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