cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

A127356 a(n) is the smallest k > 0 such that k^2 + prime(n) is prime.

Original entry on oeis.org

1, 2, 6, 2, 6, 2, 6, 2, 6, 12, 4, 2, 24, 2, 6, 6, 18, 6, 2, 6, 4, 2, 12, 12, 2, 6, 2, 12, 2, 6, 2, 6, 6, 10, 12, 4, 4, 2, 12, 12, 18, 4, 6, 2, 6, 8, 4, 2, 6, 2, 6, 12, 4, 24, 6, 18, 18, 6, 2, 6, 8, 18, 2, 6, 2, 6, 4, 4, 6, 2, 6, 12, 4, 4, 2, 6, 30, 2, 24, 10
Offset: 1

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Author

J. M. Bergot, Mar 30 2007

Keywords

Comments

All terms apart from the first need to be even because all primes but the first one have the same parity. Record values 1, 2, 6, 12, 24, 30, 42, 54, 60, 66, 90, 132, 138, 210, 270, ... are set at n=1, 2, 3, 10, 13, 77, 92, 152, 294, 484, 517, 964, 1203, 2876, 14118, ... - R. J. Mathar, Apr 02 2007
a(n) exists for all n on the Hardy-Littlewood conjecture F. - Charles R Greathouse IV, Jul 26 2012

Examples

			17 = prime(7); 17 + 1^2 = 18, 17 + 2^2 = 21, 17 + 3^2 = 26, 17 + 4^2 = 33, 17 + 5^2 = 42 are all composite, but 17 + 6^2 = 53 is prime. Hence a(7) = 6.
		

Crossrefs

Cf. A000040 (the primes), A000290 (the squares).

Programs

  • Maple
    a:=proc(n) local A,j: A:={}: for j from 1 to 50 do if isprime(ithprime(n)+j^2)=true then A:=A union {j} else A:=A fi od: A[1]: end: seq(a(n),n=1..120); # Emeric Deutsch, Apr 01 2007
    A127356 := proc(n) local p,a; p := ithprime(n) ; a := 1 ; while not isprime(p+a^2) do a := a+1 ; od ; RETURN(a) ; end: for n from 1 to 120 do printf("%d,",A127356(n)) ; od ; # R. J. Mathar, Apr 02 2007
  • Mathematica
    Join[{1},Table[p=Prime[n];x=2;While[!PrimeQ[a=p+x^2],x=x+2]; x,{n,2,100}]] (* Zak Seidov, Oct 12 2012 *)
    sk[n_]:=Module[{k=2},While[!PrimeQ[n+k^2],k=k+2];k]; Join[{1},Table[sk[n],{n,Prime[Range[2,80]]}]] (* Harvey P. Dale, Jul 26 2017 *)
  • PARI
    {for(n=1, 93, p=prime(n); k=1; while(!isprime(p+k^2), k++); print1(k, ","))} /* Klaus Brockhaus, Apr 05 2007 */
    
  • Python
    from sympy import isprime, nextprime, prime
    def a(n):
        if n == 1: return 1
        k, pn = 2, prime(n)
        while not isprime(pn + k*k): k += 2
        return k
    print([a(n) for n in range(1, 81)]) # Michael S. Branicky, Nov 11 2022

Extensions

Edited, corrected and extended by Emeric Deutsch, R. J. Mathar and Klaus Brockhaus, Apr 01 2007