A194459 -id:A194459 - cp's OEIS frontend

A270774 a(n) = (A005706(n) - A194459(n))/5.

0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 3, 3, 3, 3, 3, 6, 6, 6, 6, 6, 10, 10, 10, 10, 10, 16, 17, 18, 19, 20, 23, 24, 25, 26, 27, 32, 33, 34, 35, 36, 43, 44, 45, 46, 47, 56, 57, 58, 59, 60, 73, 76, 79, 82, 85, 91, 94, 97, 100, 103, 112, 115, 118, 121

Offset: 0

Tom Edgar, Mar 22 2016

nonn

A combinatorial interpretation is given in the Edgar link.

G. E. Andrews, A. S. Fraenkel, and J. A. Sellers, Characterizing the number of m-ary partitions modulo m, The American Mathematical Monthly, Vol. 122, No. 9 (November 2015), pp. 880-885.
G. E. Andrews, A. S. Fraenkel, and J. A. Sellers, Characterizing the number of m-ary partitions modulo m.
Tom Edgar, The distribution of the number of parts of m-ary partitions modulo m., arXiv:1603.00085 [math.CO], 2016.

Cf. A005706, A194459, A268127, A268128, A268443.

b[0] = 1; b[n_] := b[n] = b[n-1] + b[Floor[n/5]];
c[n_] := If[OddQ[n], 2 Count[Table[Binomial[n, k], {k, 0, (n-1)/2}], c_ /; !Divisible[c, 5]], 2 Count[Table[Binomial[n, k], {k, 0, (n-2)/2}], c_ /; !Divisible[c, 5]] + Boole[!Divisible[Binomial[n, n/2], 5]]];
a[n_] := (b[n] - c[n])/5;
Table[a[n], {n, 0, 63}] (* Jean-François Alcover, Feb 15 2019 *)

def b(n):
    A=[1]
    for i in [1..n]:
        A.append(A[i-1] + A[i//5])
    return A[n]
print([(b(n)-prod(x+1 for x in n.digits(5)))/5 for n in [0..63]])

Let b(0) = 1 and b(n) = b(n-1) + b(floor(n/5)) and let c(n) = Product_{i=0..k}(n_i+1) where n = Sum_{i=0..k}n_i*5^i is the base 5 representation of n. Then a(n) = (1/5)*(b(n) - c(n)).

A382720 Number of entries in the n-th row of Pascal's triangle not divisible by 7.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 2, 4, 6, 8, 10, 12, 14, 3, 6, 9, 12, 15, 18, 21, 4, 8, 12, 16, 20, 24, 28, 5, 10, 15, 20, 25, 30, 35, 6, 12, 18, 24, 30, 36, 42, 7, 14, 21, 28, 35, 42, 49, 2, 4, 6, 8, 10, 12, 14, 4, 8, 12, 16, 20, 24, 28, 6, 12, 18, 24, 30, 36, 42, 8, 16, 24, 32, 40, 48, 56, 10, 20, 30, 40

Offset: 0

N. J. A. Sloane, Apr 23 2025

nonn

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.

Cf. A001316, A006047, A194459, A382721-A382722, A382726.

a[n_] := Times @@ (IntegerDigits[n, 7] + 1);Array[a,81,0] (* James C. McMahon, Aug 16 2025 *)

from math import prod
from gmpy2 import digits
def A382720(n): return prod(int(d)+1 for d in digits(n,7)) # Chai Wah Wu, Aug 10 2025

A194458 Total number of entries in rows 0,1,...,n of Pascal's triangle not divisible by 5.

Original entry on oeis.org

1, 3, 6, 10, 15, 17, 21, 27, 35, 45, 48, 54, 63, 75, 90, 94, 102, 114, 130, 150, 155, 165, 180, 200, 225, 227, 231, 237, 245, 255, 259, 267, 279, 295, 315, 321, 333, 351, 375, 405, 413, 429, 453, 485, 525, 535, 555, 585, 625, 675, 678, 684, 693, 705, 720, 726

Offset: 0

Paul Weisenhorn, Aug 24 2011

nonn

The number of zeros in the first n rows is binomial(n+1,2) - a(n).

			n = 38: n+1 = 39 = 124_5, thus a(38) = (C(5,2)*15^0*3 + C(3,2)*15^1)*2 + C(2,2)*15^2 = (10*1*3 + 3*15)*2 + 1*225 = 375.

