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

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, 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, 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

Offset: 1

Author

Altug Alkan, Apr 03 2018

Keywords

• nonn
• easy

Comments

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, ...

Numbers n such that a(n) = 1 are 4, 8, 9, 16, 32, 64, 128, 256, 400, 512, 1024, 2048, 4096, 8192, 16384, ...

Examples

			a(21) = 4 because sigma(21) = 32 and phi(21) = 12; 12*3 - 32 = 4 is the smallest corresponding distance.
		

Links

• Harvey P. Dale, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A020492, A051612, A063514.

Programs

Mathematica

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 *)

PARI

a(n) = {my(k=0, s=sigma(n), p=eulerphi(n)); while((s+k) % p != 0 && (s-k) % p != 0, k++); k;}

Formula

a(A020492(n)) = 0.

a(2^k) = 1 for k > 1.

a(p) = 2 for prime p > 3.