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

1,1

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.

Links

Table of n, a(n) for n=1..54.

Example

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.

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 *)

Prog

(Magma) f:=func<n |n le 1 select 0 else n*(&+[Factorisation(n)[i][2] / Factorisation(n)[i][1]: i in  [1..#Factorisation(n)]])>; [p:p in [2..1200]| IsPrime(Floor(f(p))-1) and IsPrime(Floor(f(p))+1)];

Crossrefs

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

Sequence in context: A355063 A162752 A076966 * A237607 A035132 A077572

Adjacent sequences: A360755 A360756 A360757 * A360759 A360760 A360761

Keyword

nonn

Author

Marius A. Burtea, Mar 01 2023

Status

approved