A194459 Number of entries in the n-th row of Pascal's triangle not divisible by 5.

1, 2, 3, 4, 5, 2, 4, 6, 8, 10, 3, 6, 9, 12, 15, 4, 8, 12, 16, 20, 5, 10, 15, 20, 25, 2, 4, 6, 8, 10, 4, 8, 12, 16, 20, 6, 12, 18, 24, 30, 8, 16, 24, 32, 40, 10, 20, 30, 40, 50, 3, 6, 9, 12, 15, 6, 12, 18, 24, 30, 9, 18, 27, 36, 45, 12, 24, 36, 48, 60, 15, 30

Offset: 0

Author

Paul Weisenhorn, Aug 24 2011

Keywords

• nonn
• look

Comments

Pascal triangles modulo p with p prime have the dimension D = log(p*(p+1)/2)/log(p). [Corrected by Connor Lane, Nov 28 2022]

Also number of ones in row n of triangle A254609. - Reinhard Zumkeller, Feb 04 2015

Examples

			n = 32 = 112|_5: b(32,1) = 2, b(32,2) = 1, thus a(32) = 2^2 * 3^1 = 12.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..10000
• Hsien-Kuei Hwang, Svante Janson, and Tsung-Hsi Tsai, Periodic minimum in the count of binomial coefficients not divisible by a prime, arXiv:2408.06817 [math.NT], 2024.

Crossrefs

Cf. A006046, A001316 (for p=2).

Cf. A006048, A006047 (for p=3).

Cf. A194458 (for p=5).

Cf. A007318, A254609.

Programs

Haskell

a194459 = sum . map (signum . flip mod 5) . a007318_row
-- Reinhard Zumkeller, Feb 04 2015

Maple

a:= proc(n) local l, m, t;
      m:= n;
      l:= [0$5];
      while m>0 do t:= irem(m, 5, 'm')+1; l[t]:=l[t]+1 od;
      mul(r^l[r], r=2..5)
    end:
seq(a(n), n=0..100);

Mathematica

Nest[Join[#, 2#, 3#, 4#, 5#]&, {1}, 4] (* Jean-François Alcover, Apr 12 2017, after code by Robert G. Wilson v in A006047 *)

Python

from math import prod
from sympy.ntheory import digits
def A194459(n):
    s = digits(n,5)[1:]
    return prod((d+1)**s.count(d) for d in range(1,5)) # Chai Wah Wu, Jul 23 2025

Formula

a(n) = Product_{d=1..4} (d+1)^b(n,d) with b(n,d) = number of digits d in base 5 expansion of n. The formula generalizes to other prime bases p.

a(n) = A194458(n) - A194458(n-1).

Extensions

Edited by Alois P. Heinz, Sep 06 2011