A340105 - OEIS

%I #10 Mar 17 2021 08:02:15

%S 1,7,13,19,23,29,37,43,47,53,61,71,73,79,89,91,97,101,103,107,113,131,

%T 133,137,139,149,151,161,163,167,173,181,193,197,199,203,223,227,229,

%U 233,239,247,251,257,259,263,269,271,281,293,299,301,307,311,313,317

%N Odd products of distinct primes of nonprime index (A007821).

%C 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.

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

%e The sequence of terms together with the corresponding sets of multisets begins:

%e      1: {}              91: {{1,1},{1,2}}      173: {{1,1,1,3}}

%e      7: {{1,1}}         97: {{3,3}}            181: {{1,2,4}}

%e     13: {{1,2}}        101: {{1,6}}            193: {{1,1,5}}

%e     19: {{1,1,1}}      103: {{2,2,2}}          197: {{2,2,3}}

%e     23: {{2,2}}        107: {{1,1,4}}          199: {{1,9}}

%e     29: {{1,3}}        113: {{1,2,3}}          203: {{1,1},{1,3}}

%e     37: {{1,1,2}}      131: {{1,1,1,1,1}}      223: {{1,1,1,1,2}}

%e     43: {{1,4}}        133: {{1,1},{1,1,1}}    227: {{4,4}}

%e     47: {{2,3}}        137: {{2,5}}            229: {{1,3,3}}

%e     53: {{1,1,1,1}}    139: {{1,7}}            233: {{2,7}}

%e     61: {{1,2,2}}      149: {{3,4}}            239: {{1,1,6}}

%e     71: {{1,1,3}}      151: {{1,1,2,2}}        247: {{1,2},{1,1,1}}

%e     73: {{2,4}}        161: {{1,1},{2,2}}      251: {{1,2,2,2}}

%e     79: {{1,5}}        163: {{1,8}}            257: {{3,5}}

%e     89: {{1,1,1,2}}    167: {{2,6}}            259: {{1,1},{1,1,2}}

%t Select[Range[1,100,2],SquareFreeQ[#]&&FreeQ[If[#==1,{},FactorInteger[#]],{p_,k_}/;PrimeQ[PrimePi[p]]]&]

%Y These primes (of nonprime index) are listed by A007821.

%Y The non-strict version is A320629, with not necessarily odd version A320628.

%Y The not necessarily odd version is A340104.

%Y The prime instead of odd nonprime version:

%Y   primes: A006450

%Y   products: A076610

%Y   strict: A302590

%Y The squarefree semiprime instead of odd nonprime version:

%Y   strict: A309356

%Y   primes: A322551

%Y   products: A339113

%Y The semiprime instead of odd nonprime version:

%Y   primes: A106349

%Y   products: A339112

%Y   strict: A340020

%Y A001358 lists semiprimes.

%Y A056239 gives the sum of prime indices, which are listed by A112798.

%Y A257994 counts prime prime indices.

%Y A302242 is the weight of the multiset of multisets with MM-number n.

%Y A305079 is the number of connected components for MM-number n.

%Y A330944 counts nonprime prime indices.

%Y A330945 lists numbers with a nonprime prime index (nonprime case: A330948).

%Y A339561 lists products of distinct squarefree semiprimes.

%Y 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).

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

%K nonn

%O 1,2

%A _Gus Wiseman_, Mar 12 2021