cp's OEIS Frontend

A325800 Numbers whose sum of prime indices is equal to the number of distinct subset-sums of their prime indices.

%I A325800 #11 Oct 30 2024 12:12:55
%S A325800 3,10,28,66,88,156,208,306,340,408,544,570,684,760,912,966,1216,1242,
%T A325800 1288,1380,1656,1840,2208,2436,2610,2900,2944,3132,3248,3480,3906,
%U A325800 4092,4176,4340,4640,4650,5022,5208,5456,5568,5580,6200,6696,6944,7326,7424,7440
%N A325800 Numbers whose sum of prime indices is equal to the number of distinct subset-sums of their prime indices.
%C A325800 First differs from A325793 in lacking 70.
%C A325800 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). A subset-sum of an integer partition is any sum of a submultiset of it.
%C A325800 The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so these are Heinz numbers of integer partitions whose sum is equal to their number of distinct subset-sums. The enumeration of these partitions by sum is given by A126796 interlaced with zeros.
%H A325800 Robert Israel, <a href="/A325800/b325800.txt">Table of n, a(n) for n = 1..10000</a>
%F A325800 A056239(a(n)) = A299701(a(n)) = A304793(a(n)) + 1.
%e A325800 340 has prime indices {1,1,3,7} which sum to 12 and have 12 distinct subset-sums: {0, 1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 12}, so 340 is in the sequence.
%e A325800 The sequence of terms together with their prime indices begins:
%e A325800      3: {2}
%e A325800     10: {1,3}
%e A325800     28: {1,1,4}
%e A325800     66: {1,2,5}
%e A325800     88: {1,1,1,5}
%e A325800    156: {1,1,2,6}
%e A325800    208: {1,1,1,1,6}
%e A325800    306: {1,2,2,7}
%e A325800    340: {1,1,3,7}
%e A325800    408: {1,1,1,2,7}
%e A325800    544: {1,1,1,1,1,7}
%e A325800    570: {1,2,3,8}
%e A325800    684: {1,1,2,2,8}
%e A325800    760: {1,1,1,3,8}
%e A325800    912: {1,1,1,1,2,8}
%e A325800    966: {1,2,4,9}
%e A325800   1216: {1,1,1,1,1,1,8}
%e A325800   1242: {1,2,2,2,9}
%e A325800   1288: {1,1,1,4,9}
%e A325800   1380: {1,1,2,3,9}
%p A325800 filter:= proc(n) local F,t,S,i,r;
%p A325800   F:= map(t -> [numtheory:-pi(t[1]),t[2]], ifactors(n)[2]);
%p A325800   S:= {0}:
%p A325800   for t in F do
%p A325800    S:= map(s -> seq(s + i*t[1],i=0..t[2]),S);
%p A325800   od;
%p A325800   nops(S) = add(t[1]*t[2],t=F)
%p A325800 end proc:
%p A325800 select(filter, [$1..10000]); # _Robert Israel_, Oct 30 2024
%t A325800 hwt[n_]:=Total[Cases[FactorInteger[n],{p_,k_}:>PrimePi[p]*k]];
%t A325800 Select[Range[1000],hwt[#]==Length[Union[hwt/@Divisors[#]]]&]
%Y A325800 Positions of 1's in A325799.
%Y A325800 Includes A239885 except for 1.
%Y A325800 Cf. A002033, A056239, A108917, A112798, A126796, A299701, A299702.
%Y A325800 Cf. A325694, A325780, A325781, A325792, A325793, A325801, A325802.
%K A325800 nonn
%O A325800 1,1
%A A325800 _Gus Wiseman_, May 23 2019