A360758 Numbers k for which k' - 1 and k' + 1 are twin primes, where the prime denotes the arithmetic derivative.

4, 8, 9, 35, 36, 64, 65, 68, 77, 81, 112, 160, 161, 185, 188, 208, 209, 221, 225, 236, 335, 341, 371, 377, 428, 437, 441, 485, 515, 576, 596, 611, 671, 707, 731, 736, 756, 767, 779, 783, 792, 851, 868, 899, 917, 952, 965, 972, 1007, 1028, 1067, 1115, 1152, 1157

Offset: 1

Author

Marius A. Burtea, Mar 01 2023

Keywords

• nonn

Comments

Numbers that have an arithmetic derivative equal to the average of twin prime pairs (A014574).

If p is in A022005 then m = 5*p is a term. Indeed, p is prime and m' = (5*p)' = p + 5 and m' - 1 = p + 4 and m' + 1 = p + 6 which are twin prime numbers.

If p is in A046138 then m = 7*p is a term. Indeed, p is prime and m' = (7*p)' = p + 7 and m' - 1 = p + 6 and m' + 1 = p + 8 which are twin prime numbers.

If p is in A212492 then m = 11*p is a term. Indeed, p is prime and m' = (11*p)' = p + 11 and m' - 1 = p + 10 and m' + 1 = p + 12 which are twin prime numbers.

Examples

			4' =  4, 4' - 1 =  4 - 1 =  3, 4' + 1 =  4 + 1 =  5, so 4 is a term.
8' = 12, 8' - 1 = 12 - 1 = 11, 8' + 1 = 12 + 1 = 13, so 8 is a term.
9' =  6, 9' - 1 =  6 - 1 =  5, 9' + 1 =  6 + 1 =  7, so 9 is a term.
		

Crossrefs

Cf. A003415, A014574, A022005, A046138, A212492, A077800.

Programs

Magma

f:=func; [p:p in [2..1200]| IsPrime(Floor(f(p))-1) and IsPrime(Floor(f(p))+1)];

Mathematica

d[1] = 0; d[n_] := n * Plus @@ ((Last[#]/First[#]) & /@ FactorInteger[n]); Select[Range[1200], And @@ PrimeQ[d[#] + {-1, 1}] &] (* Amiram Eldar, Mar 01 2023 *)