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A360758
Numbers k for which k' - 1 and k' + 1 are twin primes, where the prime denotes the arithmetic derivative.
0
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.
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
KEYWORD
nonn
AUTHOR
Marius A. Burtea, Mar 01 2023
STATUS
approved