A006048 -id:A006048 - cp's OEIS frontend

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

Paul Weisenhorn, Aug 24 2011

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

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

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.

Cf. A006046, A001316 (for p=2).
Cf. A006048, A006047 (for p=3).
Cf. A194458 (for p=5).
Cf. A007318, A254609.

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

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);

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

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

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).

Edited by Alois P. Heinz, Sep 06 2011

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

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

A382726 Total number of entries in rows 0,1,...,n of Pascal's triangle not divisible by 7.

Original entry on oeis.org

1, 3, 6, 10, 15, 21, 28, 30, 34, 40, 48, 58, 70, 84, 87, 93, 102, 114, 129, 147, 168, 172, 180, 192, 208, 228, 252, 280, 285, 295, 310, 330, 355, 385, 420, 426, 438, 456, 480, 510, 546, 588, 595, 609, 630, 658, 693, 735, 784, 786, 790, 796, 804, 814, 826, 840, 844, 852, 864, 880, 900, 924, 952, 958, 970, 988, 1012, 1042, 1078

Offset: 0

N. J. A. Sloane, Apr 23 2025

nonn

Partial sums of A382720. - James C. McMahon, Aug 15 2025

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, A006046-A006048, A194458, A194459, A382720-A382731.

a[n_]:=(n^2+3n+2)/2-Count[Mod[Flatten[Table[Binomial[m, k], {m, 0,n}, {k, 0,m}]] ,7],0];Array[a,69,0] (* James C. McMahon, Aug 15 2025 *)

from math import prod
from gmpy2 import digits
def A382726(n): return sum(prod(int(d)+1 for d in digits(m,7)) for m in range(n+1)) # Chai Wah Wu, Aug 10 2025

from math import prod
from gmpy2 import digits
def A382726(n):
    d = list(map(lambda x:int(x)+1,digits(n+1,7)[::-1]))
    return sum((b-1)*prod(d[a:])*28**a for a, b in enumerate(d))>>1 # Chai Wah Wu, Aug 13 2025

A382730 Total number of entries in rows 0,1,...,n of Pascal's triangle not divisible by 6.

Original entry on oeis.org

1, 3, 6, 10, 14, 20, 25, 33, 42, 46, 52, 60, 66, 76, 90, 106, 118, 136, 141, 151, 162, 174, 190, 212, 223, 243, 270, 286, 296, 314, 332, 364, 376, 384, 398, 416, 422, 434, 450, 470, 488, 514, 532, 562, 598, 616, 640, 678, 692, 720, 758, 786, 824, 878, 895, 929, 944, 964, 990, 1030, 1053, 1095, 1140, 1204, 1216, 1234, 1248, 1274

Offset: 0

N. J. A. Sloane, Apr 23 2025

nonn

Chai Wah Wu, Table of n, a(n) for n = 0..10000

Cf. A001316, A006046-A006048, A194458, A194459, A382720-A382731.

A386953 Total number of entries in rows 0,1,...,n of Pascal's triangle not divisible by 9.

Original entry on oeis.org

1, 3, 6, 10, 15, 21, 28, 36, 45, 49, 57, 69, 78, 90, 105, 119, 135, 153, 160, 174, 195, 209, 228, 252, 273, 297, 324, 328, 336, 348, 360, 378, 402, 422, 450, 486, 495, 513, 540, 560, 588, 624, 655, 693, 738, 752, 780, 822, 850, 888, 936, 978, 1026, 1080, 1087

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, A006046-A006048, A194458, A194459, A382720-A382731.

Partial sums of A386952.

import re
from gmpy2 import digits
def A386953(n):
    c = 0
    for m in range(n+1):
        s = digits(m,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')
        c += ((3*((1+n01<<2)+n11)+((n02<<2)+n12<<2))*3**n2<>2
    return c

A382727 Total number of entries in rows 0,1,...,n of Pascal's triangle not divisible by 11.

Original entry on oeis.org

1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 68, 72, 78, 86, 96, 108, 122, 138, 156, 176, 198, 201, 207, 216, 228, 243, 261, 282, 306, 333, 363, 396, 400, 408, 420, 436, 456, 480, 508, 540, 576, 616, 660, 665, 675, 690, 710, 735, 765, 800, 840, 885, 935, 990, 996, 1008, 1026, 1050, 1080, 1116, 1158, 1206, 1260, 1320, 1386, 1393, 1407

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, A006046-A006048, A194458, A194459, A382720-A382731.

from math import prod
from gmpy2 import digits
def A382727(n): return sum(prod(int(d,11)+1 for d in digits(m,11)) for m in range(n+1)) # Chai Wah Wu, Aug 10 2025

from math import prod
from gmpy2 import digits
def A382727(n):
    d = list(map(lambda x:int(x,11)+1,digits(n+1,11)[::-1]))
    return sum((b-1)*prod(d[a:])*66**a for a, b in enumerate(d))>>1 # Chai Wah Wu, Aug 13 2025

A382728 Total number of entries in rows 0,1,...,n of Pascal's triangle not divisible by 13.

