cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Previous Showing 11-18 of 18 results.

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

Views

Author

Chai Wah Wu, Aug 10 2025

Keywords

Links

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

Crossrefs

Cf. A001316, A006046-A006048, A194458, A194459, A382720-A382731.
Partial sums of A386952.

Programs

  • Python
    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

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

Views

Author

Chai Wah Wu, Aug 16 2025

Keywords

Links

Crossrefs

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

Programs

  • Python
    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

Views

Author

N. J. A. Sloane, Apr 23 2025

Keywords

Crossrefs

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

Programs

  • PARI
    a(n) = sum(k=0, n, (binomial(n, k) % 4) != 0); \\ Michel Marcus, Apr 23 2025
  • Python
    def A382723(n): return bin(n)[2:].count('10')+2<Chai Wah Wu, Aug 10 2025

Formula

a(n) = (A033264(n)+2)*2^(A000120(n)-1). - Chai Wah Wu, Aug 10 2025

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

Views

Author

N. J. A. Sloane, Apr 23 2025

Keywords

Links

Crossrefs

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

Programs

  • Python
    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
  • Python
    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

Views

Author

N. J. A. Sloane, Apr 23 2025

Keywords

Links

Crossrefs

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

Programs

  • Python
    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
  • Python
    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

Views

Author

N. J. A. Sloane, Apr 23 2025

Keywords

Crossrefs

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

Programs

  • Python
    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

Views

Author

Jean-Marc Rebert, Aug 15 2025

Keywords

Examples

			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.

Links

Crossrefs

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

Programs

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

Formula

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

Views

Author

Chai Wah Wu, Aug 16 2025

Keywords

Links

Crossrefs

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

Programs

  • Python
    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
Previous Showing 11-18 of 18 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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