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A326837
Heinz numbers of integer partitions whose length and maximum both divide their sum.
30
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
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.
Sequence in context: A086486 A071139 A327473 * A326847 A298538 A326534
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jul 26 2019
STATUS
approved