A133519 Smallest k such that p(n)^4 - k is prime where p(n) is the n-th prime.

3, 2, 6, 2, 2, 2, 24, 14, 18, 2, 8, 8, 2, 2, 12, 2, 2, 24, 24, 38, 2, 8, 2, 54, 12, 2, 12, 12, 44, 18, 14, 18, 12, 32, 12, 24, 38, 14, 12, 12, 54, 2, 50, 8, 32, 8, 12, 14, 24, 8, 8, 2, 2, 12, 18, 30, 50, 12, 2, 24, 12, 2, 32, 2, 84, 12, 8, 12, 8, 74, 14, 18, 2, 20, 24, 14, 2, 14, 14, 2, 18

Offset: 1

Author

Carl R. White, Sep 14 2007

Keywords

• easy
• nonn

Examples

			p(3)=5, 5^4 = 625; for odd k and n > 1, p(n)^r - k is even and thus not prime, so we only need consider even k.
for k = 2: 625 - 2 = 623, which is 7 * 89 and not prime.
for k = 4: 625 - 4 = 621, which is 3^3 * 23, also not prime.
for k = 6: 625 - 6 = 619, which is prime, so 6 is the smallest number that can be subtracted from 625 to make another prime.
Hence a(3) = 6.
		

Links

• Harvey P. Dale, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A030814, A054271, A091666, A133517, A133518, A133520, A133521, A133522, (A001223).

Programs

Mathematica

sk[p_]:=Module[{k=1,c=p^4},While[CompositeQ[c-k],k++];k]; sk/@Prime[Range[100]] (* Harvey P. Dale, Nov 19 2023 *)
Table[With[{c=p^4},c-NextPrime[c,-1]],{p,Prime[Range[100]]}] (* Harvey P. Dale, Nov 20 2023 *)