A006047 -id:A006047 - cp's OEIS frontend

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

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

A387051 Number of entries in the n-th row of Pascal's triangle not divisible by 32.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 17, 34, 27, 36, 29, 38, 35, 40, 33, 42, 39, 44, 41, 46, 45, 48, 41, 50, 47, 52, 49, 54, 53, 56, 53, 58, 57, 60, 59, 62, 62, 64, 17, 34, 43, 68

Offset: 0

Chai Wah Wu, Aug 15 2025

nonn

Cf. A001316, A006047, A194459, A382720-A382725, A386952, A387050.

def A387051(n):
    n1 = n>>1
    n2 = n1>>1
    n3 = n2>>1
    n4 = n3>>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 = (k5:=n3&k2).bit_count(), (k4:=(k2>>1)&np).bit_count(), (k6:=(k1>>2)&k1).bit_count(), (k7:=n3&k3).bit_count()
    n10000, n11000, n10100, n11100 = ((k4>>1)&np).bit_count(), (n4&k4).bit_count(), ((k6>>1)&np).bit_count(), (n4&k5).bit_count()
    n10010, n11010, n10110, n11110 = ((k2>>2)&k1).bit_count(), (n4&k6).bit_count(), ((k1>>3)&k3).bit_count(), (n4&k7).bit_count()
    c = n10*(n10*(n10*(n10+2)+((n100<<2)+n110)*12+35)+((((((n1000<<2)+n1010+n1100<<1)+n100<<1)+n1110<<1)+n110)*12+154))//24
    c += n100*((n100<<1)+n110+1<<2)+(((n10000<<2)+n1000+n10010+n10100+n11000+1<<2)+n10110+n11010+n11100<<2)+n1110+n11110+(n110*(n110+5)>>1)
    return c<>4

A206427 Square array 2^(m-1)*(3^n+1), read by antidiagonals.

Original entry on oeis.org

1, 2, 2, 5, 4, 4, 14, 10, 8, 8, 41, 28, 20, 16, 16, 122, 82, 56, 40, 32, 32, 365, 244, 164, 112, 80, 64, 64, 1094, 730, 488, 328, 224, 160, 128, 128, 3281, 2188, 1460, 976, 656, 448, 320, 256, 256, 9842, 6562, 4376, 2920, 1952, 1312, 896, 640, 512, 512

Offset: 0

Marcus Jaiclin, Feb 07 2012

Rectangular array giving the number of 1's in any row of Pascal's Triangle (mod 3) whose row number has exactly m 1's and n 2's in its ternary expansion (listed by antidiagonals).
a(m,n) is independent of the number of zeros in the ternary expansion of the row number.
a(m,n) gives a non-recursive formula for A206424.

			Initial 5 X 5 block of entries (upper corner is (m,n)=(0,0), m increases down, n increases across):
1    2    5   14   41
2    4   10   28   82
4    8   20   56  164
8   16   40  112  328
16  32   80  224  656
Pascal's Triangle (mod 3), row numbers in ternary:
1     <=  Row 0, m = 0, n = 0, 2^(-1)(3^0 + 1) = #1's = 1
1 1     <=  Row 1, m = 1, n = 0, 2^0(3^0 + 1) = #1's = 2
1 2 1     <=  Row 2, m = 0, n = 1, 2^(-1)(3^1 + 1) = #1's = 2
1 0 0 1     <=  Row 10, m = 1, n = 0, 2^0(3^0 + 1) = #1's = 2
1 1 0 1 1     <=  Row 11, m = 2, n = 0, 2^1(3^0 + 1) = #1's = 4
1 2 1 1 2 1     <=  Row 12, m = 1, n = 1, 2^0(3^1 + 1) = #1's = 4
1 0 0 2 0 0 1     <=  Row 20, m = 0, n = 1, 2^(-1)(3^1 + 1) = #1's = 2
1 1 0 2 2 0 1 1     <=  Row 21, m = 1, n = 1, 2^0(3^1 + 1) = #1's = 4
1 2 1 2 1 2 1 2 1     <=  Row 22, m = 0, n = 2, 2^(-1)(3^2 + 1) = #1's = 5
1 0 0 0 0 0 0 0 0 1     <=  Row 100, m = 1, n = 0, 2^0(3^0 + 1) = #1's = 2

Marcus Jaiclin, et al. Pascal's Triangle, Mod 2,3,5

Cf. A206428, A206424, A227428, A083093, A077267, A062756, A081603.

Cf. A062296, A006047, A007318.

a(m, n) = 2^(m - 1)(3^n + 1).

A206428 Rectangular array, a(m,n) = 2^(m-1)*(3^n-1), read by antidiagonals.

