OFFSET
1,1
COMMENTS
In other words, partitions whose average is an integer.
The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
LINKS
Alois P. Heinz, Table of n, a(n) for n = 1..20000 (first 1327 terms from R. J. Mathar)
EXAMPLE
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).
MAPLE
isA326413 := proc(n)
psigsu := A056239(n) ;
psigle := numtheory[bigomega](n) ;
if modp(psigsu, psigle) = 0 then
true;
else
false;
end if;
end proc:
n := 1:
for i from 2 to 3000 do
if isA326413(i) then
printf("%d %d\n", n, i);
n := n+1 ;
end if;
end do: # R. J. Mathar, Aug 09 2019
# second Maple program:
q:= n-> (l-> nops(l)>0 and irem(add(i, i=l), nops(l))=0)(map
(i-> numtheory[pi](i[1])$i[2], ifactors(n)[2])):
select(q, [$1..110])[]; # Alois P. Heinz, Nov 19 2021
MATHEMATICA
Select[Range[2, 100], Divisible[Total[Cases[FactorInteger[#], {p_, k_}:>k*PrimePi[p]]], PrimeOmega[#]]&]
CROSSREFS
KEYWORD
nonn,changed
AUTHOR
Gus Wiseman, Jul 02 2018
STATUS
approved