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

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

0,2

Comments

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

Links

Vincenzo Librandi, 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.

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 *)
Table[Sum[Times@@(IntegerDigits[m, 7]+1), {m, 0, n}], {n, 0, 68}] (* Vincenzo Librandi, Feb 17 2026 *)

Prog

(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
(Magma) [&+[ &*[ d+1 : d in (m eq 0 select [0] else IntegerToSequence(m, 7))]: m in [0..n]]: n in [0..68] ]; // Vincenzo Librandi, Feb 17 2026

Crossrefs

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

Sequence in context: A390807 A054636 A231680 * A124158 A161208 A109444

Adjacent sequences: A382723 A382724 A382725 * A382727 A382728 A382729

Keyword

nonn

Author

N. J. A. Sloane, Apr 23 2025

Status

approved