A165561 Primes that are the sum of an integer n and its arithmetic derivative

3, 11, 17, 23, 31, 41, 47, 53, 59, 61, 71, 79, 83, 89, 107, 113, 127, 131, 149, 151, 167, 179, 191, 193, 197, 227, 239, 251, 263, 269, 271, 293, 311, 313, 347, 359, 383, 401, 419, 431, 439, 443, 449, 457, 479, 491, 503, 521, 523, 587, 593, 599, 607, 617, 631

Offset

1,1

Comments

Some primes are the sum of an integer and its derivative in more than one way (e.g., 23, 71, 191 (not a complete listing within the range shown)). Just calculating this sequence from A165562 gives a list that is not sorted in ascending order and contains duplicate items. However, since in the range from 1 to 10000 only the number 1 and the primes have arithmetic derivatives that are less than their square roots, I feel confident that the list given above is not missing some term that corresponds to a large value in A165562. In other words, for a term to be missing from the list above, its corresponding value in A165562 would have to be less than 625. - Alonso del Arte, Oct 30 2009

Links

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

Formula

{p in A000040: p in A129283}. - R. J. Mathar, Feb 04 2022

Example

71 is in the list because: n=46 -> n'=25 -> n+n'=71; n=51 -> n'=20 -> n+n'=71; n=55 -> n'=16 -> n+n'=71.

Maple

P:= proc(p) local a, b, m, n, i, ok, t1, t2, t3; a:=0; for n from 0 by 1 to p do b:=1000000000039; ok:=0; if n<=1 then a:=0; ok:=1; fi; if isprime(n) then a:=1; ok:=1; fi; if ok=0 then t1:=ifactor(b*n); m:=nops(t1); t2:=0; for i from 1 to m do t3:=op(i, t1); if nops(t3)=1 then t2:=t2+1/ op(t3); else t2:=t2+op(2, t3)/op(op(1, t3)); fi; od; t2:=t2-1/b; a:=n*t2; fi; if isprime(n+a) then print(n+a); fi; od; end: P(1000);
#alternative
isA165561 := proc(n)
    if isprime(n) then
        for i from 1 to n do
            if n = A129283(i) then
                return true ;
            end if;
        end do:
        false ;
    else
        false;
    end if;
end proc:
for n from 2 to 1000 do
    if isA165561(n) then
        printf("%d, ", n) ;
    end if;
end do: # R. J. Mathar, Feb 04 2022

Mathematica

(*First run the program given in A165562*) SetAttributes[a, Listable]; A165561 = Union[A165562 + a[A165562]]

Crossrefs

Cf. A003415, A165562

Sequence in context: A062284 A141339 A069348 * A038960 A019397 A108328

Adjacent sequences: A165558 A165559 A165560 * A165562 A165563 A165564

Keyword

easy,nonn

Author

Paolo P. Lava and Giorgio Balzarotti, Sep 25 2009

Extensions

Terms verified by Alonso del Arte, Oct 30 2009

Status

approved