Original entry on oeis.org

0, 1, 0, 4, 2, 0, 13, 8, 4, 0, 40, 26, 16, 8, 0, 121, 80, 52, 32, 16, 0, 364, 242, 160, 104, 64, 32, 0, 1093, 728, 484, 320, 208, 128, 64, 0, 3280, 2186, 1456, 968, 640, 416, 256, 128, 0, 9841, 6560, 4372, 2912, 1936, 1280, 832, 512, 256, 0

Offset: 0

Marcus Jaiclin, Feb 07 2012

Number of 2's in any row of Pascal's triangle (mod 3) whose row number has exactly m 1's and n 2's in its ternary expansion.
a(m,n) is independent of the number of zeros in the ternary expansion of the row number.
a(m,n) gives a non-recursive formula for A227428.

			Initial 5 X 5 block of array (upper left corner is (0,0), row index m, column index n):
0    1    4   13   40
0    2    8   26   80
0    4   16   52  160
0    8   32  104  320
0   16   64  208  640
Pascal's Triangle (mod 3), row numbers in ternary:
1     <= Row 0, m=0, n=0, 2^(-1)(3^0-1) = #2's = 0
1 1     <= Row 1, m=1, n=0, 2^0(3^0-1) = #2's = 0
1 2 1     <= Row 2, m=0, n=1, 2^(-1)(3^1-1) = #2's = 1
1 0 0 1     <= Row 10, m=1, n=0, 2^0(3^0-1) = #2's = 0
1 1 0 1 1     <= Row 11, m=2, n=0, 2^1(3^0-1) = #2's = 0
1 2 1 1 2 1     <= Row 12, m=1, n=1, 2^0(3^1-1) = #2's = 2
1 0 0 2 0 0 1     <= Row 20, m=0, n=1, 2^(-1)(3^1-1) = #2's = 1
1 1 0 2 2 0 1 1     <= Row 21, m=1, n=1, 2^0(3^1-1) = #2's = 2
1 2 1 2 1 2 1 2 1     <= Row 22, m=0, n=2, 2^(-1)(3^2-1) = #2's = 4
1 0 0 0 0 0 0 0 0 1     <= Row 100, m=1, n=0, 2^0(3^0-1) = #2's = 0

Marcus Jaiclin, et al. Pascal's Triangle, Mod 2,3,5

Cf. A206427, A206424, A227428, A083093, A077267, A062756, A081603.

Cf. A062296, A006047, A007318.

A382724 Number of entries in the n-th row of Pascal's triangle not divisible by 6.

Original entry on oeis.org

1, 2, 3, 4, 4, 6, 5, 8, 9, 4, 6, 8, 6, 10, 14, 16, 12, 18, 5, 10, 11, 12, 16, 22, 11, 20, 27, 16, 10, 18, 18, 32, 12, 8, 14, 18, 6, 12, 16, 20, 18, 26, 18, 30, 36, 18, 24, 38, 14, 28, 38, 28, 38, 54, 17, 34, 15, 20, 26, 40, 23, 42, 45, 64, 12, 18, 14, 26, 36, 24, 38, 54, 11, 20, 29, 28, 38, 56, 37, 64, 81

Offset: 0

N. J. A. Sloane, Apr 23 2025

nonn

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

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

A387108 Number of entries in the n-th row of Pascal's triangle not divisible by 25.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 6, 12, 18, 24, 30, 15, 20, 25, 30, 35, 24, 28, 32, 36, 40, 33, 36, 39, 42, 45, 42, 44, 46, 48, 50, 11, 22, 33, 44, 55, 24, 33, 42, 51, 60, 37, 44, 51, 58, 65, 50, 55, 60

Offset: 0

Chai Wah Wu, Aug 16 2025

nonn

Eric Rowland, The number of nonzero binomial coefficients modulo p^alpha, arXiv:1001.1783 [math.NT]

Cf. A001316, A006047, A194459, A382720-A382725, A386952, A387050, A387051.

import re
from gmpy2 import digits
def A387108(n):
    s = digits(n,5)
    n1, n2, n3, n4 = s.count('1'), s.count('2'), s.count('3'), s.count('4')
    n10, n12, n13, n42, n43, n11 = s.count('10'), s.count('12'), s.count('13'), s.count('42'), s.count('43'), len(re.findall('(?=11)',s))
    n20, n21, n23, n30, n22 = s.count('20'), s.count('21'), s.count('23'), s.count('30'), len(re.findall('(?=22)',s))
    n31, n32, n40, n41, n33 = s.count('31'), s.count('32'), s.count('40'), s.count('41'), len(re.findall('(?=33)',s))
    return ((1440*n10+540*n11+240*n12+90*n13+1920*n20+720*(n21+1)+320*n22+120*n23+2160*n30+810*n31+360*n32+135*n33+2304*n40+864*n41+384*n42+144*n43)*3**n2*5**n4<<(n1+(n3<<1)))//45>>4

A382723 Number of entries in the n-th row of Pascal's triangle not divisible by 4.