Original entry on oeis.org

1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 93, 97, 103, 111, 121, 133, 147, 163, 181, 201, 223, 247, 273, 276, 282, 291, 303, 318, 336, 357, 381, 408, 438, 471, 507, 546, 550, 558, 570, 586, 606, 630, 658, 690, 726, 766, 810, 858, 910, 915, 925, 940, 960, 985, 1015, 1050, 1090, 1135, 1185, 1240, 1300, 1365, 1371, 1383, 1401

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, A006046-A006048, A194458, A194459, A382720-A382731.

from math import prod
from gmpy2 import digits
def A382728(n): return sum(prod(int(d,13)+1 for d in digits(m,13)) for m in range(n+1)) # Chai Wah Wu, Aug 10 2025

from math import prod
from gmpy2 import digits
def A382728(n):
    d = list(map(lambda x:int(x,13)+1,digits(n+1,13)[::-1]))
    return sum((b-1)*prod(d[a:])*91**a for a, b in enumerate(d))>>1 # Chai Wah Wu, Aug 13 2025

A382729 Total number of entries in rows 0,1,...,n of Pascal's triangle not divisible by 4.

Original entry on oeis.org

1, 3, 6, 10, 13, 19, 25, 33, 36, 42, 50, 62, 68, 80, 92, 108, 111, 117, 125, 137, 145, 161, 177, 201, 207, 219, 235, 259, 271, 295, 319, 351, 354, 360, 368, 380, 388, 404, 420, 444, 452, 468, 488, 520, 536, 568, 600, 648, 654, 666, 682, 706, 722, 754, 786, 834, 846, 870, 902, 950, 974, 1022, 1070, 1134, 1137, 1143, 1151

Offset: 0

N. J. A. Sloane, Apr 23 2025

nonn

Cf. A001316, A006046-A006048, A194458, A194459, A382720-A382731.

def A382729(n): return 1+sum(bin(m)[2:].count('10')+2<Chai Wah Wu, Aug 10 2025

A387064 Total number of entries in rows 0 to n of Pascal's triangle multiple of n.

Original entry on oeis.org

0, 3, 1, 2, 2, 4, 3, 6, 4, 6, 10, 10, 12, 12, 21, 22, 8, 16, 18, 18, 30, 42, 47, 22, 38, 20, 74, 18, 65, 28, 81, 30, 16, 113, 136, 132, 94, 36, 147, 195, 140, 40, 162, 42, 199, 210, 217, 46, 126, 42, 146, 302, 261, 52, 110, 335, 243, 374, 394, 58, 363, 60, 465, 416

Offset: 0

Jean-Marc Rebert, Aug 15 2025

			The first two rows of Pascal's triangle are [1] and [1, 1]. Since all elements are divisible by 1, a(1) equals the total number of such divisible terms: 1 + 2 = 3.

Chai Wah Wu, Table of n, a(n) for n = 0..10000

Cf. A001316, A006046, A006047, A006048, A007318, A194458, A194459, A382720-A382731, A386953.

a[n_] := Sum[Boole[Divisible[Binomial[k, i], n]], {k, 0, n}, {i, 0, k}]; a[0] = 0; Array[a, 100, 0] (* Amiram Eldar, Aug 17 2025 *)

a(n) = if (n, sum(r=0, n, sum(k=0, r, !(binomial(r,k) % n))), 0); \\ Michel Marcus, Aug 15 2025

from sympy import isprime, integer_nthroot
def A387064(n):
    if isprime(n): return n-1
    a, b = integer_nthroot(n,2)
    if b and isprime(a): return n-a
    r, c = [1], n==1
    for m in range(n):
        s = [1]
        for i in range(m):
            s.append((r[i]+r[i+1])%n)
            c += s[-1]==0
        r = s+[1]
        c += (n==1)<<1
    return int(c) # Chai Wah Wu, Aug 21 2025

a(p) = p-1, a(p^2) = p*(p-1) for p prime. Conjecture: a(p^k) = (p-1)*p^(k-1) for p prime. - Chai Wah Wu, Aug 21 2025

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

Python

Formula

Extensions

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

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

A382726 Total number of entries in rows 0,1,...,n of Pascal's triangle not divisible by 7.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Python

Python

A382730 Total number of entries in rows 0,1,...,n of Pascal's triangle not divisible by 6.

Views

Author

Keywords

Links

Crossrefs

A386953 Total number of entries in rows 0,1,...,n of Pascal's triangle not divisible by 9.

Views

Author

Keywords

Links

Crossrefs

Programs

Python

A382727 Total number of entries in rows 0,1,...,n of Pascal's triangle not divisible by 11.

Views

Author

Keywords

Links

Crossrefs

Programs

Python

Python

A382728 Total number of entries in rows 0,1,...,n of Pascal's triangle not divisible by 13.

Views

Author

Keywords

Links

Crossrefs

Programs