Alois P. Heinz, Table of n, a(n) for n = 0..10000
Hsien-Kuei Hwang, Svante Janson, and Tsung-Hsi Tsai, Identities and periodic oscillations of divide-and-conquer recurrences splitting at half, arXiv:2210.10968 [cs.DS], 2022, p. 53.
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. See p. 1.

A006046(n+1) = A006046(n) + A001316(n) for p=2.
A006048(n+1) = A006048(n) + A006047(n+1) for p=3.
a(n+1) = a(n) + A194459(n+1) for p=5.

a:= proc(n) local l, m, h, j;
      m:= n+1;
      l:= [];
      while m>0 do l:= [l[], irem (m, 5, 'm')+1] od;
      h:= 0;
      for j to nops(l) do h:= h*l[j] +binomial (l[j], 2) *15^(j-1) od:
      h
    end:
seq(a(n), n=0..100);

a[n_] := Module[{l, m, r, h, j}, m = n+1; l = {}; While[m>0, l = Append[l, {m, r} = QuotientRemainder[m, 5]; r+1]]; h = 0; For[j = 1, j <= Length[l], j++, h = h*l[[j]] + Binomial [l[[j]], 2] *15^(j-1)]; h]; Table [a[n], {n, 0, 100}] (* Jean-François Alcover, Feb 26 2017, translated from Maple *)

from math import prod
from gmpy2 import digits
def A194458(n): return sum(prod(int(d)+1 for d in digits(m,5)) for m in range(n+1)) # Chai Wah Wu, Aug 10 2025

from math import prod
from gmpy2 import digits
def A194458(n):
    d = list(map(lambda x:int(x)+1,digits(n+1,5)[::-1]))
    return sum((b-1)*prod(d[a:])*15**a for a, b in enumerate(d))>>1 # Chai Wah Wu, Aug 13 2025

a(n) = ((C(d0+1,2)*15^0*(d1+1) + C(d1+1,2)*15^1)*(d1+1) + C(d1+1,2)*15^1)*(d2+1) + C(d2+1,2)*15^2 ..., where d_k...d_1d_0 is the base 5 expansion of n+1 and 15 = binomial(5+1,2). The formula generalizes to other prime bases p.

Edited by Alois P. Heinz, Sep 06 2011

A382725 Number of entries in the n-th row of Pascal's triangle not divisible by 8.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 5, 10, 9, 12, 11, 14, 14, 16, 5, 10, 13, 20, 13, 18, 20, 24, 11, 22, 20, 28, 22, 28, 28, 32, 5, 10, 13, 20, 17, 26, 28, 40, 13, 26, 26, 36, 28, 40, 40, 48, 11, 22, 28, 44, 28, 40, 44, 56, 22, 44, 40, 56, 44, 56, 56, 64, 5, 10, 13, 20, 17, 26, 28, 40, 17, 34, 34, 52, 36, 56, 56, 80, 13

Offset: 0

N. J. A. Sloane, Apr 23 2025

nonn

James G. Huard, Blair K. Spearman, and Kenneth S. Williams, Pascal's triangle (mod 8), European Journal of Combinatorics 19:1 (1998), pp. 45-62.

Cf. A001316, A006047, A194459, A382720-A382724.

def A382725(n):
    n1 = n>>1
    n2 = n1>>1
    np = ~n
    n100 = (n2&(~n1)&np).bit_count()
    n110 = (n2&n1&np).bit_count()
    n10 = (n1&np).bit_count()
    return ((n100+1<<3)+(n110<<1)+n10*(n10+3))<>3 # Chai Wah Wu, Aug 10 2025

A382731 Total number of entries in rows 0,1,...,n of Pascal's triangle not divisible by 8.

Original entry on oeis.org

1, 3, 6, 10, 15, 21, 28, 36, 41, 51, 60, 72, 83, 97, 111, 127, 132, 142, 155, 175, 188, 206, 226, 250, 261, 283, 303, 331, 353, 381, 409, 441, 446, 456, 469, 489, 506, 532, 560, 600, 613, 639, 665, 701, 729, 769, 809, 857, 868, 890, 918, 962, 990, 1030, 1074, 1130, 1152, 1196, 1236, 1292, 1336, 1392, 1448, 1512, 1517, 1527

Offset: 0

N. J. A. Sloane, Apr 23 2025

nonn

James G. Huard, Blair K. Spearman, and Kenneth S. Williams, Pascal's triangle (mod 8), European Journal of Combinatorics 19:1 (1998), pp. 45-62.

