cp's OEIS Frontend

A379301 Positive integers whose prime indices include a unique composite number.

%I A379301 #10 Dec 26 2024 23:02:57
%S A379301 7,13,14,19,21,23,26,28,29,35,37,38,39,42,43,46,47,52,53,56,57,58,61,
%T A379301 63,65,69,70,71,73,74,76,77,78,79,84,86,87,89,92,94,95,97,101,103,104,
%U A379301 105,106,107,111,112,113,114,115,116,117,119,122,126,129,130,131
%N A379301 Positive integers whose prime indices include a unique composite number.
%C A379301 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 A379301 The prime indices of 70 are {1,3,4}, so 70 is in the sequence.
%e A379301 The prime indices of 98 are {1,4,4}, so 98 is not in the sequence.
%t A379301 prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A379301 Select[Range[100],Length[Select[prix[#],CompositeQ]]==1&]
%Y A379301 For no composite parts we have A302540, counted by A034891 (strict A036497).
%Y A379301 For all composite parts we have A320629, counted by A023895 (strict A204389).
%Y A379301 For a unique prime part we have A331915, counted by A379304 (strict A379305).
%Y A379301 Positions of one in A379300.
%Y A379301 Partitions of this type are counted by A379302 (strict A379303).
%Y A379301 A000040 lists the prime numbers, differences A001223.
%Y A379301 A002808 lists the composite numbers, nonprimes A018252, differences A073783 or A065310.
%Y A379301 A055396 gives least prime index, greatest A061395.
%Y A379301 A056239 adds up prime indices, row sums of A112798, counted by A001222.
%Y A379301 A066247 is the characteristic function for the composite numbers.
%Y A379301 A377033 gives k-th differences of composite numbers, see A073445, A377034-A377037.
%Y A379301 Other counts of prime indices:
%Y A379301 - A087436 postpositive, see A038550.
%Y A379301 - A257991 odd, see A000041, A000070, A066207, A349158.
%Y A379301 - A257992 even, see A000009, A038348, A066208, A379317.
%Y A379301 - A257994 prime, see A000586, A000607, A076610, A331386.
%Y A379301 - A330944 nonprime, see A002095, A096258, A320628, A330945.
%Y A379301 - A379306 squarefree, see A302478, A379308, A379309, A379316.
%Y A379301 - A379310 nonsquarefree, see A114374, A256012, A379307.
%Y A379301 - A379311 old prime, see A379312-A379315.
%Y A379301 Cf. A113646, A376680.
%K A379301 nonn
%O A379301 1,1
%A A379301 _Gus Wiseman_, Dec 25 2024