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
(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]
print([k for k in range(300) if ok(k)]) # Michael S. Branicky, Apr 16 2022 after David A. Corneth
CROSSREFS
KEYWORD
nonn
AUTHOR
Gus Wiseman, May 17 2019
STATUS
approved