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A308029
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Numbers whose sum of coreful divisors is equal to the sum of non-coreful divisors.
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4
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OFFSET
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1,1
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COMMENTS
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A coreful divisor d of a number k is a divisor with the same set of distinct prime factors as k (see LINKS).
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LINKS
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G. E. Hardy and M. V. Subbarao, Highly powerful numbers, Congress. Numer. 37 (1983), 277-307. (Annotated scanned copy)
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FORMULA
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EXAMPLE
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Divisors of 1638 are 1, 2, 3, 6, 7, 9, 13, 14, 18, 21, 26, 39, 42, 63, 78, 91, 117, 126, 182, 234, 273, 546, 819, 1638. The coreful ones are 546, 1638 and 1 + 2 + 3 + 6 + 7 + 9 + 13 + 14 + 18 + 21 + 26 + 39 + 42 + 63 + 78 + 91 + 117 + 126 + 182 + 234 + 273 + 819 = 546 + 1638 = 2184.
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MAPLE
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with(numtheory): P:=proc(q) local a, k, n; for n from 1 to q do
a:=mul(k, k=factorset(n)); if sigma(n)=2*a*sigma(n/a)
then print(n); fi; od; end: P(10^7);
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MATHEMATICA
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f[p_, e_] := (p^(e + 1) - 1)/(p - 1); fc[p_, e_] := f[p, e] - 1; csigmaQ[n_] := Times @@ (fc @@@ FactorInteger[n]) == Times @@ (f @@@ FactorInteger[n])/2; Select[Range[2, 10^5], csigmaQ] (* Amiram Eldar, May 11 2019 *)
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PROG
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(PARI) rad(n) = factorback(factorint(n)[, 1]); \\ A007947
s(n) = my(rn=rad(n)); rn*sigma(n/rn); \\ A057723
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CROSSREFS
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KEYWORD
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nonn,more
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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