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%I #20 Nov 11 2019 18:36:15
%S 1,4,9,16,27,35,49,63,65,85,95,105,121,135,145,169,175,187,203,209,
%T 221,253,265,273,289,301,305,319,351,369,387,403,407,425,445,473,485,
%U 495,517,529,545,551,567,611,615,635,639,671,679,693,703,725,747,781,793
%N 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.
%C 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, ..., .
%H Charles R Greathouse IV, <a href="/A180306/b180306.txt">Table of n, a(n) for n = 1..2000</a>
%H Michael Hellus, Anton Rechenauer, Rolf Waldi, <a href="https://arxiv.org/abs/1908.09483">Numerical Semigroups generated by Primes</a>, arXiv:1908.09483 [math.NT], 2019.
%F 2p-2 <= a(n) << p^2, where p is the n-th prime, for n > 1. - _Charles R Greathouse IV_, Apr 03 2012
%F a(n) <= A007414(n), so conjecturally a(n) ~ 3*prime(n). - _Charles R Greathouse IV_, Apr 03 2012
%t f[n_] := FrobeniusNumber[ Prime@ Range[n, n + 100]]; Array[f, 55]
%t FrobeniusNumber/@Partition[Prime[Range[300]],100,1] (* _Harvey P. Dale_, Jun 01 2017 *)
%o (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
%o 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
%Y Cf. A007414.
%K nonn
%O 1,2
%A _Robert G. Wilson v_, Aug 25 2010
%E Edited by _N. J. A. Sloane_, Aug 26 2010
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