A326847 Heinz numbers of integer partitions of m >= 0 using divisors of m whose length also divides m.

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

First differs from A071139, A089352 and A086486 in lacking 60. First differs from A326837 in lacking 268.

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 A326842.

Links

R. J. Mathar, Table of n, a(n) for n = 1..489

Formula

Intersection of A326841 and A316413.

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

isA326847 := proc(n)
    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:
    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 isA326847(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], Length[y]]&&And@@IntegerQ/@(Total[y]/y)]&]

Crossrefs

Intersection of A326841 and A316413.

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

Sequence in context: A071139 A327473 A326837 * A298538 A326534 A046686

Adjacent sequences: A326844 A326845 A326846 * A326848 A326849 A326850

Keyword

nonn

Author

Gus Wiseman, Jul 26 2019

Status

approved