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A326841 Heinz numbers of integer partitions of m >= 0 using divisors of m. 16
1, 2, 3, 4, 5, 7, 8, 9, 11, 12, 13, 16, 17, 19, 23, 25, 27, 29, 30, 31, 32, 36, 37, 40, 41, 43, 47, 48, 49, 53, 59, 61, 63, 64, 67, 71, 73, 79, 81, 83, 84, 89, 97, 101, 103, 107, 108, 109, 112, 113, 121, 125, 127, 128, 131, 137, 139, 144, 149, 151, 157, 163 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

The enumeration of these partitions by sum is given by A018818.

LINKS

R. J. Mathar, Table of n, a(n) for n = 1..543

EXAMPLE

The sequence of terms together with their prime indices begins:

    1: {}

    2: {1}

    3: {2}

    4: {1,1}

    5: {3}

    7: {4}

    8: {1,1,1}

    9: {2,2}

   11: {5}

   12: {1,1,2}

   13: {6}

   16: {1,1,1,1}

   17: {7}

   19: {8}

   23: {9}

   25: {3,3}

   27: {2,2,2}

   29: {10}

   30: {1,2,3}

   31: {11}

MAPLE

isA326841 := proc(n)

    local ifs, psigsu, p, psig ;

    psigsu := A056239(n) ;

    for ifs in ifactors(n)[2] do

        p := op(1, ifs) ;

        psig := numtheory[pi](p) ;

        if modp(psigsu, psig) <> 0 then

            return false;

        end if;

    end do:

    true;

end proc:

for i from 1 to 3000 do

    if isA326841(i) then

        printf("%d %d\n", n, i);

        n := n+1 ;

    end if;

end do: # R. J. Mathar, Aug 09 2019

MATHEMATICA

Select[Range[100], With[{y=If[#==1, {}, Flatten[Cases[FactorInteger[#], {p_, k_}:>Table[PrimePi[p], {k}]]]]}, And@@IntegerQ/@(Total[y]/y)]&]

CROSSREFS

The case where the length also divides m is A326847.

Cf. A001222, A018818, A056239, A067538, A112798, A316413, A326836, A326842.

Sequence in context: A085233 A133813 A326836 * A274222 A300273 A219301

Adjacent sequences:  A326838 A326839 A326840 * A326842 A326843 A326844

KEYWORD

nonn

AUTHOR

Gus Wiseman, Jul 26 2019

STATUS

approved

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Last modified July 25 14:49 EDT 2021. Contains 346290 sequences. (Running on oeis4.)