OFFSET
1,2
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. The prime signature of a number is the multiset of multiplicities (or exponents) in its prime factorization.
Also Heinz numbers of integer partitions in which every part divides its multiplicity (counted by A001156). The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).
Also products of elements of A062457.
LINKS
Amiram Eldar, Table of n, a(n) for n = 1..10000
FORMULA
Closed under multiplication.
Sum_{n>=1} 1/a(n) = Product_{k>=1} 1/(1-prime(k)^(-k)) = 2.26910478689594012492... - Amiram Eldar, Sep 30 2020
EXAMPLE
The sequence of terms together with their prime indices begins as follows. For example, we have 18: {1,2,2} because 18 = prime(1) * prime(2) * prime(2).
1: {}
2: {1}
4: {1,1}
8: {1,1,1}
9: {2,2}
16: {1,1,1,1}
18: {1,2,2}
32: {1,1,1,1,1}
36: {1,1,2,2}
64: {1,1,1,1,1,1}
72: {1,1,1,2,2}
81: {2,2,2,2}
125: {3,3,3}
128: {1,1,1,1,1,1,1}
144: {1,1,1,1,2,2}
162: {1,2,2,2,2}
250: {1,3,3,3}
256: {1,1,1,1,1,1,1,1}
MAPLE
q:= n-> andmap(i-> irem(i[2], numtheory[pi](i[1]))=0, ifactors(n)[2]):
select(q, [$1..10000])[]; # Alois P. Heinz, Mar 08 2019
MATHEMATICA
Select[Range[1000], And@@Cases[If[#==1, {}, FactorInteger[#]], {p_, k_}:>Divisible[k, PrimePi[p]]]&]
v = Join[{1}, Prime[(r = Range[10])]^r]; n = Length[v]; vmax = 10^4; s = {1}; Do[v1 = v[[k]]; rmax = Floor[Log[v1, vmax]]; s1 = v1^Range[0, rmax]; s2 = Select[Union[Flatten[Outer[Times, s, s1]]], # <= vmax &]; s = Union[s, s2], {k, 2, n}]; Length[s] (* Amiram Eldar, Sep 30 2020 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Gus Wiseman, Mar 07 2019
STATUS
approved