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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.

%I A229996 #25 Nov 03 2024 17:37:10
%S A229996 1,10,65,130,260,340,1105,1972,2210,4420,8840,9860,15650,20737,32045,
%T A229996 41474,44200,51272,55250,64090,75140,82948,103685,128180,207370,
%U A229996 207553,221000,256360,352529,414740,415106,512720,532100,705058,759025,813800,829480,830212
%N 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.
%C A229996 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 A229996 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 A229996 Amiram Eldar, <a href="/A229996/b229996.txt">Table of n, a(n) for n = 1..333</a> (terms below 10^10)
%e A229996 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 A229996 z = 1000; r[n_] := Select[Divisors[n], GCD[#, n/#] == 1 &]; k[n_] := Length[r[n]];
%t A229996 t[n_] := Table[r[n][[k[n] + 1 - i]]/r[n][[k[1] + i - 1]], {i, 1, k[n]}];
%t A229996 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 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 *)
%o A229996 (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 A229996 Cf. A034676, A046762, A229994, A077610, A229999.
%K A229996 nonn,easy
%O A229996 1,2
%A A229996 _Clark Kimberling_, Oct 31 2013
%E A229996 Definition corrected by _Clark Kimberling_, Jun 16 2018