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Sum of odd prime indices of n.
+10
24
0, 1, 0, 2, 3, 1, 0, 3, 0, 4, 5, 2, 0, 1, 3, 4, 7, 1, 0, 5, 0, 6, 9, 3, 6, 1, 0, 2, 0, 4, 11, 5, 5, 8, 3, 2, 0, 1, 0, 6, 13, 1, 0, 7, 3, 10, 15, 4, 0, 7, 7, 2, 0, 1, 8, 3, 0, 1, 17, 5, 0, 12, 0, 6, 3, 6, 19, 9, 9, 4, 0, 3, 21, 1, 6, 2, 5, 1, 0, 7, 0, 14, 23, 2
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, sum A056239(n).
EXAMPLE
The prime indices of 198 are {1,2,2,5}, so a(198) = 1+5 = 6.
MATHEMATICA
Table[Total[Cases[FactorInteger[n], {p_?(OddQ@*PrimePi), k_}:>PrimePi[p]*k]], {n, 100}]
CROSSREFS
The triangle for this rank statistic is A113685, without zeros A365067. A053253 = partitions with all odd parts and conjugate parts, ranks A352143.
A066967 adds up sums of odd parts over all partitions.
Coefficients of the '3rd-order' mock theta function omega(q).
+10
18
1, 2, 3, 4, 6, 8, 10, 14, 18, 22, 29, 36, 44, 56, 68, 82, 101, 122, 146, 176, 210, 248, 296, 350, 410, 484, 566, 660, 772, 896, 1038, 1204, 1391, 1602, 1846, 2120, 2428, 2784, 3182, 3628, 4138, 4708, 5347, 6072, 6880, 7784, 8804, 9940, 11208, 12630
COMMENTS
Empirical: a(n) is the number of integer partitions mu of 2n+1 such that the diagram of mu has an odd number of cells in each row and in each column. - John M. Campbell, Apr 24 2020 By Campbell's conjecture above that a(n) is the number of partitions of 2n+1 with all odd parts and all odd conjugate parts, the a(0) = 1 through a(5) = 8 partitions are (B = 11):
(1) (3) (5) (7) (9) (B)
(111) (311) (511) (333) (533)
(11111) (31111) (711) (911)
(1111111) (51111) (33311)
(3111111) (71111)
(111111111) (5111111)
(311111111)
(11111111111)
These partitions are ranked by A352143. (End)
REFERENCES
Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, pp. 15, 17, 31.
FORMULA
G.f.: omega(q) = Sum_{n>=0} q^(2*n*(n+1))/((1-q)*(1-q^3)*...*(1-q^(2*n+1)))^2.
G.f.: Sum_{k>=0} x^k/((1-x)(1-x^3)...(1-x^(2k+1))). - Michael Somos, Aug 18 2006 G.f.: (1 - G(0))/(1-x) where G(k) = 1 - 1/(1-x^(2*k+1))/(1-x/(x-1/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jan 18 2013
MATHEMATICA
Series[Sum[q^(2n(n+1))/Product[1-q^(2k+1), {k, 0, n}]^2, {n, 0, 6}], {q, 0, 100}]
PROG
(PARI) {a(n)=local(A); if(n<0, 0, A=1+x*O(x^n); polcoeff( sum(k=0, (sqrtint(2*n+1)-1)\2, A*=(x^(4*k)/(1-x^(2*k+1))^2 +x*O(x^(n-2*(k^2-k))))), n))} /* Michael Somos, Aug 18 2006 */ (PARI) {a(n)=local(A); if(n<0, 0, n++; A=1+x*O(x^n); polcoeff( sum(k=0, n-1, A*=(x/(1-x^(2*k+1)) +x*O(x^(n-k)))), n))} /* Michael Somos, Aug 18 2006 */
CROSSREFS
Conjectured to count the partitions ranked by A352143. A117958 = partitions w/ all odd parts and multiplicities, ranked by A352142.
Numbers whose prime factorization has all odd indices and all odd exponents.
