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 A072109 Numbers k such that Sum_{i=1..k} gcd(k,i) divides Sum_{i=1..k} lcm(k,i). 2
 1, 4, 36, 125, 469, 536, 882, 1156, 8532, 8775, 25012, 32000, 34749, 36324, 37179, 61952, 147456, 405224, 451584, 644304, 954084, 1185921, 1560546, 1562500, 1982464, 3080025, 5229378, 5784025, 6138868, 9231327, 12806144, 22108500, 25509168, 25562264, 29762208, 40894464, 45001899, 47397636, 49242375 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 LINKS Amiram Eldar, Table of n, a(n) for n = 1..100 FORMULA Numbers k such that A018804(k) divides A051193(k). MAPLE with(numtheory): for n from 1 to 10^6 do a := divisors(n): s1 := add(a[m]*phi(a[m]), m=1..nops(a)): s2 := add(phi(a[m])/a[m], m=1..nops(a)): if type((s1+1)/(2*s2), integer) then printf(`%d, `, n); fi: od: MATHEMATICA f[n_] := (k = n; While[ !IntegerQ[ Sum[ LCM[k, i], {i, 1, k}] / Sum[ GCD[k, i], {i, 1, k}]], k++ ]; k); j = 1; Do[ m = f[j]; Print[m]; j = m + 1, {n, 1, 9}] f1[p_, e_] := (p^(2*e + 1) + 1)/(p + 1); f2[p_, e_] := e*(p - 1)/p + 1; q[n_] := IntegerQ[(1 + Times @@ f1 @@@ (fct = FactorInteger[n]))/(2 * Times @@ f2 @@@ fct)]; Select[Range[10^5], q] (* Amiram Eldar, May 02 2023 *) PROG (PARI) for(n=1, 1156, if(sum(i=1, n, lcm(n, i))%sum(i=1, n, gcd(n, i))==0, print1(n, ", "))) (PARI) is(n) = {my(f = factor(n)); (1 + prod(i = 1, #f~, (f[i, 1]^(2*f[i, 2] + 1) + 1)/(f[i, 1] + 1))) % (2*prod(i = 1, #f~, (f[i, 2]*(f[i, 1] - 1)/f[i, 1] + 1))) == 0; } \\ Amiram Eldar, May 02 2023 CROSSREFS Cf. A018804, A051193. Sequence in context: A362751 A076830 A144298 * A045490 A318150 A275133 Adjacent sequences: A072106 A072107 A072108 * A072110 A072111 A072112 KEYWORD nonn AUTHOR Benoit Cloitre, Jun 19 2002 EXTENSIONS Edited by Robert G. Wilson v, Jun 22 2002 More terms from Vladeta Jovovic, Jun 22 2002 More terms from Sean A. Irvine, Feb 01 2011 Corrected definition - Richard L. Ollerton, May 06 2021 STATUS approved

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Last modified November 29 06:17 EST 2023. Contains 367422 sequences. (Running on oeis4.)