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A326841
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Heinz numbers of integer partitions of m >= 0 using divisors of m.
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22
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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
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OFFSET
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1,2
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COMMENTS
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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.
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LINKS
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EXAMPLE
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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}
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MAPLE
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isA326841 := proc(n)
local ifs, psigsu, p, psig ;
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;
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MATHEMATICA
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Select[Range[100], With[{y=If[#==1, {}, Flatten[Cases[FactorInteger[#], {p_, k_}:>Table[PrimePi[p], {k}]]]]}, And@@IntegerQ/@(Total[y]/y)]&]
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CROSSREFS
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The case where the length also divides m is A326847.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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