A072527 Number of values of k such that n divided by k leaves a remainder 3.

0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 2, 1, 2, 1, 3, 1, 2, 2, 3, 1, 3, 1, 4, 2, 2, 1, 5, 2, 2, 2, 4, 1, 5, 1, 4, 2, 2, 3, 6, 1, 2, 2, 6, 1, 5, 1, 4, 4, 2, 1, 7, 2, 4, 2, 4, 1, 5, 3, 6, 2, 2, 1, 9, 1, 2, 4, 5, 3, 5, 1, 4, 2, 6, 1, 9, 1, 2, 4, 4, 3, 5, 1, 8, 3, 2, 1, 9, 3, 2, 2, 6, 1, 9, 3, 4, 2, 2, 3, 9, 1, 4, 4, 7, 1, 5

Offset: 1

Author

Amarnath Murthy, Aug 01 2002

Keywords

• nonn
• easy

Comments

For n > 3, the number of divisors of (n - 3) that are greater than 3; equivalently, those that are less than (n - 3)/3. - Peter Munn, May 18 2017

Examples

			a(15) = 3 as 15 divided by exactly three numbers 4, 6 and 12 leaves a remainder 3.
		

Links

• Paolo Xausa, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A001620, A023645, A072528.

Programs

Mathematica

A072527[n_] := If[n>6, DivisorSum[n-3, 1&, #>3&], 0];
Array[A072527, 150] (* Paolo Xausa, Jan 18 2024 *)

PARI

a(n) = sum(k=1, n-1, (n % k) == 3); \\ Michel Marcus, May 25 2017

PARI

a(n)=if(n>6, numdiv(n-3) - if(n%6==3, 3, if(n%6==2 || n%6==4, 1, 2)), 0) \\ Charles R Greathouse IV, May 27 2017

Formula

a(n) = tau(n-3)-1 if n is congruent to {2, 4} mod 6, tau(n-3)-2 if n is congruent to {0, 1, 5} mod 6, tau(n-3)-3 if n is congruent to 3 mod 6; n<>3. - Vladeta Jovovic, Aug 06 2002

G.f.: Sum_{k>0} x^(4*k+3)/(1-x^k). - Vladeta Jovovic, Dec 15 2002

Sum_{k=1..n} a(k) ~ n * (log(n) + 2*gamma - 17/6), where gamma is Euler's constant (A001620). - Amiram Eldar, Jan 18 2024

Extensions

More terms from Matthew Conroy, Sep 09 2002

Incorrect comment deleted by Peter Munn, May 25 2017