime factors.
A355731 counts choices of a divisor of each prime index, firsts A355732.
Number of integer partitions of n such that the number of parts is equal to the number of distinct divisors of parts.
+10
18
1, 1, 0, 1, 2, 0, 4, 2, 4, 5, 5, 11, 10, 16, 17, 21, 26, 32, 44, 53, 69, 71, 101, 110, 148, 168, 205, 249, 289, 356, 418, 502, 589, 716, 812, 999, 1137, 1365, 1566, 1873, 2158, 2537, 2942, 3449, 4001, 4613, 5380, 6193, 7220, 8224, 9575, 10926, 12683, 14430
COMMENTS
The Heinz numbers of these partitions are given by A370802.
EXAMPLE
The partition (6,2,2,1) has 4 parts and 4 distinct divisors of parts {1,2,3,6} so is counted under a(11).
The a(1) = 1 through a(11) = 11 partitions:
(1) . (21) (22) . (33) (322) (71) (441) (55) (533)
(31) (51) (421) (332) (522) (442) (722)
(321) (422) (531) (721) (731)
(411) (521) (4311) (4321) (911)
(6111) (6211) (4322)
(4331)
(5321)
(5411)
(6221)
(6311)
(8111)
MATHEMATICA
Table[Length[Select[IntegerPartitions[n], Length[#]==Length[Union@@Divisors/@#]&]], {n, 0, 30}]
CROSSREFS
These partitions are ranked by A370802. For (greater than) instead of (equal to) we have A371171, ranks A370348. For submultisets instead of parts on the LHS we have A371172. For (less than) instead of (equal to) we have A371173, ranked by A371168. A008284 counts partitions by length.
Number of strict integer partitions of n containing all distinct divisors of all parts.
+10
17
1, 1, 0, 1, 1, 0, 2, 1, 2, 1, 2, 2, 3, 3, 3, 5, 3, 5, 6, 7, 7, 8, 8, 9, 12, 13, 13, 14, 15, 16, 19, 23, 25, 26, 26, 27, 36, 37, 40, 42, 46, 50, 55, 66, 65, 71, 71, 82, 90, 102, 103, 114, 117, 130, 147, 154, 166, 176, 182, 194, 228, 239, 259, 267, 287, 307, 336
COMMENTS
Also strict integer partitions such that the number of parts is equal to the number of distinct divisors of all parts.
EXAMPLE
The a(9) = 1 through a(19) = 7 partitions (A..H = 10..17):
531 721 731 B1 751 D1 B31 D21 B51 H1 B71
4321 5321 5421 931 B21 7521 7531 D31 9531 D51
6321 7321 7421 8421 64321 B321 A521 B521
9321 65321 B421 D321
54321 74321 75321 75421
84321 76321
94321
MATHEMATICA
Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&SubsetQ[#, Union@@Divisors/@#]&]], {n, 0, 30}]
CROSSREFS
A008284 counts partitions by length.
Number of integer partitions of n with more parts than distinct divisors of parts.
+10
15
0, 0, 1, 1, 2, 4, 5, 9, 12, 18, 26, 34, 50, 65, 92, 121, 161, 209, 274, 353, 456, 590, 745, 950, 1195, 1507, 1885, 2350, 2923, 3611, 4465, 5485, 6735, 8223, 10050, 12195, 14822, 17909, 21653, 26047, 31340, 37557, 44990, 53708, 64068, 76241, 90583, 107418
COMMENTS
The Heinz numbers of these partitions are given by A370348.
EXAMPLE
The partition (3,2,1,1) has 4 parts {1,2,3,4} and 3 distinct divisors of parts {1,2,3}, so is counted under a(7).
The a(0) = 0 through a(8) = 12 partitions:
. . (11) (111) (211) (221) (222) (331) (2222)
(1111) (311) (2211) (511) (3221)
(2111) (3111) (2221) (3311)
(11111) (21111) (3211) (4211)
(111111) (4111) (5111)
(22111) (22211)
(31111) (32111)
(211111) (41111)
(1111111) (221111)
(311111)
(2111111)
(11111111)
MATHEMATICA
Table[Length[Select[IntegerPartitions[n], Length[#] > Length[Union@@Divisors/@#]&]], {n, 0, 30}]
CROSSREFS
The partitions are ranked by A370348. For submultisets instead of parts on the LHS we get ranks A371167.
Positive integers whose prime indices include all distinct divisors of all prime indices.
