

A325698


Numbers with as many even as odd prime indices, counted with multiplicity.


6



1, 6, 14, 15, 26, 33, 35, 36, 38, 51, 58, 65, 69, 74, 77, 84, 86, 90, 93, 95, 106, 119, 122, 123, 141, 142, 143, 145, 156, 158, 161, 177, 178, 185, 196, 198, 201, 202, 209, 210, 214, 215, 216, 217, 219, 221, 225, 226, 228, 249, 262, 265, 278, 287, 291, 299
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OFFSET

1,2


COMMENTS

These are Heinz numbers of the integer partitions counted by A045931.
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.
The integers in the multiplicative subgroup of positive rational numbers generated by the products of two consecutive primes (A006094). The sequence is closed under multiplication, prime shift (A003961), and  where the result is an integer  under division. Using these closures, all the terms can be derived from the presence of 6. For example, A003961(6) = 15, A003961(15) = 35, 6 * 35 = 210, 210/15 = 14. Closed also under A297845, since A297845 can be defined using squaring, prime shift and multiplication.  Peter Munn, Oct 05 2020


LINKS

David A. Corneth, Table of n, a(n) for n = 1..10000


EXAMPLE

The sequence of terms together with their prime indices begins:
1: {}
6: {1,2}
14: {1,4}
15: {2,3}
26: {1,6}
33: {2,5}
35: {3,4}
36: {1,1,2,2}
38: {1,8}
51: {2,7}
58: {1,10}
65: {3,6}
69: {2,9}
74: {1,12}
77: {4,5}
84: {1,1,2,4}
86: {1,14}
90: {1,2,2,3}
93: {2,11}
95: {3,8}


MATHEMATICA

Select[Range[100], Total[Cases[If[#==1, {}, FactorInteger[#]], {p_, k_}:>k*(1)^PrimePi[p]]]==0&]


PROG

(PARI) is(n) = {my(v = vector(2), f = factor(n)); for(i = 1, #f~, v[1 + primepi(f[i, 1])%2]+=f[i, 2]); v[1] == v[2]} \\ David A. Corneth, Oct 06 2020


CROSSREFS

Positions of 0's in A195017.
A257992(n) = A257991(n).
Cf. A000712, A001222, A001405, A006094, A026010, A045931, A063886, A097613, A112798, A130780, A171966, A239241, A241638, A325700.
Closed under: A003961, A003991, A297845.
Subsequence of A028260, A332820.
Sequence in context: A107982 A341448 A081535 * A338907 A218005 A337384
Adjacent sequences: A325695 A325696 A325697 * A325699 A325700 A325701


KEYWORD

nonn


AUTHOR

Gus Wiseman, May 17 2019


STATUS

approved



