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A326847
Heinz numbers of integer partitions of m >= 0 using divisors of m whose length also divides m.
11
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
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
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jul 26 2019
STATUS
approved