cp's OEIS Frontend

A325793 Positive integers whose number of divisors is equal to their sum of prime indices.

%I A325793 #13 Oct 16 2023 23:26:10
%S A325793 3,10,28,66,70,88,208,228,306,340,364,490,495,525,544,550,675,744,870,
%T A325793 966,1160,1216,1242,1254,1288,1326,1330,1332,1672,1768,1785,1870,2002,
%U A325793 2064,2145,2295,2457,2900,2944,3250,3280,3430,3468,3540,3724,4125,4144,4248
%N A325793 Positive integers whose number of divisors is equal to their sum of prime indices.
%C A325793 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, with sum A056239(n).
%H A325793 Robert Israel, <a href="/A325793/b325793.txt">Table of n, a(n) for n = 1..10000</a>
%e A325793 The term 70 is in the sequence because it has 8 divisors {1, 2, 5, 7, 10, 14, 35, 70} and its sum of prime indices is also 1 + 3 + 4 = 8.
%e A325793 The sequence of terms together with their prime indices begins:
%e A325793      3: {2}
%e A325793     10: {1,3}
%e A325793     28: {1,1,4}
%e A325793     66: {1,2,5}
%e A325793     70: {1,3,4}
%e A325793     88: {1,1,1,5}
%e A325793    208: {1,1,1,1,6}
%e A325793    228: {1,1,2,8}
%e A325793    306: {1,2,2,7}
%e A325793    340: {1,1,3,7}
%e A325793    364: {1,1,4,6}
%e A325793    490: {1,3,4,4}
%e A325793    495: {2,2,3,5}
%e A325793    525: {2,3,3,4}
%e A325793    544: {1,1,1,1,1,7}
%e A325793    550: {1,3,3,5}
%e A325793    675: {2,2,2,3,3}
%e A325793    744: {1,1,1,2,11}
%e A325793    870: {1,2,3,10}
%e A325793    966: {1,2,4,9}
%p A325793 filter:= proc(n) local F,t;
%p A325793   F:= ifactors(n)[2];
%p A325793   add(numtheory:-pi(t[1])*t[2],t=F) = mul(t[2]+1,t=F)
%p A325793 end proc:
%p A325793 select(filter, [$1..10000]); # _Robert Israel_, Oct 16 2023
%t A325793 Select[Range[100],DivisorSigma[0,#]==Total[Cases[FactorInteger[#],{p_,k_}:>PrimePi[p]*k]]&]
%Y A325793 Positions of 0's in A325794.
%Y A325793 Contains A239885 except for 1.
%Y A325793 Cf. A000005, A056239, A112798, A299702, A304793.
%Y A325793 Cf. A325694, A325780, A325781, A325792, A325795, A325796, A325797, A325798.
%K A325793 nonn
%O A325793 1,1
%A A325793 _Gus Wiseman_, May 23 2019