Cf. A001316, A006046-A006048, A194458, A194459, A382720-A382730.

def A382731(n):
    c = 0
    for m in range(n+1):
        n1 = m>>1
        n2 = n1>>1
        np = ~m
        n100 = (n2&(~n1)&np).bit_count()
        n110 = (n2&n1&np).bit_count()
        n10 = (n1&np).bit_count()
        c += ((n100+1<<3)+(n110<<1)+n10*(n10+3))<>3
    return c # Chai Wah Wu, Aug 10 2025

A386952 Number of entries in the n-th row of Pascal's triangle not divisible by 9.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 4, 8, 12, 9, 12, 15, 14, 16, 18, 7, 14, 21, 14, 19, 24, 21, 24, 27, 4, 8, 12, 12, 18, 24, 20, 28, 36, 9, 18, 27, 20, 28, 36, 31, 38, 45, 14, 28, 42, 28, 38, 48, 42, 48, 54, 7, 14, 21, 20, 31, 42, 33, 48, 63, 14, 28, 42, 31, 44, 57, 48

Offset: 0

Chai Wah Wu, Aug 10 2025

nonn

James G. Huard, Blair K. Spearman, and Kenneth S. Williams, Pascal's triangle (mod 9), Acta Arithmetica, 78 (1997), 331-349.

Cf. A001316, A006047, A194459, A382720-A382725.

import re
from gmpy2 import digits
def A386952(n):
    s = digits(n,3)
    n1 = s.count('1')
    n2 = s.count('2')
    n01 = s.count('10')
    n02 = s.count('20')
    n11 = len(re.findall('(?=11)',s))
    n12 = s.count('21')
    return ((3*((1+n01<<2)+n11)+((n02<<2)+n12<<2))*3**n2<>2

A387050 Number of entries in the n-th row of Pascal's triangle not divisible by 16.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 9, 18, 15, 20, 17, 22, 21, 24, 21, 26, 25, 28, 27, 30, 30, 32, 9, 18, 23, 36, 21, 30, 33, 40, 25, 34, 35, 44, 37, 42, 44, 48, 21, 42, 37, 52, 37, 50, 48, 56, 43, 54, 52, 60, 54, 60, 60, 64, 9, 18, 23, 36, 29

Offset: 0

Chai Wah Wu, Aug 15 2025

nonn

Cf. A001316, A006047, A194459, A382720-A382725, A386952.

def A387050(n):
    n1 = n>>1
    n2 = n1>>1
    n3 = n2>>1
    np = ~n
    n10, n100, n110 = (k1:=n1&np).bit_count(), (k2:=(k1>>1)&np).bit_count(), (k3:=n2&k1).bit_count()
    n1100, n1000, n1010, n1110 = (n3&k2).bit_count(), ((k2>>1)&np).bit_count(), ((k1>>2)&k1).bit_count(), (n3&k3).bit_count()
    return n10*(n10*(n10+3)+6*((n100<<2)+n110)+20)//6+((n1000<<2)+n100+n1010+n1100<<2)+n110+n1110+8<>3

A254609 Triangle read by rows: T(n,k) = A243757(n)/(A243757(k)*A243757(n-k)).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 5, 5, 5, 1, 1, 1, 5, 5, 5, 1, 1, 1, 1, 1, 5, 5, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 5, 5, 5, 1, 5, 5, 5, 5, 1, 1, 1, 5, 5, 5, 1, 1, 5, 5, 5, 1, 1, 1, 1, 1, 5, 5, 1, 1, 1, 5, 5

Offset: 0

Tom Edgar, Feb 02 2015

These are the generalized binomial coefficients associated with A060904.
The exponent of T(n,k) is the number of 'carries' that occur when adding k and n-k in base 5 using the traditional addition algorithm.
If T(n,k) != 0 mod 5, then n dominates k in base 5.
A194459(n) = number of ones in row n. - Reinhard Zumkeller, Feb 04 2015

