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.
EXAMPLE
The sequence of terms together with the corresponding prime indices of prime indices begins:
1: {} 58: {{},{1,3}} 113: {{1,2,3}}
2: {{}} 61: {{1,2,2}} 122: {{},{1,2,2}}
7: {{1,1}} 71: {{1,1,3}} 131: {{1,1,1,1,1}}
13: {{1,2}} 73: {{2,4}} 133: {{1,1},{1,1,1}}
14: {{},{1,1}} 74: {{},{1,1,2}} 137: {{2,5}}
19: {{1,1,1}} 79: {{1,5}} 139: {{1,7}}
23: {{2,2}} 86: {{},{1,4}} 142: {{},{1,1,3}}
26: {{},{1,2}} 89: {{1,1,1,2}} 146: {{},{2,4}}
29: {{1,3}} 91: {{1,1},{1,2}} 149: {{3,4}}
37: {{1,1,2}} 94: {{},{2,3}} 151: {{1,1,2,2}}
38: {{},{1,1,1}} 97: {{3,3}} 158: {{},{1,5}}
43: {{1,4}} 101: {{1,6}} 161: {{1,1},{2,2}}
46: {{},{2,2}} 103: {{2,2,2}} 163: {{1,8}}
47: {{2,3}} 106: {{},{1,1,1,1}} 167: {{2,6}}
53: {{1,1,1,1}} 107: {{1,1,4}} 173: {{1,1,1,3}}
MATHEMATICA
Select[Range[100], SquareFreeQ[#]&&FreeQ[If[#==1, {}, FactorInteger[#]], {p_, k_}/; PrimeQ[PrimePi[p]]]&]
CROSSREFS
These primes (of nonprime index) are listed by A007821.
The odd case is A340105.
The prime instead of nonprime version:
primes: A006450
products: A076610
strict: A302590
The semiprime instead of nonprime version:
primes: A106349
products: A339112
strict: A340020
The squarefree semiprime instead of nonprime version:
strict: A309356
primes: A322551
products: A339113
A257994 counts prime prime indices.
A302242 is the weight of the multiset of multisets with MM-number n.
A305079 is the number of connected components for MM-number n.
A330944 counts nonprime prime indices.
MM-numbers: A255397 (normal), A302478 (set multisystems), A320630 (set multipartitions), A302494 (sets of sets), A305078 (connected), A316476 (antichains), A318991 (chains), A320456 (covers), A328514 (connected sets of sets), A329559 (clutters), A340019 (half-loop graphs).
KEYWORD
nonn
AUTHOR
Gus Wiseman, Mar 12 2021
STATUS
approved