cp's OEIS Frontend

A379316 Positive integers whose prime indices include a unique squarefree number.

%I A379316 #7 Dec 29 2024 14:07:54
%S A379316 2,3,5,11,13,14,17,21,29,31,35,38,41,43,46,47,57,59,67,69,73,74,77,79,
%T A379316 83,91,95,98,101,106,109,111,113,115,119,122,127,137,139,142,147,149,
%U A379316 157,159,163,167,178,179,181,183,185,191,194,199,203,206,209,211
%N A379316 Positive integers whose prime indices include a unique squarefree number.
%C A379316 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 A379316 The terms together with their prime indices begin:
%e A379316     2: {1}
%e A379316     3: {2}
%e A379316     5: {3}
%e A379316    11: {5}
%e A379316    13: {6}
%e A379316    14: {1,4}
%e A379316    17: {7}
%e A379316    21: {2,4}
%e A379316    29: {10}
%e A379316    31: {11}
%e A379316    35: {3,4}
%e A379316    38: {1,8}
%e A379316    41: {13}
%e A379316    43: {14}
%e A379316    46: {1,9}
%t A379316 prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A379316 Select[Range[1000],Length[Select[prix[#],SquareFreeQ]]==1&]
%Y A379316 For all squarefree parts we have A302478, zeros of A379310.
%Y A379316 Positions of 1 in A379306.
%Y A379316 For no squarefree parts we have A379307, counted by A114374, strict A256012.
%Y A379316 Partitions of this type are counted by A379308, strict A379309.
%Y A379316 A000040 lists the primes, differences A001223.
%Y A379316 A005117 lists the squarefree numbers, differences A076259.
%Y A379316 A008966 is the characteristic function for the squarefree numbers.
%Y A379316 A013929 lists the nonsquarefree numbers, differences A078147.
%Y A379316 A055396 gives least prime index, greatest A061395.
%Y A379316 A056239 adds up prime indices, row sums of A112798, counted by A001222.
%Y A379316 Other counts of prime indices:
%Y A379316 - A257991 odd, see A000041, A000070, A066207, A349158.
%Y A379316 - A257992 even, see A000009, A038348, A066208, A379317.
%Y A379316 - A257994 prime, see A002095, A096258, A320628, A331386, A331915, A379304, A379305.
%Y A379316 - A330944 nonprime, see A000586, A000607, A076610, A330945.
%Y A379316 - A379300 composite, see A023895, A034891, A036497, A302540, A379301.
%Y A379316 - A379311 prime or 1, see A204389, A320629, A379312-A379315.
%Y A379316 Cf. A000720, A013928, A038550, A057627, A068361, A070321, A072284, A087436, A112929, A377430.
%K A379316 nonn
%O A379316 1,1
%A A379316 _Gus Wiseman_, Dec 29 2024