A326837 Heinz numbers of integer partitions whose length and maximum both divide their sum.

2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, 25, 27, 29, 30, 31, 32, 37, 41, 43, 47, 49, 53, 59, 61, 64, 67, 71, 73, 79, 81, 83, 84, 89, 97, 101, 103, 107, 109, 113, 121, 125, 127, 128, 131, 137, 139, 149, 151, 157, 163, 167, 169, 173, 179, 181, 191, 193, 197

Offset

1,1

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 A326843.

Links

R. J. Mathar, Table of n, a(n) for n = 1..505

Example

The sequence of terms together with their prime indices begins:
    2: {1}
    3: {2}
    4: {1,1}
    5: {3}
    7: {4}
    8: {1,1,1}
    9: {2,2}
   11: {5}
   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}
   32: {1,1,1,1,1}
   37: {12}

Maple

isA326837 := proc(n)
    psigsu := A056239(n) ;
    psigma := A061395(n) ;
    psigle := numtheory[bigomega](n) ;
    if modp(psigsu, psigma) = 0 and modp(psigsu, psigle) = 0 then
        true;
    else
        false;
    end if;
end proc:
n := 1:
for i from 2 to 3000 do
    if isA326837(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], With[{y=Flatten[Cases[FactorInteger[#], {p_, k_}:>Table[PrimePi[p], {k}]]]}, Divisible[Total[y], Max[y]]&&Divisible[Total[y], Length[y]]]&]

Crossrefs

The non-constant case is A326838.

The strict case is A326851.

Cf. A001222, A047993, A056239, A061395, A067538, A112798, A316413, A326836, A326843, A326847, A326848.

Sequence in context: A086486 A071139 A327473 * A326847 A298538 A326534

Adjacent sequences: A326834 A326835 A326836 * A326838 A326839 A326840

Keyword

nonn

Author

Gus Wiseman, Jul 26 2019

Status

approved