A325817 a(n) is the least k >= 0 such that n-k and n-(sigma(n)-k) are relatively prime.

0, 0, 0, 0, 0, 5, 0, 0, 0, 1, 0, 1, 0, 1, 2, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 3, 0, 27, 0, 1, 0, 0, 2, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 2, 3, 0, 1, 0, 0, 2, 3, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 2, 1, 0, 1, 0, 1, 0, 3, 0, 1, 0, 1, 0, 1, 0, 1, 0, 3, 2, 1, 0, 1, 2, 1, 0, 1, 2, 5, 0, 0, 2, 0, 0, 1, 0, 1, 2

Offset: 1

Author

Antti Karttunen, May 29 2019

Keywords

• nonn

Comments

a(n) is the least k >= 0 such that -n + k and (n-sigma(n))+k are coprime.

Examples

			For n=15, gcd(15-0, 15-(24-0)) = 3, gcd(15-1, 15-(24-1)) = 2 and gcd(15-2, 15-(24-2)) = 1, thus a(15) = 2.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..65537
• Index entries for sequences related to sigma(n)

Crossrefs

Cf. A000203, A000396, A001065, A009194, A014567 (positions of zeros), A324213, A325818, A325826, A325962, A325965, A325967, A325976.

Programs

Mathematica

lk[n_]:=Module[{k=0},While[!CoprimeQ[n-k,n-(DivisorSigma[1,n]-k)],k++];k]; Array[lk,110] (* Harvey P. Dale, Nov 24 2024 *)

PARI

A325817(n) = { my(s=sigma(n)); for(k=0, s, if(1==gcd(-n + k, (n-s)+k), return(k))); };

PARI

A325817(n) = { my(s=sigma(n)); for(i=0, s, if(1==gcd(n-i, n-(s-i)), return(i))); };

Formula

a(n) = A000203(n) - A325818(n) = A001065(n) - A325826(n) = n - A325976(n).

For all n:

a(A000396(n)) = A000396(n)-1.

a(n) <= n-1.

a(n) <= A325965(n).

a(n) <= A325967(n).