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A316413 Heinz numbers of integer partitions whose length divides their sum. 78
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

R. J. Mathar, Table of n, a(n) for n = 1..1327

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

MATHEMATICA

Select[Range[2, 100], Divisible[Total[Cases[FactorInteger[#], {p_, k_}:>k*PrimePi[p]]], PrimeOmega[#]]&]

CROSSREFS

Cf. A056239, A067538, A074761, A143773, A237984, A289508, A289509, A290103, A296150, A298423, A316428, A316431.

Sequence in context: A326621 A324758 A305504 * A316465 A004764 A128649

Adjacent sequences:  A316410 A316411 A316412 * A316414 A316415 A316416

KEYWORD

nonn

AUTHOR

Gus Wiseman, Jul 02 2018

STATUS

approved

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Last modified July 26 15:49 EDT 2021. Contains 346294 sequences. (Running on oeis4.)