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A302172 Distance from sigma(n) to nearest multiple of phi(n).

%I A302172 #18 Apr 29 2018 15:00:08
%S A302172 0,0,0,1,2,0,2,1,1,2,2,0,2,0,0,1,2,3,2,2,4,4,2,4,9,6,4,4,2,0,2,1,8,6,
%T A302172 0,5,2,6,8,6,2,0,2,4,6,6,2,4,15,7,8,2,2,6,8,0,8,6,2,8,2,6,4,1,12,4,2,
%U A302172 2,8,0,2,3,2,6,4,4,24,0,2,6,13,6,2,8,20,6,8,20,2,6,32,8,8,6,24,4,2,3,24,17
%N A302172 Distance from sigma(n) to nearest multiple of phi(n).
%C A302172 Numbers n such that a(n) <> A063514(n) are 8, 16, 21, 22, 25, 28, 32, 36, 40, 48, 50, 54, 55, 63, 64, 65, 68, 76, 77, 80, ...
%C A302172 Numbers n such that a(n) = 1 are 4, 8, 9, 16, 32, 64, 128, 256, 400, 512, 1024, 2048, 4096, 8192, 16384, ...
%H A302172 Harvey P. Dale, <a href="/A302172/b302172.txt">Table of n, a(n) for n = 1..1000</a>
%F A302172 a(A020492(n)) = 0.
%F A302172 a(2^k) = 1 for k > 1.
%F A302172 a(p) = 2 for prime p > 3.
%e A302172 a(21) = 4 because sigma(21) = 32 and phi(21) = 12; 12*3 - 32 = 4 is the smallest corresponding distance.
%t A302172 dsp[n_]:=Module[{s=DivisorSigma[1,n],p=EulerPhi[n],m},m=Floor[s/p];Abs[ Nearest[ {m*p,(m+1)p},s]-s]]; Array[dsp,100][[All,1]] (* _Harvey P. Dale_, Apr 29 2018 *)
%o A302172 (PARI) a(n) = {my(k=0, s=sigma(n), p=eulerphi(n)); while((s+k) % p != 0 && (s-k) % p != 0, k++); k;}
%Y A302172 Cf. A020492, A051612, A063514.
%K A302172 nonn,easy
%O A302172 1,5
%A A302172 _Altug Alkan_, Apr 03 2018