A229996 For every positive integer m, let u(m) = (d(1),d(2),...,d(k)) be the unitary divisors of m. The sequence (a(n)) consists of successive numbers m which d(k)/d(1) + d(k-1)/d(2) + ... + d(k)/d(1) is an integer.

1, 10, 65, 130, 260, 340, 1105, 1972, 2210, 4420, 8840, 9860, 15650, 20737, 32045, 41474, 44200, 51272, 55250, 64090, 75140, 82948, 103685, 128180, 207370, 207553, 221000, 256360, 352529, 414740, 415106, 512720, 532100, 705058, 759025, 813800, 829480, 830212

Offset

1,2

Comments

The integer sums d(k)/d(1) + d(k-1)/d(2) + ... + d(k)/d(1) are given by A229999. - Clark Kimberling, Jun 16 2018

Also numbers m such that the sum of the squares of the unitary divisors of m is divisible by m (the unitary version of A046762). - Amiram Eldar, Jun 16 2018

Links

Amiram Eldar, Table of n, a(n) for n = 1..333 (terms below 10^10)

Example

The first 10 sums: 1, 5/2, 10/3, 17/4, 26/5, 25/3, 50/7, 65/8, 82/9, 13, so that a(1) = 1 and a(10) = 13.

Mathematica

z = 1000; r[n_] := Select[Divisors[n], GCD[#, n/#] == 1 &]; k[n_] := Length[r[n]];
t[n_] := Table[r[n][[k[n] + 1 - i]]/r[n][[k[1] + i - 1]], {i, 1, k[n]}];
s = Table[Plus @@ t[n], {n, 1, z}]; a[n_] := If[IntegerQ[s[[n]]], 1, 0]; u = Table[a[n], {n, 1, z}]; Flatten[Position[u, 1]]  (* A229996 *)
usigma2[n_] :=  If[n == 1, 1, Times @@ (1 + Power @@@ FactorInteger[n]^2)]; seqQ[n_] := Divisible[usigma2[n], n]; Select[Range[10^6], seqQ] (* Amiram Eldar, Jun 16 2018 *)

Prog

(PARI) is(n) = {my(f = factor(n)); !(prod(i = 1, #f~, f[i, 1]^(2*f[i, 2]) + 1) % n); } \\ Amiram Eldar, Jun 16 2024

Crossrefs

Cf. A034676, A046762, A229994, A077610, A229999.

Sequence in context: A033908 A033863 A233246 * A255245 A210369 A058920

Adjacent sequences: A229993 A229994 A229995 * A229997 A229998 A229999

Keyword

nonn,easy

Author

Clark Kimberling, Oct 31 2013

Extensions

Definition corrected by Clark Kimberling, Jun 16 2018

Status

approved