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Numbers k such that Sum_{i=1..k} i' == 0 (mod k), where i' is the arithmetic derivative of i.
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%I #18 Feb 11 2021 01:26:10

%S 1,6,344,1475,3816,5463,18468,78894,515108,566932,1600370,14380856,

%T 27129564,28669993,31401775,39638108,2245196680,2878016306,5890364987,

%U 7838325300,23168759538,63226475740,121869542099

%N Numbers k such that Sum_{i=1..k} i' == 0 (mod k), where i' is the arithmetic derivative of i.

%C Next term > 10^7. - _M. F. Hasler_, Sep 25 2013

%C a(21) > 10^10. - _Donovan Johnson_, Sep 25 2013

%C a(24) > 10^12. - _Giovanni Resta_, Mar 13 2014

%F A229501 = { n | A190121(n) = 0 (mod n) }. - _M. F. Hasler_, Sep 25 2013

%e 1' + 2' + 3' + 4' + 5' + 6' = 0 + 1 + 1 + 4 + 1 + 5 = 12, and 12 mod 6 = 0.

%p with(numtheory); P:= proc(q) local a,n,p; a:=0;

%p for n from 1 to q do a:=a+n*add(op(2,p)/op(1,p),p=ifactors(n)[2]);

%p if a mod n=0 then print(n); fi; od; end: P(10^6);

%o (PARI) s=0;for(n=1,1e7,(s+=A003415(n))%n||print1(n",")) \\ - _M. F. Hasler_, Sep 25 2013

%Y Cf. A003415, A227848.

%K nonn,more

%O 1,2

%A _Paolo P. Lava_, Sep 25 2013

%E Double-checked below 10^6 and extended up to 10^7 by _M. F. Hasler_, Sep 25 2013

%E a(12)-a(20) from _Donovan Johnson_, Sep 25 2013

%E a(21)-a(23) from _Giovanni Resta_, Mar 13 2014