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.

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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

Views

Author

Chai Wah Wu, Aug 15 2025

Keywords

Crossrefs

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

Programs

  • Python
    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

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

Views

Author

N. J. A. Sloane, Apr 23 2025

Keywords

Links

Crossrefs

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

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

Views

Author

N. J. A. Sloane, Apr 23 2025

Keywords

Comments

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

Links

Crossrefs

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

Programs

  • Mathematica
    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 *)
  • Python
    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
  • Python
    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

Views

Author

N. J. A. Sloane, Apr 23 2025

Keywords

Links

Crossrefs

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

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

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