cp's OEIS Frontend

A325802 Numbers with one more divisor than distinct subset-sums of their prime indices.

%I A325802 #10 Oct 30 2024 08:05:33
%S A325802 12,30,40,63,70,112,154,165,198,220,273,286,325,351,352,364,442,525,
%T A325802 550,561,595,646,675,714,741,748,765,832,850,874,918,931,952,988,1045,
%U A325802 1173,1254,1334,1425,1495,1539,1564,1653,1666,1672,1771,1794,1798,1870,1900
%N A325802 Numbers with one more divisor than distinct subset-sums of their prime indices.
%C A325802 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. A subset-sum of an integer partition is any sum of a submultiset of it.
%C A325802 The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so these are Heinz numbers of the partitions counted by A325835.
%H A325802 Robert Israel, <a href="/A325802/b325802.txt">Table of n, a(n) for n = 1..10000</a>
%F A325802  A000005(a(n)) = 1 + A299701(a(n)).
%e A325802 The sequence of terms together with their prime indices begins:
%e A325802    12: {1,1,2}
%e A325802    30: {1,2,3}
%e A325802    40: {1,1,1,3}
%e A325802    63: {2,2,4}
%e A325802    70: {1,3,4}
%e A325802   112: {1,1,1,1,4}
%e A325802   154: {1,4,5}
%e A325802   165: {2,3,5}
%e A325802   198: {1,2,2,5}
%e A325802   220: {1,1,3,5}
%e A325802   273: {2,4,6}
%e A325802   286: {1,5,6}
%e A325802   325: {3,3,6}
%e A325802   351: {2,2,2,6}
%e A325802   352: {1,1,1,1,1,5}
%e A325802   364: {1,1,4,6}
%e A325802   442: {1,6,7}
%e A325802   525: {2,3,3,4}
%e A325802   550: {1,3,3,5}
%e A325802   561: {2,5,7}
%p A325802 filter:= proc(n) local F,t,S,i;
%p A325802   F:= map(t -> [numtheory:-pi(t[1]),t[2]], ifactors(n)[2]);
%p A325802   S:= {0}:
%p A325802   for t in F do
%p A325802    S:= map(s -> seq(s + i*t[1],i=0..t[2]),S);
%p A325802   od;
%p A325802   nops(S) = mul(t[2]+1,t=F)-1
%p A325802 end proc:
%p A325802 select(filter, [$1..2000]); # _Robert Israel_, Oct 30 2024
%t A325802 Select[Range[100],DivisorSigma[0,#]==1+Length[Union[hwt/@Divisors[#]]]&]
%Y A325802 Positions of 1's in A325801.
%Y A325802 Cf. A000005, A056239, A108917, A112798, A276024, A299701, A299702.
%Y A325802 Cf. A325694, A325780, A325781, A325799, A325800, A325835.
%K A325802 nonn
%O A325802 1,1
%A A325802 _Gus Wiseman_, May 23 2019