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A075833 Least k such that for any p prime dividing n, p does not divide binomial((n+1)*k, k+1) or 0 if no k was found. 0
1, 1, 2, 3, 4, 11, 6, 7, 8, 29, 10, 0, 12, 55, 29, 15, 16, 0, 18, 259, 62, 131, 22, 71, 24, 519, 26, 55, 28, 0, 30, 31, 32, 305, 34, 0, 36, 0 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

It seems that if p is prime a(6p) doesn't exist.

LINKS

Table of n, a(n) for n=1..38.

FORMULA

a(p^m) = p^m - 1 for prime p and m > 0. [Proof: if k in base p is x_1 x_2 ... x_t and t <= m, then (p^m+1)*k in base p is x_1 x_2 ... x_t 0 0 ... 0 x_1 x_2 ... x_t. Let k+1 in base p be y_1 y_2 ... y_r, where r = t or t+1. By Lucas theorem, we have y_r <= x_t, y_(r-1) <= x_(t-1), y_(r-2) <= x_(t-2), ... Therefore, x_1 = x_2 = ... = x_m = p-1 and k in base 10 is p^m - 1. - Jinyuan Wang, Apr 06 2020]

PROG

(PARI) D(k, n) = binomial((n+1)*k, k+1);

a(n) = {my(d=divisors(n), k=1); while(prod(i=1, numdiv(n), D(k, n)%if(isprime(component(d, i)), component(d, i), D(k, n)+1)) == 0, k++); k; }

CROSSREFS

Sequence in context: A160652 A131485 A325692 * A265904 A117351 A343602

Adjacent sequences:  A075830 A075831 A075832 * A075834 A075835 A075836

KEYWORD

nonn,more

AUTHOR

Benoit Cloitre, Oct 14 2002

STATUS

approved

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Last modified December 7 12:17 EST 2021. Contains 349581 sequences. (Running on oeis4.)