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A340105
Odd products of distinct primes of nonprime index (A007821).
3
1, 7, 13, 19, 23, 29, 37, 43, 47, 53, 61, 71, 73, 79, 89, 91, 97, 101, 103, 107, 113, 131, 133, 137, 139, 149, 151, 161, 163, 167, 173, 181, 193, 197, 199, 203, 223, 227, 229, 233, 239, 247, 251, 257, 259, 263, 269, 271, 281, 293, 299, 301, 307, 311, 313, 317
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.
FORMULA
EXAMPLE
The sequence of terms together with the corresponding sets of multisets begins:
1: {} 91: {{1,1},{1,2}} 173: {{1,1,1,3}}
7: {{1,1}} 97: {{3,3}} 181: {{1,2,4}}
13: {{1,2}} 101: {{1,6}} 193: {{1,1,5}}
19: {{1,1,1}} 103: {{2,2,2}} 197: {{2,2,3}}
23: {{2,2}} 107: {{1,1,4}} 199: {{1,9}}
29: {{1,3}} 113: {{1,2,3}} 203: {{1,1},{1,3}}
37: {{1,1,2}} 131: {{1,1,1,1,1}} 223: {{1,1,1,1,2}}
43: {{1,4}} 133: {{1,1},{1,1,1}} 227: {{4,4}}
47: {{2,3}} 137: {{2,5}} 229: {{1,3,3}}
53: {{1,1,1,1}} 139: {{1,7}} 233: {{2,7}}
61: {{1,2,2}} 149: {{3,4}} 239: {{1,1,6}}
71: {{1,1,3}} 151: {{1,1,2,2}} 247: {{1,2},{1,1,1}}
73: {{2,4}} 161: {{1,1},{2,2}} 251: {{1,2,2,2}}
79: {{1,5}} 163: {{1,8}} 257: {{3,5}}
89: {{1,1,1,2}} 167: {{2,6}} 259: {{1,1},{1,1,2}}
MATHEMATICA
Select[Range[1, 100, 2], SquareFreeQ[#]&&FreeQ[If[#==1, {}, FactorInteger[#]], {p_, k_}/; PrimeQ[PrimePi[p]]]&]
CROSSREFS
These primes (of nonprime index) are listed by A007821.
The non-strict version is A320629, with not necessarily odd version A320628.
The not necessarily odd version is A340104.
The prime instead of odd nonprime version:
primes: A006450
products: A076610
strict: A302590
The squarefree semiprime instead of odd nonprime version:
strict: A309356
primes: A322551
products: A339113
The semiprime instead of odd nonprime version:
primes: A106349
products: A339112
strict: A340020
A001358 lists semiprimes.
A056239 gives the sum of prime indices, which are listed by A112798.
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.
A330945 lists numbers with a nonprime prime index (nonprime case: A330948).
A339561 lists products of distinct squarefree semiprimes.
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).
Sequence in context: A336373 A320629 A371166 * A309649 A088982 A354460
KEYWORD
nonn
AUTHOR
Gus Wiseman, Mar 12 2021
STATUS
approved