A326841 Heinz numbers of integer partitions of m >= 0 using divisors of m.

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

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: A357861 A133813 A326836 * A274222 A300273 A353844

Adjacent sequences: A326838 A326839 A326840 * A326842 A326843 A326844

Keyword

nonn

Author

Gus Wiseman, Jul 26 2019

Status

approved