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A188775 Numbers k such that Sum_{j=1..k} j^j == -1 (mod k).

%I A188775 #45 May 20 2021 18:29:47
%S A188775 1,2,3,6,14,42,46,1806,2185,4758,5266,10895,24342,26495,44063,52793,
%T A188775 381826,543026,547311,805002
%N A188775 Numbers k such that Sum_{j=1..k} j^j == -1 (mod k).
%C A188775 Numbers k such that A001923(k) == -1 (mod k).
%C A188775 a(21) > 10^7. - _Hiroaki Yamanouchi_, Aug 25 2015
%C A188775 Numbers k such that k divides A062970(k). - _Jianing Song_, Feb 03 2019
%e A188775 6 is a term because 1^1 + 2^2 + 3^3 + 4^4 + 5^5 + 6^6 = 50069 and 50069 + 1 = 6 * 8345. - _Bernard Schott_, Feb 03 2019
%p A188775 isA188775 := proc(n) add( modp(k &^ k,n),k=1..n) ; if modp(%,n) = n-1 then true; else false; end if; end proc:
%p A188775 for n from 1 do if isA188775(n) then printf("%d\n",n) ; end if; end do: # _R. J. Mathar_, Apr 10 2011
%t A188775 Union@Table[If[Mod[Sum[PowerMod[i,i,n],{i,1,n}],n]==n-1,Print[n];n],{n,1,10000}]
%o A188775 (PARI)
%o A188775 f(n)=lift(sum(k=1,n,Mod(k,n)^k));
%o A188775 for(n=1,10^6,if(f(n)==n-1,print1(n,", "))) \\ _Joerg Arndt_, Apr 10 2011
%o A188775 (PARI) m=0;for(n=1,1000,m=m+n^n;if((m+1)%n==0,print1(n,", "))) \\ _Jinyuan Wang_, Feb 04 2019
%o A188775 (Python)
%o A188775 sum = 0
%o A188775 for n in range(10000):
%o A188775     sum += n**n
%o A188775     if sum % (n+1) == 0:
%o A188775         print(n+1, end=',')
%o A188775 # _Alex Ratushnyak_, May 13 2013
%Y A188775 Cf. A128981 (sum == 0 (mod n)), A188776 (sum == 1 (mod n)).
%Y A188775 Cf. A057245.
%Y A188775 Cf. A001923, A062970.
%K A188775 nonn,hard,more
%O A188775 1,2
%A A188775 _José María Grau Ribas_, Apr 10 2011
%E A188775 a(12)-a(16) from _Joerg Arndt_, Apr 10 2011
%E A188775 a(17)-a(20) from _Lars Blomberg_, May 10 2011