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

1,11

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

Links

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

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

Example

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

Mathematica

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

Prog

(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

Crossrefs

Cf. A001620, A023645, A072528.

Sequence in context: A211271 A124768 A321014 * A345138 A343945 A081373

Adjacent sequences: A072524 A072525 A072526 * A072528 A072529 A072530

Keyword

nonn,easy

Author

Amarnath Murthy, Aug 01 2002

Extensions

More terms from Matthew Conroy, Sep 09 2002

Incorrect comment deleted by Peter Munn, May 25 2017

Status

approved