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Numbers with one more divisor than distinct subset-sums of their prime indices.
5

%I #10 Oct 30 2024 08:05:33

%S 12,30,40,63,70,112,154,165,198,220,273,286,325,351,352,364,442,525,

%T 550,561,595,646,675,714,741,748,765,832,850,874,918,931,952,988,1045,

%U 1173,1254,1334,1425,1495,1539,1564,1653,1666,1672,1771,1794,1798,1870,1900

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

%C 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 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 Robert Israel, <a href="/A325802/b325802.txt">Table of n, a(n) for n = 1..10000</a>

%F A000005(a(n)) = 1 + A299701(a(n)).

%e The sequence of terms together with their prime indices begins:

%e 12: {1,1,2}

%e 30: {1,2,3}

%e 40: {1,1,1,3}

%e 63: {2,2,4}

%e 70: {1,3,4}

%e 112: {1,1,1,1,4}

%e 154: {1,4,5}

%e 165: {2,3,5}

%e 198: {1,2,2,5}

%e 220: {1,1,3,5}

%e 273: {2,4,6}

%e 286: {1,5,6}

%e 325: {3,3,6}

%e 351: {2,2,2,6}

%e 352: {1,1,1,1,1,5}

%e 364: {1,1,4,6}

%e 442: {1,6,7}

%e 525: {2,3,3,4}

%e 550: {1,3,3,5}

%e 561: {2,5,7}

%p filter:= proc(n) local F,t,S,i;

%p F:= map(t -> [numtheory:-pi(t[1]),t[2]], ifactors(n)[2]);

%p S:= {0}:

%p for t in F do

%p S:= map(s -> seq(s + i*t[1],i=0..t[2]),S);

%p od;

%p nops(S) = mul(t[2]+1,t=F)-1

%p end proc:

%p select(filter, [$1..2000]); # _Robert Israel_, Oct 30 2024

%t Select[Range[100],DivisorSigma[0,#]==1+Length[Union[hwt/@Divisors[#]]]&]

%Y Positions of 1's in A325801.

%Y Cf. A000005, A056239, A108917, A112798, A276024, A299701, A299702.

%Y Cf. A325694, A325780, A325781, A325799, A325800, A325835.

%K nonn

%O 1,1

%A _Gus Wiseman_, May 23 2019