A054900 -id:A054900 - cp's OEIS frontend

A090623 Triangle of T(n,k) = [n/k] + [n/k^2] + [n/k^3] + [n/k^4] + ... for n, k > 1.

1, 1, 1, 3, 1, 1, 3, 1, 1, 1, 4, 2, 1, 1, 1, 4, 2, 1, 1, 1, 1, 7, 2, 2, 1, 1, 1, 1, 7, 4, 2, 1, 1, 1, 1, 1, 8, 4, 2, 2, 1, 1, 1, 1, 1, 8, 4, 2, 2, 1, 1, 1, 1, 1, 1, 10, 5, 3, 2, 2, 1, 1, 1, 1, 1, 1, 10, 5, 3, 2, 2, 1, 1, 1, 1, 1, 1, 1, 11, 5, 3, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 11, 6, 3, 3, 2, 2, 1, 1, 1

Offset: 2

Henry Bottomley, Dec 06 2003

			Rows start:
  1;
  1,1;
  3,1,1;
  3,1,1,1;
  4,2,1,1,1;
  4,2,1,1,1,1;
  7,2,2,1,1,1,1;
  7,4,2,1,1,1,1,1;
  8,4,2,2,1,1,1,1,1;
  ...

Zhuorui He, Table of n, a(n) for n = 2..11326
Wenguang Zhai, On the prime power factorization of n!, Journal of Number Theory, Volume 129, Issue 8, August 2009, Pages 1820-1836.

Columns include A011371, A054861, A054893, A027868, A054895, A054896, A054897, A054898, A054899, A064458, A064459, A090620, A054900.
Row sums: A078632.
Cf. A240236.

A090623[n_, k_] := Quotient[n - DigitSum[n, k], k - 1];
Table[A090623[n, k], {n, 2, 15}, {k, 2, n}] (* Paolo Xausa, Sep 02 2025 *)

T(n,k) = {my(s = 0, j = 1); while(p=n\k^j, s += p; j++); s;} \\ Michel Marcus, Feb 02 2016

T(n,k) = (n - sumdigits(n,k))/(k-1) \\ Zhuorui He, Aug 25 2025

For p prime, T(n, p) = A090622(n, p) is the number of times that p is a factor of n!.

T(n,k) = (n - A240236(n, k))/(k - 1). - Zhuorui He, Aug 25 2025

a(41) onward corrected by Zhuorui He, Aug 25 2025

A093391 a(n) = floor(n/16) + floor((n+1)/16) + floor((n+2)/16) + floor((n+3)/16).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 6, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 9, 10, 11, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 13, 14, 15, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 17, 18, 19, 20, 20, 20, 20

Offset: 0

Reinhard Zumkeller, Mar 28 2004

nonn

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,-1,0,-1,2,-1,0,-1,2,-1,0,-1,2,-1).

Cf. A004524, A093390, A093392.

Cf. A054900.

a093391 n = sum $ map ((flip div 16) . (+ n)) [0..3] -- Reinhard Zumkeller, Oct 10 2013

[Floor(n/16)+Floor((n+1)/16)+Floor((n+2)/16)+Floor((n+3)/16): n in [0..100]]; // Vincenzo Librandi, Feb 16 2018

Total/@(Floor/@(Partition[Range[0,90],4,1]/16)) (* Harvey P. Dale, Sep 21 2013 *)
CoefficientList[Series[x^13 / ((1 + x^4) (x^8 + 1) (x - 1)^2), {x, 0, 100}], x] (* Vincenzo Librandi, Feb 16 2018 *)

a(n) = n\16 + (n+1)\16 + (n+2)\16 + (n+3)\16 \\ Andrew Howroyd, Feb 15 2018

From R. J. Mathar, Mar 22 2011: (Start)
a(n) = +2*a(n-1) -a(n-2) -a(n-4) +2*a(n-5) -a(n-6) -a(n-8) +2*a(n-9) -a(n-10) -a(n-12) +2*a(n-13) -a(n-14).
G.f.: x^13/ ( (1+x^4)*(x^8+1)*(x-1)^2 ).
(End)

cp's OEIS Frontend

A090623 Triangle of T(n,k) = [n/k] + [n/k^2] + [n/k^3] + [n/k^4] + ... for n, k > 1.

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A093391 a(n) = floor(n/16) + floor((n+1)/16) + floor((n+2)/16) + floor((n+3)/16).

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