+10
8
1, 2, 5, 8, 10, 11, 17, 22, 23, 31, 32, 34, 40, 41, 46, 47, 55, 59, 62, 67, 73, 82, 83, 85, 88, 94, 97, 103, 109, 110, 115, 118, 125, 127, 128, 134, 136, 137, 146, 149, 155, 157, 160, 166, 167, 170, 179, 184, 187, 191, 194, 197, 205, 206, 211, 218, 227, 230
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, sum A056239, length A001222. A number's prime signature is the sequence of positive exponents in its prime factorization, which is row n of A124010, length A001221, sum A001222. These are the Heinz numbers of integer partitions with all odd parts and all odd multiplicities, counted by A117958.
EXAMPLE
The terms together with their prime indices begin:
1 = 1
2 = prime(1)
5 = prime(3)
8 = prime(1)^3
10 = prime(1) prime(3)
11 = prime(5)
17 = prime(7)
22 = prime(1) prime(5)
23 = prime(9)
31 = prime(11)
32 = prime(1)^5
34 = prime(1) prime(7)
40 = prime(1)^3 prime(3)
MATHEMATICA
Select[Range[100], #==1||And@@OddQ/@PrimePi/@First/@FactorInteger[#]&&And@@OddQ/@Last/@FactorInteger[#]&]
PROG
(Python)
from itertools import count, islice
from sympy import primepi, factorint
def A352142_gen(startvalue=1): # generator of terms >= startvalue return filter(lambda k:all(map(lambda x:x[1]%2 and primepi(x[0])%2, factorint(k).items())), count(max(startvalue, 1)))
CROSSREFS
The restriction to primes is A031368. These partitions are counted by A117958. A352140 = even indices with odd exponents, counted by A055922 aerated.
A352143 = odd indices with odd conjugate indices, counted by A053253 aerated.
Cf. A000720, A028260, A055396, A061395, A106529, A181819, A195017, A241638, A276078, A324517, A324524, A324525, A325698, A325700.
Numbers whose prime factorization has all even indices and all even exponents.
+10
7
1, 9, 49, 81, 169, 361, 441, 729, 841, 1369, 1521, 1849, 2401, 2809, 3249, 3721, 3969, 5041, 6241, 6561, 7569, 7921, 8281, 10201, 11449, 12321, 12769, 13689, 16641, 17161, 17689, 19321, 21609, 22801, 25281, 26569, 28561, 29241, 29929, 32761, 33489, 35721
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, sum A056239, length A001222. A number's prime signature is the sequence of positive exponents in its prime factorization, which is row n of A124010, length A001221, sum A001222. These are the Heinz numbers of partitions with all even parts and all even multiplicities, counted by A035444.
FORMULA
Sum_{n>=1} 1/a(n) = 1/Product_{k>=1} (1 - 1/prime(2*k)^2) = 1.163719... . - Amiram Eldar, Sep 19 2022
EXAMPLE
The terms together with their prime indices begin:
1 = 1
9 = prime(2)^2
49 = prime(4)^2
81 = prime(2)^4
169 = prime(6)^2
361 = prime(8)^2
441 = prime(2)^2 prime(4)^2
729 = prime(2)^6
841 = prime(10)^2
1369 = prime(12)^2
1521 = prime(2)^2 prime(6)^2
1849 = prime(14)^2
2401 = prime(4)^4
2809 = prime(16)^2
3249 = prime(2)^2 prime(8)^2
3721 = prime(18)^2
3969 = prime(2)^4 prime(4)^2
MATHEMATICA
Select[Range[1000], #==1||And@@EvenQ/@PrimePi/@First/@FactorInteger[#]&&And@@EvenQ/@Last/@FactorInteger[#]&]
PROG
(Python)
from itertools import count, islice
from sympy import factorint, primepi
def A352141_gen(startvalue=1): # generator of terms >= startvalue return filter(lambda k:all(map(lambda x: not (x[1]%2 or primepi(x[0])%2), factorint(k).items())), count(max(startvalue, 1)))
CROSSREFS
The second condition alone (all even exponents) is A000290, counted by A035363. The restriction to primes is A031215. These partitions are counted by A035444. A352140 = even indices with odd exponents, counted by A055922 aerated.
Cf. A000720, A028260, A055396, A061395, A181819, A195017, A241638, A268335, A276078, A324524, A324525, A324588, A325698, A325700, A352143.
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