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

0,2

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

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.

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).

Example

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

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 *)
Table[Times@@(IntegerDigits[n, 5]+1), {n, 0, 80}] (* Vincenzo Librandi, Feb 21 2026 *)

Prog

(Haskell)
a194459 = sum . map (signum . flip mod 5) . a007318_row
-- Reinhard Zumkeller, Feb 04 2015
(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
(Magma) [&*[ d+1 : d in (n eq 0 select [0] else IntegerToSequence(n, 5))] : n in [0..70]]; // Vincenzo Librandi, Feb 21 2026

Crossrefs

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

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

Cf. A194458 (for p=5).

Cf. A007318, A254609.

Sequence in context: A141809 A309435 A043265 * A143120 A353277 A386630

Adjacent sequences: A194456 A194457 A194458 * A194460 A194461 A194462

Keyword

nonn,look

Author

Paul Weisenhorn, Aug 24 2011

Extensions

Edited by Alois P. Heinz, Sep 06 2011

Status

approved