A368077 Numbers k such that row k of Pascal's triangle mod 10 contains all the numbers 0 to 9.

47, 59, 89, 94, 117, 118, 119, 123, 147, 173, 189, 198, 214, 219, 221, 222, 223, 233, 237, 238, 239, 243, 244, 247, 248, 297, 298, 309, 313, 317, 318, 319, 323, 339, 344, 345, 346, 347, 348, 363, 366, 367, 368, 369, 373, 397, 398, 409, 413, 414, 417, 418, 421, 422, 423, 429, 433, 437, 438, 439

Offset: 1

Author

Robert Israel, Dec 10 2023

Keywords

• nonn
• base

Comments

Numbers k such that A208280(k) = 10.

Examples

			a(3) = 89 is a term because
  binomial(89,15) = 38163061637050680 == 0 (mod 10),
  binomial(89,0) = 1 == 1 (mod 10),
  binomial(89,5) = 41507642 == 2 (mod 10),
  binomial(89,8) = 70625252863 == 3 (mod 10),
  binomial(89,3) = 113564 == 4 (mod 10),
  binomial(89,16) = 176504160071359395 == 5 (mod 10),
  binomial(89,2) = 3916 == 6 (mod 10),
  binomial(89,9) = 635627275767 == 7 (mod 10),
  binomial(89,6) = 581106988 == 8 (mod 10), and
  binomial(89,1) = 89 == 9 (mod 10).
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A208280.

Programs

Maple

filter:= proc(n) local k,S;
  S:= {$0..9}:
  for k from 0 to n/2 do
    S:= S minus {(binomial(n,k) mod 10)};
    if S = {} then return true fi
  od;
  false
end proc:
select(filter, [$1..1000]); # Robert Israel, Dec 10 2023

Python

from itertools import count, islice
def A368077_gen(): # generator of terms
    a, b = [], set(range(10))
    for i in count(0):
        c, d = 0, []
        for k in a:
            d.append((c+k)%10)
            c = k
        a = d+[1]
        if b.issubset(set(a)): yield i
A368077_list = list(islice(A368077_gen(),30)) # Chai Wah Wu, May 01 2025