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A325698
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Numbers with as many even as odd prime indices, counted with multiplicity.
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64
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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
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1,2
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COMMENTS
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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
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LINKS
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EXAMPLE
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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}
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MATHEMATICA
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Select[Range[100], Total[Cases[If[#==1, {}, FactorInteger[#]], {p_, k_}:>k*(-1)^PrimePi[p]]]==0&]
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PROG
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(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
(Python)
from sympy import factorint, primepi
def ok(n):
v = [0, 0]
for p, e in factorint(n).items(): v[primepi(p)%2] += e
return v[0] == v[1]
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CROSSREFS
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Cf. A000712, A001222, A001405, A006094, A026010, A045931, A063886, A097613, A112798, A130780, A171966, A239241, A241638, A325700.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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