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A357976
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Numbers with a divisor having the same sum of prime indices as their quotient.
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38
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1, 4, 9, 12, 16, 25, 30, 36, 40, 48, 49, 63, 64, 70, 81, 84, 90, 100, 108, 112, 120, 121, 144, 154, 160, 165, 169, 192, 196, 198, 210, 220, 225, 252, 256, 264, 270, 273, 280, 286, 289, 300, 324, 325, 336, 351, 352, 360, 361, 364, 390, 400, 432, 441, 442, 448
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OFFSET
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1,2
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COMMENTS
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A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
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LINKS
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EXAMPLE
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The terms together with their prime indices begin:
1: {}
4: {1,1}
9: {2,2}
12: {1,1,2}
16: {1,1,1,1}
25: {3,3}
30: {1,2,3}
36: {1,1,2,2}
40: {1,1,1,3}
48: {1,1,1,1,2}
49: {4,4}
For example, 40 has factorization 8*5, and both factors have the same sum of prime indices 3, so 40 is in the sequence.
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MAPLE
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filter:= proc(n) local F, s, t, i, R;
F:= ifactors(n)[2];
F:= map(t -> [numtheory:-pi(t[1]), t[2]], F);
s:= add(t[1]*t[2], t=F)/2;
if not s::integer then return false fi;
try
R:= Optimization:-Maximize(0, [add(F[i][1]*x[i], i=1..nops(F)) = s, seq(x[i]<= F[i][2], i=1..nops(F))], assume=nonnegint, depthlimit=20);
catch "no feasible integer point found; use feasibilitytolerance option to adjust tolerance": return false;
end try;
true
end proc:
filter(1):= true:
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MATHEMATICA
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sumprix[n_]:=Total[Cases[FactorInteger[n], {p_, k_}:>k*PrimePi[p]]];
Select[Range[100], MemberQ[sumprix/@Divisors[#], sumprix[#]/2]&]
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CROSSREFS
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The partitions with these Heinz numbers are counted by A002219.
Positions of nonzero terms in A357879.
Cf. A033879, A033880, A064914, A181819, A213086, A235130, A237194, A276107, A300273, A321144, A357975.
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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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