Search: a067317 -id:a067317
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A067316
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a(n) is the number of values of j, 0 <= j <= n, such that 1 + binomial(n,j) is prime.
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+10
2
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1, 2, 3, 2, 5, 4, 4, 2, 5, 6, 6, 6, 6, 4, 5, 2, 6, 8, 8, 6, 6, 4, 4, 2, 11, 4, 4, 8, 8, 8, 4, 2, 6, 4, 8, 14, 8, 4, 5, 6, 12, 10, 4, 6, 9, 8, 8, 4, 6, 8, 6, 10, 6, 6, 12, 6, 8, 4, 12, 2, 6, 8, 4, 2, 8, 18, 8, 2, 6, 14, 10, 16, 10, 6, 4, 10, 13, 8, 12, 4, 8, 2, 8, 14, 2, 6, 4, 10, 10, 16, 10, 10, 9
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OFFSET
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0,2
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LINKS
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EXAMPLE
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For n = 8, the primes are 2, 29, 71, 29, 2, so a(n) = 5.
a(n) = 6 for n = 9, 10, 11, 12. Also, a(n) = 10 for n = 149, ..., 154.
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MATHEMATICA
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a[n_] := Count[Table[PrimeQ[Binomial[n, w]+1], {w, 0, n}], True]
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PROG
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(PARI) a(n) = sum(j=0, n, isprime(1 + binomial(n, j))); \\ Michel Marcus, Oct 30 2018
(PARI) a(n) = 2 * sum(k=0, (n-1)\2, isprime(binomial(n, k) + 1)) + if(!(n%2), isprime(binomial(n, n/2) + 1)); \\ Amiram Eldar, Jul 18 2024
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A106790
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Smallest k>0 such that binomial(n,k) + 1 is prime.
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+10
1
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1, 1, 3, 1, 2, 1, 7, 2, 2, 1, 4, 1, 2, 5, 15, 1, 2, 1, 4, 2, 2, 1, 23, 2, 6, 4, 4, 1, 7, 1, 31, 10, 6, 10, 4, 1, 18, 15, 9, 1, 2, 1, 17, 2, 2, 1, 23, 2, 4, 20, 6, 1, 6, 8, 10, 6, 2, 1, 59, 1, 2, 25, 63, 2, 2, 1, 67, 8, 2, 1
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OFFSET
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1,3
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COMMENTS
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a(n) <= n; for primes p: a(p-1) = 1.
If a(n) > n/2 then a(n) = n. a(n) = floor(n/2) for n = 2, 5, 37, 47, 124. Are there others? - Robert Israel, Mar 09 2020
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LINKS
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MAPLE
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a:=proc(n) local k: for k while isprime(1+binomial(n, k))=false do end do: k end proc: seq(a(n), n=1..70); # Emeric Deutsch, Aug 27 2007
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MATHEMATICA
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a[n_] := For[k = 1, True, k++, If[PrimeQ[Binomial[n, k] + 1], Return[k]]];
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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