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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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified April 17 02:26 EDT 2021. Contains 343059 sequences. (Running on oeis4.)