+10
15
1, 2, 4, 6, 8, 10, 12, 16, 18, 20, 22, 24, 30, 32, 34, 36, 40, 42, 44, 48, 50, 54, 60, 62, 64, 66, 68, 72, 80, 82, 84, 88, 90, 96, 100, 102, 108, 110, 118, 120, 124, 126, 128, 132, 134, 136, 144, 150, 160, 162, 164, 166, 168, 170, 176, 180, 186, 192, 198, 200
COMMENTS
A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798. Also positive integers with as many distinct prime factors ( A001221) as distinct divisors of prime indices ( A370820).
EXAMPLE
The terms together with their prime indices begin:
1: {}
2: {1}
4: {1,1}
6: {1,2}
8: {1,1,1}
10: {1,3}
12: {1,1,2}
16: {1,1,1,1}
18: {1,2,2}
20: {1,1,3}
22: {1,5}
24: {1,1,1,2}
30: {1,2,3}
32: {1,1,1,1,1}
34: {1,7}
36: {1,1,2,2}
40: {1,1,1,3}
42: {1,2,4}
44: {1,1,5}
48: {1,1,1,1,2}
MATHEMATICA
Select[Range[100], PrimeNu[#]==Length[Union @@ Divisors/@PrimePi/@First/@If[#==1, {}, FactorInteger[#]]]&]
CROSSREFS
A008284 counts partitions by length.
Positive integers with as many divisors ( A000005) as distinct divisors of prime indices ( A370820).
+10
14
3, 5, 11, 17, 26, 31, 35, 38, 39, 41, 49, 57, 58, 59, 65, 67, 69, 77, 83, 86, 87, 94, 109, 119, 127, 129, 133, 146, 148, 157, 158, 179, 191, 202, 206, 211, 217, 235, 237, 241, 244, 253, 274, 277, 278, 283, 284, 287, 291, 298, 303, 319, 326, 331, 333, 334, 353
COMMENTS
A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
EXAMPLE
The terms together with their prime indices begin:
3: {2} 67: {19} 158: {1,22}
5: {3} 69: {2,9} 179: {41}
11: {5} 77: {4,5} 191: {43}
17: {7} 83: {23} 202: {1,26}
26: {1,6} 86: {1,14} 206: {1,27}
31: {11} 87: {2,10} 211: {47}
35: {3,4} 94: {1,15} 217: {4,11}
38: {1,8} 109: {29} 235: {3,15}
39: {2,6} 119: {4,7} 237: {2,22}
41: {13} 127: {31} 241: {53}
49: {4,4} 129: {2,14} 244: {1,1,18}
57: {2,8} 133: {4,8} 253: {5,9}
58: {1,10} 146: {1,21} 274: {1,33}
59: {17} 148: {1,1,12} 277: {59}
65: {3,6} 157: {37} 278: {1,34}
MATHEMATICA
Select[Range[100], Length[Divisors[#]] == Length[Union@@Divisors/@PrimePi/@First/@If[#==1, {}, FactorInteger[#]]]&]
CROSSREFS
For prime factors instead of divisors on both sides we get A319899. For prime factors on LHS we get A370802, for distinct prime factors A371177. For (greater than) instead of (equal) we get A371166. For (less than) instead of (equal) we get A371167. Partitions of this type are counted by A371172. A001221 counts distinct prime factors.
A355731 counts choices of a divisor of each prime index, firsts A355732.
Positive integers with fewer prime factors ( A001222) than distinct divisors of prime indices ( A370820).
+10
14
3, 5, 7, 11, 13, 14, 15, 17, 19, 21, 23, 26, 29, 31, 33, 35, 37, 38, 39, 41, 43, 46, 47, 49, 51, 52, 53, 55, 57, 58, 59, 61, 65, 67, 69, 70, 71, 73, 74, 76, 77, 78, 79, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97, 101, 103, 105, 106, 107, 109, 111, 113, 114, 115
COMMENTS
A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
EXAMPLE
The prime indices of 105 are {2,3,4}, and there are 3 prime factors (3,5,7) and 4 distinct divisors of prime indices (1,2,3,4), so 105 is in the sequence.
