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 A343507 a(n) is the smallest nonnegative integer k such that (2*k)! / (k+n)!^2 is an integer. 1
 0, 208, 3475, 8174, 252965, 3648835, 72286092, 159329607, 2935782889 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS So far, all the numbers a(n) + n are squarefree. LINKS EXAMPLE a(1) = 0: (2*0)! / (0+1)!^2 = 1/1 = 1. From Jon E. Schoenfield, Apr 18 2021: (Start) Let f(n,k) = (2*k)!/(k+n)!^2. Then a(n) is the smallest nonnegative k such that f(n,k) is an integer. f(n,0) = (2*0)!/(0+n)!^2 = 1/n!^2, so the fraction begins at k=0 with a value of 1/n!^2, and each time k is incremented by 1, the fraction is multiplied by (2*k)*(2*k-1) and divided by (k+n)^2. Whenever k+n is a prime p, this division will cause the reduced fraction to have p in its denominator with multiplicity 1 (since the multiplicity of p in (k+n)!^2 is 2, but its multiplicity in (2*k)! is only 1). Further multiplications by (2*k)*(2*k-1)/(k+n)^2 using successive values of k will not remove the prime factor p from the reduced fraction's denominator until k reaches p. As a result, the interval [a(n)+1, a(n)+n] can never contain a prime. For n=2, the factorizations of the numerator and denominator of the reduced fraction (2k)!/(k+n)!^2 are shown in the table below for the first several values of k and for the last few through k = 208. Large blocks of consecutive primes (denoted by ellipses), each with multiplicity one, accumulate in the numerator as k gets larger. .       |           Reduced fraction (2k)!/(k+2)!^2       +-------------------------------------+---------------     k | numerator                           | denominator   ----+-------------------------------------+---------------     0 | 1                                   | 2^2     1 | 1                                   | 2 * 3^2     2 | 1                                   | 2^3 * 3     3 | 1                                   | 2^2 * 5     4 | 7                                   | 2 * 3^2 * 5     5 | 1                                   | 7     6 | 3 * 11                              | 2^4 * 7     7 | 11 * 13                             | 2^3 * 3^3     8 | 11 * 13                             | 2 * 3^2 * 5     9 | 13 * 17                             | 5 * 11    10 | 13 * 17 * 19                        | 2^2 * 3^2 * 11     . |                                     |     . |                                     |     . |                                     |   205 | 2^3 * 5 * 7 * 11^2 * 13 * 17 * 19^2 | 23 * 103       | * 31 * 37 * 43 * 53 * 71 * 73 * 79  |       | * 107 * ... * 131 * 211 * ... * 409 |       |                                     |   206 | 3 * 5 * 7 * 11^2 * 17 * 19^2 * 31   | 2^3 * 13 * 23       | * 37* 43 * 53 * 71 * 73 * 79        |       | * 107 * ... * 137 * 211 * ... * 409 |       |                                     |   207 | 3^3 * 5 * 7^2 * 17 * 31 * 37        |       | * 43 * 53 * 59 * 71 * 73 * 79       |       | * 107 * ... * 137 * 211 * ... * 409 | 2^2 * 13       |                                     |   208 | 2 * 3 * 17 * 31 * 37 * 43           |       | * 53 * 59 * 71 * ... * 83           |       | * 107 * ... * 137 * 211 * ... * 409 | 1 . The denominator first reaches 1 at k=208, so a(2)=208. (End) PROG (PARI) f(n, k) = (2*k)! / (k+n)!^2; isok(n, k) = denominator(f(n, k)) == 1; a(n) = my(k=0); while (!isok(n, k), k++); k; \\ Michel Marcus, May 03 2021 (Python) from fractions import Fraction from sympy import factorial def A343507(n):     k, f = 0, Fraction(1, int(factorial(n))**2)     while f.denominator != 1:         k += 1         f *= Fraction(2*k*(2*k-1), (k+n)**2)     return k # Chai Wah Wu, May 03 2021 (Python) from math import gcd n = 0 while n >= 0:     num, den, i, n = 1, 1, 1, n+1     while i <= n:         den, i = den*i*i, i+1     k, kk = 0, 0     while den > 1:         k, kk = k+1, kk+2         d = gcd(num, (n+k)*(n+k))*gcd(den, (kk-1)*kk)         num, den = num*(kk-1)*kk//d, den*(n+k)*(n+k)//d     print(n, k) # A.H.M. Smeets, May 03 2021 CROSSREFS Cf. A001044, A010050. Sequence in context: A235444 A255074 A172967 * A231111 A339762 A223252 Adjacent sequences:  A343504 A343505 A343506 * A343508 A343509 A343510 KEYWORD nonn,hard,more AUTHOR Daniel Mizrahi, Apr 17 2021 STATUS approved

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Last modified August 4 14:26 EDT 2021. Contains 346447 sequences. (Running on oeis4.)