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 A308029 Numbers whose sum of coreful divisors is equal to the sum of non-coreful divisors. 4
 6, 1638, 55860, 168836850, 12854283750 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS A coreful divisor d of a number k is a divisor with the same set of distinct prime factors as k (see LINKS). Sequence is a subset of A083207. Tested up to 10^12. - Giovanni Resta, May 10 2019 LINKS G. E. Hardy and M. V. Subbarao, Highly powerful numbers, Congress. Numer. 37 (1983), 277-307. (Annotated scanned copy) FORMULA Solutions of A000203(k) = 2*A057723(k). EXAMPLE 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. MAPLE 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); MATHEMATICA 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 *) PROG (PARI) rad(n) = factorback(factorint(n)[, 1]); \\ A007947 s(n) = my(rn=rad(n)); rn*sigma(n/rn); \\ A057723 isok(n) = 2*s(n) == sigma(n); \\ Michel Marcus, May 11 2019 CROSSREFS Cf. A000203, A007947, A057723, A083207, A307888, A307958, A307962, A307963, A307986. Sequence in context: A265862 A281255 A216934 * A160226 A209609 A034841 Adjacent sequences:  A308026 A308027 A308028 * A308030 A308031 A308032 KEYWORD nonn,more AUTHOR Paolo P. Lava, May 10 2019 EXTENSIONS a(4)-a(5) from Giovanni Resta, May 10 2019 STATUS approved

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Last modified August 4 16:10 EDT 2021. Contains 346447 sequences. (Running on oeis4.)