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Search: id:A037153
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| A037153 |
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a(n)=p-n!, where p is the smallest prime > n!+1. |
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11
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| 2, 3, 5, 5, 7, 7, 11, 23, 17, 11, 17, 29, 67, 19, 43, 23, 31, 37, 89, 29, 31, 31, 97, 131, 41, 59, 47, 67, 223, 107, 127, 79, 37, 97, 61, 131, 311, 43, 97, 53, 61, 97, 71, 47, 239, 101, 233, 53, 83, 61, 271, 53, 71, 223, 71, 149, 107, 283, 293, 271, 769, 131, 271, 67, 193
(list; graph; listen)
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OFFSET
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1,1
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COMMENT
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Analogous to Fortunate numbers and like them, the entries appear to be primes. In fact, the first 541 terms are primes. Are all terms prime?
a(n) is the first (smallest) m such that m > 1 and n!+ m is prime. The second such m is A087202(n). a(n) must be greater than nextprime(n)-1. - Farideh Firoozbakht (f.firoozbakht(AT)math.ui.ac.ir), Sep 01 2003
All a(n) are primes. [Proof by reductio at absurdum: if a(n) were composite, say a(n)=r*s with 1<r<=s<a(n), we had p=a(n)+n!=r*s+n!. Since n! contains r<=n as a factor, this cannot be true because p then could be factored r*(s+n!/r). This needs r<=n as a lemma which follows from Bertrand's postulate, here in the sense of p<2n!.] - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Aug 22 2007
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MATHEMATICA
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NextPrime[ n_Integer ] := (k=n+1; While[ !PrimeQ[ k ], k++ ]; Return[ k ]); f[ n_Integer ] := (p = n! + 1; q = NextPrime[ p ]; Return[ q - p + 1 ]); Table[ f[ n ], {n, 1, 75} ] (from Robert G. Wilson v)
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PROGRAM
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(Mupad) for n from 1 to 65 do f := n!:a := nextprime(f+2)-f:print(a) end_for; - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Feb 22 2007
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CROSSREFS
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Cf. A087202, A005235.
Sequence in context: A123318 A111060 A082432 this_sequence A077724 A023838 A089625
Adjacent sequences: A037150 A037151 A037152 this_sequence A037154 A037155 A037156
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KEYWORD
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nonn
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AUTHOR
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Jud McCranie (j.mccranie(AT)comcast.net)
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