cp's OEIS Frontend

A357854 Squarefree numbers with a divisor having the same sum of prime indices as their quotient.

%I A357854 #6 Oct 27 2022 12:48:43
%S A357854 1,30,70,154,165,210,273,286,390,442,462,561,595,646,714,741,858,874,
%T A357854 910,1045,1155,1173,1254,1326,1330,1334,1495,1653,1771,1794,1798,1870,
%U A357854 1938,2139,2145,2294,2415,2465,2470,2530,2622,2639,2730,2926,2945,2958,3034
%N A357854 Squarefree numbers with a divisor having the same sum of prime indices as their quotient.
%C A357854 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 A357854 The terms together with their prime indices begin:
%e A357854      1: {}
%e A357854     30: {1,2,3}
%e A357854     70: {1,3,4}
%e A357854    154: {1,4,5}
%e A357854    165: {2,3,5}
%e A357854    210: {1,2,3,4}
%e A357854    273: {2,4,6}
%e A357854    286: {1,5,6}
%e A357854    390: {1,2,3,6}
%e A357854 For example, 210 has factorization 14*15, and both factors have the same sum of prime indices 5, so 210 is in the sequence.
%t A357854 sumprix[n_]:=Total[Cases[FactorInteger[n],{p_,k_}:>k*PrimePi[p]]];
%t A357854 Select[Range[1000],SquareFreeQ[#]&&MemberQ[sumprix/@Divisors[#],sumprix[#]/2]&]
%Y A357854 The partitions with these Heinz numbers are counted by A237258.
%Y A357854 A subset of A319241, squarefree case of A300061.
%Y A357854 Squarefree positions of nonzero terms in A357879.
%Y A357854 This is the squarefree case of A357976, counted by A002219.
%Y A357854 A001222 counts prime factors, distinct A001221.
%Y A357854 A056239 adds up prime indices, row sums of A112798.
%Y A357854 Cf. A033879, A033880, A064914, A181819, A235130, A237194, A276107, A300273, A321144, A357975, A357976.
%K A357854 nonn
%O A357854 1,2
%A A357854 _Gus Wiseman_, Oct 27 2022