OFFSET
1,2
COMMENTS
The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.
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.
EXAMPLE
The prime indices of 105 are {2,3,4}, with divisor sets {{1,2},{1,3},{1,2,4}}, with multiset union {1,1,1,2,2,3,4}, with Heinz number 2520, so a(105) = 2520.
The terms together with their prime indices begin:
1: {}
2: {1}
6: {1,2}
4: {1,1}
10: {1,3}
12: {1,1,2}
42: {1,2,4}
8: {1,1,1}
36: {1,1,2,2}
20: {1,1,3}
22: {1,5}
24: {1,1,1,2}
390: {1,2,3,6}
84: {1,1,2,4}
60: {1,1,2,3}
16: {1,1,1,1}
34: {1,7}
72: {1,1,1,2,2}
MATHEMATICA
prix[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
Table[Times@@Prime/@Join@@Divisors/@prix[n], {n, 100}]
CROSSREFS
Product of A275700 applied to each prime index.
The squarefree case is also A275700.
The sorted version is A371286.
A000005 counts divisors.
A001221 counts distinct prime factors.
A355741 counts choices of a prime factor of each prime index.
KEYWORD
nonn,mult
AUTHOR
Gus Wiseman, Mar 21 2024
STATUS
approved