			The first five terms in A060904 are 1, 1, 1, 1, and 5 and so T(4,2) = 1*1*1*1/((1*1)*(1*1))=1 and T(5,3) = 5*1*1*1*1/((1*1*1)*(1*1))=5.
The triangle begins:
1
1, 1
1, 1, 1
1, 1, 1, 1
1, 1, 1, 1, 1
1, 5, 5, 5, 5, 1
1, 1, 5, 5, 5, 1, 1
1, 1, 1, 5, 5, 1, 1, 1
1, 1, 1, 1, 5, 1, 1, 1, 1
1, 1, 1, 1, 1, 1, 1, 1, 1, 1
1, 5, 5, 5, 5, 1, 5, 5, 5, 5, 1
1, 1, 5, 5, 5, 1, 1, 5, 5, 5, 1, 1
1, 1, 1, 5, 5, 1, 1, 1, 5, 5, 1, 1, 1
1, 1, 1, 1, 5, 1, 1, 1, 1, 5, 1, 1, 1, 1
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Reinhard Zumkeller, Rows n = 0..124 of triangle, flattened
Tyler Ball, Tom Edgar, and Daniel Juda, Dominance Orders, Generalized Binomial Coefficients, and Kummer's Theorem, Mathematics Magazine, Vol. 87, No. 2, April 2014, pp. 135-143.
Tyler Ball and Daniel Juda, Dominance over N, Rose-Hulman Undergraduate Mathematics Journal, Vol. 13, No. 2, Fall 2013.
Tom Edgar and Michael Z. Spivey, Multiplicative functions, generalized binomial coefficients, and generalized Catalan numbers, Journal of Integer Sequences, Vol. 19 (2016), Article 16.1.6.

Cf. A060904, A243757, A234957, A242849, A082907.

Cf. A194459.

import Data.List (inits)
a254609 n k = a254609_tabl !! n !! k
a254609_row n = a254609_tabl !! n
a254609_tabl = zipWith (map . div)
   a243757_list $ zipWith (zipWith (*)) xss $ map reverse xss
   where xss = tail $ inits a243757_list
-- Reinhard Zumkeller, Feb 04 2015

T(n,k) = A243757(n)/(A243757(k)*A243757(n-k)).
T(n,k) = Product_{i=1..n} A060904(i)/(Product_{i=1..k} A060904(i)*Product_{i=1..n-k} A060904(i)).
T(n,k) = A060904(n)/n*(k/A060904(k)*T(n-1,k-1)+(n-k)/A060904(n-k)*T(n-1,k)).

A382721 Number of entries in the n-th row of Pascal's triangle not divisible by 11.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 6, 12, 18, 24, 30, 36, 42, 48, 54, 60, 66, 7, 14, 21, 28, 35, 42, 49, 56, 63, 70, 77, 8, 16, 24, 32

Offset: 0

N. J. A. Sloane, Apr 23 2025

nonn

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.

Cf. A001316, A006047, A194459, A382720-A382722.

from math import prod
from gmpy2 import digits
def A382721(n): return prod(int(d,11)+1 for d in digits(n,11)) # Chai Wah Wu, Aug 10 2025

A382722 Number of entries in the n-th row of Pascal's triangle not divisible by 13.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48, 52, 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60, 65, 6, 12, 18, 24, 30, 36, 42, 48, 54, 60, 66, 72, 78, 7, 14, 21

Offset: 0

N. J. A. Sloane, Apr 23 2025

nonn

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.

Cf. A001316, A006047, A194459, A382720-A382721.

from math import prod
from gmpy2 import digits
def A382722(n): return prod(int(d,13)+1 for d in digits(n,13)) # Chai Wah Wu, Aug 10 2025

cp's OEIS Frontend

A270774 a(n) = (A005706(n) - A194459(n))/5.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Sage

Formula

A382720 Number of entries in the n-th row of Pascal's triangle not divisible by 7.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Python

A194458 Total number of entries in rows 0,1,...,n of Pascal's triangle not divisible by 5.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Python

Python

Formula

Extensions

A382725 Number of entries in the n-th row of Pascal's triangle not divisible by 8.

Views

Author

Keywords

Links

Crossrefs

Programs

Python

A382731 Total number of entries in rows 0,1,...,n of Pascal's triangle not divisible by 8.

Views

Author

Keywords

Links

Crossrefs

Programs

Python

A386952 Number of entries in the n-th row of Pascal's triangle not divisible by 9.

Views

Author

Keywords

Links

Crossrefs

Programs

Python

A387050 Number of entries in the n-th row of Pascal's triangle not divisible by 16.

Views

Author

Keywords

Crossrefs

Programs

Python

A254609 Triangle read by rows: T(n,k) = A243757(n)/(A243757(k)*A243757(n-k)).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Formula

A382721 Number of entries in the n-th row of Pascal's triangle not divisible by 11.

Views