OFFSET
1,2
COMMENTS
The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so these are Heinz numbers of integer partitions into distinct non-consecutive parts (counted by A003114). The nonsquarefree case is A319630, which gives the Heinz numbers of integer partitions with no consecutive parts (counted by A116931).
The numbers of terms not exceeding 10^k, for k = 1, 2, ..., are 6, 52, 515, 5146, 51435, 514416, 5144232, 51442384, ... . Apparently, the asymptotic density of this sequence exists and equals 0.51442... . - Amiram Eldar, Sep 24 2022
LINKS
Amiram Eldar, Table of n, a(n) for n = 1..10000
EXAMPLE
The sequence of terms together with their prime indices begins:
1: {}
2: {1}
3: {2}
5: {3}
7: {4}
10: {1,3}
11: {5}
13: {6}
14: {1,4}
17: {7}
19: {8}
21: {2,4}
22: {1,5}
23: {9}
26: {1,6}
29: {10}
31: {11}
33: {2,5}
34: {1,7}
37: {12}
MATHEMATICA
Select[Range[100], Min@@Differences[Flatten[Cases[FactorInteger[#], {p_, k_}:>Table[PrimePi[p], {k}]]]]>1&]
PROG
(PARI) isok(k) = {if (issquarefree(k), my(v = apply(primepi, factor(k)[, 1])); ! #select(x->(v[x+1]-v[x] == 1), [1..#v-1])); } \\ Michel Marcus, Jan 09 2021
CROSSREFS
KEYWORD
nonn
AUTHOR
Gus Wiseman, Apr 05 2019
STATUS
approved