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A326841
Heinz numbers of integer partitions of m >= 0 using divisors of m.
22
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
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.
Sequence in context: A357861 A133813 A326836 * A274222 A300273 A353844
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jul 26 2019
STATUS
approved