A180306 a(n) is the largest integer k for which the Frobenius equation a_1*x_1 + a_2*x_2 + ... + a_n*x_n == k has no nonnegative integer solutions, where the a_i are consecutive primes beginning with the n-th prime.

1, 4, 9, 16, 27, 35, 49, 63, 65, 85, 95, 105, 121, 135, 145, 169, 175, 187, 203, 209, 221, 253, 265, 273, 289, 301, 305, 319, 351, 369, 387, 403, 407, 425, 445, 473, 485, 495, 517, 529, 545, 551, 567, 611, 615, 635, 639, 671, 679, 693, 703, 725, 747, 781, 793

Offset

1,2

Comments

Many terms are squares, their square roots being 1, 2, 3, 4, 7, 11, 13, 17, 23, 35, 37, 59, 69, 79, 89, 101, 103, ..., .

Links

Charles R Greathouse IV, Table of n, a(n) for n = 1..2000

Michael Hellus, Anton Rechenauer, Rolf Waldi, Numerical Semigroups generated by Primes, arXiv:1908.09483 [math.NT], 2019.

Formula

2p-2 <= a(n) << p^2, where p is the n-th prime, for n > 1. - Charles R Greathouse IV, Apr 03 2012

a(n) <= A007414(n), so conjecturally a(n) ~ 3*prime(n). - Charles R Greathouse IV, Apr 03 2012

Mathematica

f[n_] := FrobeniusNumber[ Prime@ Range[n, n + 100]]; Array[f, 55]
FrobeniusNumber/@Partition[Prime[Range[300]], 100, 1] (* Harvey P. Dale, Jun 01 2017 *)

Prog

(PARI) issum(n, x)=if(isprime(n), return(n>=x)); if(if(n%2, n<3*x, n<2*x), return(!n)); forprime(p=x, n-if(n%2, 2*x, x), if(issum(n-p, p), return(1))); 0
a(n)=if(n<2, return(1)); my(p=prime(n), k=2*p-2, lower=k, upper=2*k+2); while(upper>lower, if(issum(upper, p), upper--, lower=2*k+2; k=upper; upper=2*k+2)); k \\ Charles R Greathouse IV, Apr 03 2012

Crossrefs

Cf. A007414.

Sequence in context: A027365 A100216 A333417 * A138994 A195618 A066969

Adjacent sequences: A180303 A180304 A180305 * A180307 A180308 A180309

Keyword

nonn

Author

Robert G. Wilson v, Aug 25 2010

Extensions

Edited by N. J. A. Sloane, Aug 26 2010

Status

approved