A072109 Numbers k such that Sum_{i=1..k} gcd(k,i) divides Sum_{i=1..k} lcm(k,i).

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

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