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A165561
Primes that are the sum of an integer n and its arithmetic derivative
7
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
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
KEYWORD
easy,nonn
AUTHOR
EXTENSIONS
Terms verified by Alonso del Arte, Oct 30 2009
STATUS
approved