A368833 - OEIS

%I #6 Jan 08 2024 14:30:24

%S 19,37,38,53,57,61,71,74,76,89,95,103,106,107,111,113,114,122,131,133,

%T 142,148,151,152,159,171,173,178,181,183,185,190,193,197,206,209,212,

%U 213,214,222,223,226,228,229,239,244,247,251,259,262,263,265,266,267

%N Numbers whose prime indices are not 1, prime, or semiprime.

%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.

%e The terms together with their prime indices begin:

%e    19: {8}

%e    37: {12}

%e    38: {1,8}

%e    53: {16}

%e    57: {2,8}

%e    61: {18}

%e    71: {20}

%e    74: {1,12}

%e    76: {1,1,8}

%e    89: {24}

%e    95: {3,8}

%e   103: {27}

%e   106: {1,16}

%e   107: {28}

%e   111: {2,12}

%e   113: {30}

%e   114: {1,2,8}

%e   122: {1,18}

%e   131: {32}

%e   133: {4,8}

%e   142: {1,20}

%e   148: {1,1,12}

%t prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n], {p_,k_}:>Table[PrimePi[p],{k}]]]];

%t Select[Range[100], Max@@PrimeOmega/@prix[#]>2&]

%Y These are non-products of primes indexed by elements of A037143.

%Y The complement for just primes is A076610, strict A302590.

%Y The complement for just semiprimes is A339112, strict A340020.

%Y The complement for just squarefree semiprimes is A339113, strict A309356.

%Y The complement is A368728.

%Y The complement for just primes and semiprimes is A368729, strict A340019.

%Y A000607 counts partitions into primes, with ones allowed A034891.

%Y A001358 lists semiprimes, squarefree A006881.

%Y A006450, A106349, A322551, A368732 list selected primes.

%Y A056239 adds up prime indices, row sums of A112798.

%Y A101048 counts partitions into semiprimes.

%Y Cf. A000040, A000720, A001222, A003963, A005117, A302242, A320628.

%K nonn

%O 1,1

%A _Gus Wiseman_, Jan 08 2024