Original entry on oeis.org

1, 2, 3, 4, 3, 6, 6, 8, 3, 6, 8, 12, 6, 12, 12, 16, 3, 6, 8, 12, 8, 16, 16, 24, 6, 12, 16, 24, 12, 24, 24, 32, 3, 6, 8, 12, 8, 16, 16, 24, 8, 16, 20, 32, 16, 32, 32, 48, 6, 12, 16, 24, 16, 32, 32, 48, 12, 24, 32, 48, 24, 48, 48, 64, 3, 6, 8, 12, 8, 16, 16, 24, 8, 16, 20, 32, 16, 32, 32, 48, 8

Offset: 0

N. J. A. Sloane, Apr 23 2025

nonn

Cf. A000120, A001316, A006047, A033264, A194459, A382720-A382725.

a(n) = sum(k=0, n, (binomial(n, k) % 4) != 0); \\ Michel Marcus, Apr 23 2025

def A382723(n): return bin(n)[2:].count('10')+2<Chai Wah Wu, Aug 10 2025

a(n) = (A033264(n)+2)*2^(A000120(n)-1). - 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

nonn

			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

A387109 Number of entries in the n-th row of Pascal's triangle not divisible by 27.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 10, 20, 30, 19, 26, 33, 28, 32, 36, 25, 32, 39, 32, 37, 42, 39, 42, 45, 40, 44, 48, 45, 48, 51, 50, 52, 54, 19, 38, 57, 34, 47, 60, 49, 56, 63, 40, 53, 66, 51, 60

Offset: 0

Chai Wah Wu, Aug 16 2025

nonn

Eric Rowland, The number of nonzero binomial coefficients modulo p^alpha, arXiv:1001.1783 [math.NT]

Cf. A001316, A006047, A194459, A382720-A382725, A386952, A387050, A387051, A387108.

import re
from gmpy2 import digits
def A387109(n):
    s = digits(n,3)
    n1, n2, n10, n20, n21, n11 = s.count('1'), s.count('2'), s.count('10'), s.count('20'), s.count('21'), len(re.findall('(?=11)',s))
    n100, n110, n120, n101, n111, n121 = s.count('100'), s.count('110'), s.count('120'), len(re.findall('(?=101)',s)), len(re.findall('(?=111)',s)), len(re.findall('(?=121)',s))
    n200, n201, n210, n211, n220, n221 = s.count('200'), s.count('201'), s.count('210'), s.count('211'), s.count('220'), s.count('221')
    c = 144*n10+63*n11+128*(n20+n220)+80*n21+864*n100+216*(n101+n110)+54*n111+96*n120+24*n121+1152*n200+288*(n201+n210+1)+72*n211+32*n221
    c += (m:=4*n10+n11)*(96*n20+24*n21+9*m)+16*(4*n20+n21)**2
    return (c*3**n2<>5

A206425 Erroneous version of A227428.

Original entry on oeis.org

0, 0, 1, 0, 0, 2, 1, 2, 4, 0, 0, 2, 0, 0, 4, 2, 4, 8, 1, 2, 4, 2, 4, 8, 1, 2, 4, 2, 4, 8, 4, 8, 13, 0, 0, 2, 0, 0, 4, 2, 4, 8, 0, 0, 4, 0, 0, 8, 4, 8, 16, 2, 4, 8, 4, 8, 16, 8, 16, 26, 1, 2, 4, 2, 4, 8, 4, 8, 13, 2, 4, 8, 4, 8, 16, 8, 16, 26, 4, 8, 13, 8, 16, 26

Offset: 0

Marcus Jaiclin, Feb 07 2012

dead

A006047(n) = A206424(n) + a(n).

			Example: Rows 0-8 of Pascal's Triangle (mod 3) are:
1                   So a(0) = 0
1 1                 So a(1) = 0
1 2 1               So a(2) = 1
1 0 0 1                 .
1 1 0 1 1               .
1 2 1 1 2 1             .
1 0 0 2 0 0 1
1 1 0 2 2 0 1 1
1 2 1 2 1 2 1 2 1

Marcus Jaiclin, et al. Pascal's Triangle, Mod 2,3,5

Cf. A083093, A062296, A006047.

Table[Count[Mod[Binomial[n, Range[0, n]], 3], 2], {n, 0, 99}] (* Alonso del Arte, Feb 07 2012 *)

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

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A387051 Number of entries in the n-th row of Pascal's triangle not divisible by 32.

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A206427 Square array 2^(m-1)*(3^n+1), read by antidiagonals.

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A206428 Rectangular array, a(m,n) = 2^(m-1)*(3^n-1), read by antidiagonals.

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A382724 Number of entries in the n-th row of Pascal's triangle not divisible by 6.

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A387108 Number of entries in the n-th row of Pascal's triangle not divisible by 25.

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A382723 Number of entries in the n-th row of Pascal's triangle not divisible by 4.

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

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Mathematica

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A387109 Number of entries in the n-th row of Pascal's triangle not divisible by 27.

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A206425 Erroneous version of A227428.

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