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A075833
a(n) is the least k such that for any p prime dividing n, p does not divide binomial((n+1)*k, k+1), or -1 if no such k exists.
0
1, 1, 2, 3, 4, 11, 6, 7, 8, 29, 10, 19847, 12, 55, 29, 15, 16, 266831, 18, 259, 62, 131, 22, 71, 24, 519, 26, 55, 28
OFFSET
1,3
COMMENTS
The initial terms (with a question mark for currently unknown terms) are 1, 1, 2, 3, 4, 11, 6, 7, 8, 29, 10, 19847, 12, 55, 29, 15, 16, 266831, 18, 259, 62, 131, 22, 71, 24, 519, 26, 55, 28, ?, 30, 31, 32, 305, 34, 536579, 36, 2203, 545219, 39, 40, 140069, 42, 2067, 89, 3219, 46, 4655, 48, 328799, 305, 207, 52, 70739, 274, 356383, 398, 6785, 58, ?, ... .
a(36) = 536579. - Michel Marcus, Jan 30 2022
It seems that a(30) = -1, otherwise a(30) > 10^10. To prove it one needs to show that for every k, binomial(31*k,k+1) is divisible by 2, 3, or 5. The next values of n for which a(n) = -1 seem to be 60, 66, 78, 84, 90, 105, ... . - Pontus von Brömssen, Jan 30 2022
Either a(30) = -1 or a(30) > 10^17. - Charles R Greathouse IV, Feb 02 2022
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's 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; }
(PARI) isok(x, f) = for (i=1, #f, if (!(x % f[i]), return(0))); return(1);
a(n) = my(k=1, f=factor(n)[, 1]~); while (!isok(binomial((n+1)*k, k+1), f), k++); k; \\ Michel Marcus, Jan 29 2022
(PARI) f(x, k) = if (x, x\k + f(x\k, k)); \\ valuation(x!, k)
isoki(x, y, k) = f(x, k) - f(y, k) - f(x-y, k) == 0;
isokf(x, y, f) = for (i=1, #f, if (! isoki(x, y, f[i]), return(0))); return(1);
af(n) = my(k=1, f=factor(n)[, 1]~); while (!isokf((n+1)*k, k+1, f), k++); k; \\ Michel Marcus, Jan 30 2022
CROSSREFS
Sequence in context: A160652 A131485 A325692 * A265904 A117351 A343602
KEYWORD
nonn,more
AUTHOR
Benoit Cloitre, Oct 14 2002
EXTENSIONS
a(12), a(38) corrected by Michel Marcus, Jan 28 2022
Edited by N. J. A. Sloane, Jan 29 2022: the previous definition was unacceptable; changed escape clause value to -1; deleted terms starting at a(18).
a(18) from Michel Marcus, Jan 30 2022
STATUS
approved