A229996 - OEIS

%I #25 Nov 03 2024 17:37:10

%S 1,10,65,130,260,340,1105,1972,2210,4420,8840,9860,15650,20737,32045,

%T 41474,44200,51272,55250,64090,75140,82948,103685,128180,207370,

%U 207553,221000,256360,352529,414740,415106,512720,532100,705058,759025,813800,829480,830212

%N 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.

%C 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

%C 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

%H Amiram Eldar, <a href="/A229996/b229996.txt">Table of n, a(n) for n = 1..333</a> (terms below 10^10)

%e 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.

%t z = 1000; r[n_] := Select[Divisors[n], GCD[#, n/#] == 1 &]; k[n_] := Length[r[n]];

%t t[n_] := Table[r[n][[k[n] + 1 - i]]/r[n][[k[1] + i - 1]], {i, 1, k[n]}];

%t 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 *)

%t 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 *)

%o (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

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

%K nonn,easy

%O 1,2

%A _Clark Kimberling_, Oct 31 2013

%E Definition corrected by _Clark Kimberling_, Jun 16 2018