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

1,5

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, ...

Links

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

Formula

a(A020492(n)) = 0.

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

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

Example

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

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 *)

Prog

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

Crossrefs

Cf. A020492, A051612, A063514.

Sequence in context: A358011 A000586 A029399 * A249338 A046069 A320042

Adjacent sequences: A302169 A302170 A302171 * A302173 A302174 A302175

Keyword

nonn,easy

Author

Altug Alkan, Apr 03 2018

Status

approved