cp's OEIS Frontend

A316413 Heinz numbers of integer partitions whose length divides their sum.

%I A316413 #20 Nov 10 2024 16:09:30
%S A316413 2,3,4,5,7,8,9,10,11,13,16,17,19,21,22,23,25,27,28,29,30,31,32,34,37,
%T A316413 39,41,43,46,47,49,53,55,57,59,61,62,64,67,68,71,73,78,79,81,82,83,84,
%U A316413 85,87,88,89,90,91,94,97,98,99,100,101,103,105,107,109,110
%N A316413 Heinz numbers of integer partitions whose length divides their sum.
%C A316413 In other words, partitions whose average is an integer.
%C A316413 The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
%H A316413 Alois P. Heinz, <a href="/A316413/b316413.txt">Table of n, a(n) for n = 1..20000</a> (first 1327 terms from R. J. Mathar)
%e A316413 Sequence of partitions whose length divides their sum begins (1), (2), (11), (3), (4), (111), (22), (31), (5), (6), (1111), (7), (8), (42), (51), (9), (33), (222), (411).
%p A316413 isA326413 := proc(n)
%p A316413     psigsu := A056239(n) ;
%p A316413     psigle := numtheory[bigomega](n) ;
%p A316413     if modp(psigsu,psigle) = 0 then
%p A316413         true;
%p A316413     else
%p A316413         false;
%p A316413     end if;
%p A316413 end proc:
%p A316413 n := 1:
%p A316413 for i from 2 to 3000 do
%p A316413     if isA326413(i) then
%p A316413         printf("%d %d\n",n,i);
%p A316413         n := n+1 ;
%p A316413     end if;
%p A316413 end do: # _R. J. Mathar_, Aug 09 2019
%p A316413 # second Maple program:
%p A316413 q:= n-> (l-> nops(l)>0 and irem(add(i, i=l), nops(l))=0)(map
%p A316413         (i-> numtheory[pi](i[1])$i[2], ifactors(n)[2])):
%p A316413 select(q, [$1..110])[];  # _Alois P. Heinz_, Nov 19 2021
%t A316413 Select[Range[2,100],Divisible[Total[Cases[FactorInteger[#],{p_,k_}:>k*PrimePi[p]]],PrimeOmega[#]]&]
%Y A316413 Cf. A056239, A067538, A074761, A143773, A237984, A289508, A289509, A290103, A296150, A298423, A316428, A316431.
%K A316413 nonn
%O A316413 1,1
%A A316413 _Gus Wiseman_, Jul 02 2018