OFFSET
1,1
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.
If n is a term, then so is n^k for k > 1. - Robert Israel, Feb 08 2021
LINKS
Robert Israel, Table of n, a(n) for n = 1..10000
EXAMPLE
The sequence of terms together with their prime indices begins:
2: {1} 64: {1,1,1,1,1,1} 216: {1,1,1,2,2,2}
4: {1,1} 75: {2,3,3} 240: {1,1,1,1,2,3}
6: {1,2} 81: {2,2,2,2} 256: {1,1,1,1,1,1,1,1}
8: {1,1,1} 84: {1,1,2,4} 264: {1,1,1,2,5}
9: {2,2} 96: {1,1,1,1,1,2} 294: {1,2,4,4}
16: {1,1,1,1} 125: {3,3,3} 315: {2,2,3,4}
20: {1,1,3} 126: {1,2,2,4} 324: {1,1,2,2,2,2}
24: {1,1,1,2} 128: {1,1,1,1,1,1,1} 350: {1,3,3,4}
30: {1,2,3} 140: {1,1,3,4} 360: {1,1,1,2,2,3}
32: {1,1,1,1,1} 144: {1,1,1,1,2,2} 384: {1,1,1,1,1,1,1,2}
36: {1,1,2,2} 160: {1,1,1,1,1,3} 396: {1,1,2,2,5}
45: {2,2,3} 176: {1,1,1,1,5} 400: {1,1,1,1,3,3}
50: {1,3,3} 189: {2,2,2,4} 416: {1,1,1,1,1,6}
54: {1,2,2,2} 196: {1,1,4,4} 440: {1,1,1,3,5}
56: {1,1,1,4} 210: {1,2,3,4} 441: {2,2,4,4}
MAPLE
filter:= proc(n) local F, m, g, t;
F:= ifactors(n)[2];
m:= add(t[2], t=F);
g:= numtheory:-pi(max(seq(t[1], t=F)));
m mod g = 0;
end proc:
seelect(filter, [$2..1000]); # Robert Israel, Feb 08 2021
MATHEMATICA
Select[Range[2, 100], Divisible[PrimeOmega[#], PrimePi[FactorInteger[#][[-1, 1]]]]&]
CROSSREFS
Note: Heinz numbers are given in parentheses below.
These are the Heinz numbers of certain partitions counted by A168659.
A001222 counts prime factors.
A056239 adds up prime indices.
A061395 selects the maximum prime index.
A112798 lists the prime indices of each positive integer.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jan 27 2021
STATUS
approved