cp's OEIS Frontend

A255354 a(n) = smallest number k such that (k + n)' = k', or -1 if no such number exists, where k' is the arithmetic derivative of k.

%I A255354 #34 Nov 13 2024 08:28:15
%S A255354 2,3,2,3,2,5,110,3,2,3,2,5,50145,3,2,3,2,5,53115,3,2,7,189,5,273,3,2,
%T A255354 3,2,7,75,5,930642191642,3,2,5,165,3,2,3,2,5,12,3,2,7,99,5,182,3,2,7,
%U A255354 706,5,1523965807,3,2,3,2,7,494,5
%N A255354 a(n) = smallest number k such that (k + n)' = k', or -1 if no such number exists, where k' is the arithmetic derivative of k.
%C A255354 The sequence begins (first 100 terms):
%C A255354 2, 3, 2, 3, 2, 5, 110, 3, 2, 3, 2, 5, 50145, 3, 2, 3, 2, 5, 53115, 3, 2, 7, 189, 5, 273, 3, 2, 3, 2, 7, 75, 5, 930642191642, 3, 2, 5, 165, 3, 2, 3, 2, 5, 12, 3, 2, 7, 99, 5, 182, 3, 2, 7, 706, 5, 1523965807, 3, 2, 3, 2, 7, 494, 5, -1, 3, 2, 5, 1151559, 3, 2, 3, 2, 7, 705, 5, 20, 3, 2, 5, 4526, 3, 2, 7, 1102, 5, 1509626, 3, 2, 13, 778, 7, 226429394, 5, -1, 3, 2, 5, 1910, 3, 2, 3 where the other missing terms (designated by -1: a(63), a(93)) are > 10^12, if they exist.
%C A255354 a(91) = 226429394. - _Michel Marcus_, Feb 28 2015
%C A255354 a(63), a(93) > 10^12. - _Giovanni Resta_, Jun 22 2018
%e A255354 a(1) = 2 because (2 + 1)' = 2' = 1.
%e A255354 a(2) = 3 because (3 + 2)' = 3' = 1.
%e A255354 a(3) = 2 because (2 + 3)' = 2' = 1.
%e A255354 ...
%e A255354 a(7) = 110 because (110 + 7)' = 110' = . Etc.
%p A255354 with(numtheory): P:=proc(q) local a,b,k,n,p;
%p A255354 for n from 1 to q do for k from 1 to q do
%p A255354 a:=k*add(op(2,p)/op(1,p),p=ifactors(k)[2]); b:=(k+n)*add(op(2,p)/op(1,p),p=ifactors(k+n)[2]);
%p A255354 if a=b then print(k); break; fi; od;
%p A255354 od; end: P(10^20);
%Y A255354 Cf. A003415, A007015, A007365, A015886, A065559.
%K A255354 nonn,more
%O A255354 1,1
%A A255354 _Paolo P. Lava_, Feb 24 2015
%E A255354 a(33)-a(62) from _Giovanni Resta_, Jun 22 2018