cp's OEIS Frontend

A387119 Numbers whose prime indices all have exactly 2 divisors in common.

%I A387119 #6 Aug 23 2025 23:15:16
%S A387119 3,5,9,11,17,21,25,27,31,39,41,57,59,63,65,67,81,83,87,91,109,111,115,
%T A387119 117,121,125,127,129,147,157,159,171,179,183,185,189,191,203,211,213,
%U A387119 235,237,241,243,247,261,267,273,277,283,289,299,301,303,305,319,321
%N A387119 Numbers whose prime indices all have exactly 2 divisors in common.
%C A387119 All terms are odd.
%C A387119 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 A387119 The prime indices of 87 are {2,10}, with divisors {{1,2},{1,2,5,10}}, with intersection {1,2}, so 87 is in the sequence.
%e A387119 The prime indices of 91 are {4,6}, with divisors {{1,2,4},{1,2,3,6}}, with intersection {1,2}, so 91 is in the sequence.
%e A387119 The terms together with their prime indices begin:
%e A387119     3: {2}
%e A387119     5: {3}
%e A387119     9: {2,2}
%e A387119    11: {5}
%e A387119    17: {7}
%e A387119    21: {2,4}
%e A387119    25: {3,3}
%e A387119    27: {2,2,2}
%e A387119    31: {11}
%e A387119    39: {2,6}
%e A387119    41: {13}
%e A387119    57: {2,8}
%e A387119    59: {17}
%e A387119    63: {2,2,4}
%e A387119    65: {3,6}
%e A387119    67: {19}
%e A387119    81: {2,2,2,2}
%t A387119 prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A387119 Select[Range[2,100],Length[Intersection@@Divisors/@prix[#]]==2&]
%Y A387119 For initial intervals instead of divisors we have A016945.
%Y A387119 Positions of 1 are A289509, complement A318978.
%Y A387119 Positions of 2 in A387114, for prime factors or indices A387135.
%Y A387119 A000005 counts divisors.
%Y A387119 A001414 adds up distinct prime divisors, counted by A001221.
%Y A387119 A003963 multiplies together the prime indices of n.
%Y A387119 A112798 lists prime indices, row sums A056239 or A066328, lengths A001222.
%Y A387119 A120383 lists numbers divisible by all of their prime indices.
%Y A387119 A289508 gives greatest common divisor of prime indices.
%Y A387119 Cf. A000720, A055396, A061395, A335433, A355731, A355733, A355739, A355740, A370820.
%K A387119 nonn,new
%O A387119 1,1
%A A387119 _Gus Wiseman_, Aug 21 2025