A165562 Numbers n for which n+n' is prime, n' being the arithmetic derivative of n.

2, 6, 10, 14, 15, 21, 26, 30, 33, 34, 35, 38, 42, 46, 51, 55, 57, 58, 65, 66, 74, 78, 85, 86, 93, 102, 110, 111, 118, 123, 141, 143, 145, 155, 158, 161, 166, 177, 178, 182, 185, 186, 194, 201, 203, 205, 206, 209, 210, 215, 221, 230, 246, 254, 258, 267, 278, 282, 290

Offset: 1

Author

Paolo P. Lava and Giorgio Balzarotti, Sep 25 2009

Keywords

• easy
• nonn

Comments

The only prime in this sequence is 2. Since it is the only even prime and p' = 1, it is the only prime that added to its derivative can give an odd prime (namely 3).

Examples

			46 is in the list because: n=46 -> n'=25 -> n+n'=71 that is prime.
		

Links

• T. D. Noe, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A003415, A165561.

Programs

Maple

with(numtheory);
P:= proc(n)
local a,i,p,pfs;
for i from 1 to n do
pfs:=ifactors(i)[2]; a:=i*add(op(2,p)/op(1,p),p=pfs);
if isprime(a+i) then print(i); fi;
od;
end:
P(1000);
# alternative
isA165562 := proc(n)
    isprime(A129283(n)) ;
end proc:
for n from 1 to 1000 do
    if isA165562(n) then
        printf("%d,",n) ;
    end if;
end do: # R. J. Mathar, Feb 04 2022

Mathematica

(*First run the program given in A003415*) A165562 = Select[ Range[ 1000 ], PrimeQ[ # + a[ # ] ] & ]

Python

from sympy import isprime, factorint
A165562 = [n for n in range(1,10**5) if isprime(n+sum([int(n*e/p) for p,e in factorint(n).items()]))] # Chai Wah Wu, Aug 21 2014

Extensions

Terms verified by Alonso del Arte, Oct 30 2009