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A316413 Heinz numbers of integer partitions whose length divides their sum. 194
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, 39, 41, 43, 46, 47, 49, 53, 55, 57, 59, 61, 62, 64, 67, 68, 71, 73, 78, 79, 81, 82, 83, 84, 85, 87, 88, 89, 90, 91, 94, 97, 98, 99, 100, 101, 103, 105, 107, 109, 110 (list; graph; refs; listen; history; text; internal format)
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
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
Sequence in context: A326621 A324758 A305504 * A360009 A359905 A316465
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jul 02 2018
STATUS
approved

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Last modified March 29 00:26 EDT 2024. Contains 371264 sequences. (Running on oeis4.)