The terms together with their prime indices begin:
3: {2} 35: {3,4} 59: {17} 86: {1,14}
5: {3} 37: {12} 61: {18} 87: {2,10}
7: {4} 38: {1,8} 65: {3,6} 89: {24}
11: {5} 39: {2,6} 67: {19} 91: {4,6}
13: {6} 41: {13} 69: {2,9} 93: {2,11}
14: {1,4} 43: {14} 70: {1,3,4} 94: {1,15}
15: {2,3} 46: {1,9} 71: {20} 95: {3,8}
17: {7} 47: {15} 73: {21} 97: {25}
19: {8} 49: {4,4} 74: {1,12} 101: {26}
21: {2,4} 51: {2,7} 76: {1,1,8} 103: {27}
23: {9} 52: {1,1,6} 77: {4,5} 105: {2,3,4}
26: {1,6} 53: {16} 78: {1,2,6} 106: {1,16}
29: {10} 55: {3,5} 79: {22} 107: {28}
31: {11} 57: {2,8} 83: {23} 109: {29}
33: {2,5} 58: {1,10} 85: {3,7} 111: {2,12}
MATHEMATICA
Select[Range[100], PrimeOmega[#]<Length[Union @@ Divisors/@PrimePi/@First/@If[#==1, {}, FactorInteger[#]]]&]
CROSSREFS
For divisors instead of prime factors on the LHS we get A371166. The complement is counted by A371169. Partitions of this type are counted by A371173. A001221 counts distinct prime factors.
A355731 counts choices of a divisor of each prime index, firsts A355732.
Number of integer partitions of n with fewer parts than distinct divisors of parts.
+10
14
0, 0, 1, 1, 1, 3, 2, 4, 6, 7, 11, 11, 17, 20, 26, 34, 44, 56, 67, 84, 102, 131, 156, 195, 232, 283, 346, 411, 506, 598, 721, 855, 1025, 1204, 1448, 1689, 2018, 2363, 2796, 3265, 3840, 4489, 5242, 6104, 7106, 8280, 9595, 11143, 12862, 14926, 17197, 19862, 22841
COMMENTS
The Heinz numbers of these partitions are given by A371168.
EXAMPLE
The partition (4,3,2) has 3 parts {2,3,4} and 4 distinct divisors of parts {1,2,3,4}, so is counted under a(9).
The a(2) = 1 through a(10) = 11 partitions:
(2) (3) (4) (5) (6) (7) (8) (9) (10)
(3,2) (4,2) (4,3) (4,4) (5,4) (6,4)
(4,1) (5,2) (5,3) (6,3) (7,3)
(6,1) (6,2) (7,2) (8,2)
(4,3,1) (8,1) (9,1)
(6,1,1) (4,3,2) (4,3,3)
(6,2,1) (5,3,2)
(5,4,1)
(6,2,2)
(6,3,1)
(8,1,1)
MATHEMATICA
Table[Length[Select[IntegerPartitions[n], Length[#] < Length[Union@@Divisors/@#]&]], {n, 0, 30}]
CROSSREFS
For submultisets instead of parts on the LHS we get ranks A371166. These partitions are ranked by A371168. A355731 counts choices of a divisor of each prime index, firsts A355732.
Number of integer partitions of n with as many submultisets as distinct divisors of parts.
+10
12
0, 0, 1, 1, 0, 1, 0, 3, 2, 3, 1, 4, 2, 1, 2, 3, 4, 2, 4, 1, 5, 2, 7, 5, 9, 4, 9, 15, 18, 16, 24, 13, 17, 23, 23, 22, 34, 17, 30, 31, 36, 29, 43, 21, 30, 35, 44, 28, 47, 19, 44
COMMENTS
The Heinz numbers of these partitions are given by A371165.
EXAMPLE
The partition (8,6,6) has 6 submultisets {(8,6,6),(8,6),(6,6),(8),(6),()} and 6 distinct divisors of parts {1,2,3,4,6,8}, so is counted under a(20).
The a(17) = 2 through a(24) = 9 partitions:
(17) (9,9) (19) (11,9) (14,7) (13,9) (23) (21,3)
(13,4) (15,3) (15,5) (17,4) (21,1) (19,4) (22,2)
(6,6,6) (8,6,6) (8,8,6) (22,1) (8,8,8)
(12,3,3) (12,4,4) (10,6,6) (15,4,4) (10,8,6)
(18,1,1) (16,3,3) (12,10,1) (12,6,6)
(18,2,2) (12,7,5)
(20,1,1) (18,3,3)
(20,2,2)
(12,10,2)
MATHEMATICA
Table[Length[Select[IntegerPartitions[n], Length[Divisors[Times@@Prime/@#]] == Length[Union@@Divisors/@#]&]], {n, 0, 30}]
CROSSREFS
These partitions are ranked by A371165. A355731 counts choices of a divisor of each prime index, firsts A355732.
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