A340105 Odd products of distinct primes of nonprime index (A007821).

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.

Links

Table of n, a(n) for n=1..56.

Formula

Equals A005117 /\ A005408 /\ A320628 = A005117 /\ A320629.

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).

Cf. A000040, A000720, A001055, A001222, A003963, A005117, A007097, A018252, A289509, A320461, A320631, A320911, A320912.

Sequence in context: A336373 A320629 A371166 * A309649 A088982 A354460

Adjacent sequences: A340102 A340103 A340104 * A340106 A340107 A340108

Keyword

nonn

Author

Gus Wiseman, Mar 12 2021

Status

approved