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A196226 m such that A054024(m) (sum of divisors of m reduced modulo m) is 3 + m/2.

%I A196226 #21 Aug 24 2023 07:07:39
%S A196226 8,10,14,22,26,34,38,46,58,62,74,82,86,94,106,118,122,134,142,146,158,
%T A196226 166,178,194,202,206,214,218,226,254,262,274,278,298,302,314,326,334,
%U A196226 346,358,362,382,386,394,398,422,446,454,458,466,478,482,502,514
%N A196226 m such that A054024(m) (sum of divisors of m reduced modulo m) is 3 + m/2.
%C A196226 This sequence appears to be identical to A073582 with its first term omitted and to A161344 with its first two terms omitted.
%C A196226 Conjectures.  (1) If m>=14 is a term of this sequence, then sigma(2,m) is congruent to 5 + m/2 modulo m;  (2) If m>=22 is a term of this sequence, then sigma(3,m) is congruent to 9 + m/2 modulo m;  If m>=38 is a term of this sequence, then sigma(4,m) is congruent to 17 + m/2 modulo m. (sigma(k,m) denotes the sum of the k-th powers of the divisors of m.)
%C A196226 Similar conjectures can be made about sigma(k,m) congruent to 2^k+1 + m/2 modulo m, for m a sufficiently large term of this sequence..
%C A196226 The even semiprimes (A100484) m= 2*p with p>3, with sigma(2*p)= 3+p (mod 2p), are a subsequence. - _R. J. Mathar_, Oct 02 2011
%C A196226 The terms in this sequence which are not even semiprimes are 8, 690, 12978, 176946, ... - _R. J. Mathar_, Aug 24 2023
%p A196226 isA196226 := proc(n)
%p A196226     sigmar := modp(numtheory[sigma](n),n) ;
%p A196226     if sigmar = 3+n/2 then
%p A196226         true;
%p A196226     else
%p A196226         false;
%p A196226     end if;
%p A196226 end proc:
%p A196226 A196226 := proc(n)
%p A196226      option remember;
%p A196226      if n =1 then
%p A196226         8;
%p A196226     else
%p A196226         for a from procname(n-1)+1 do
%p A196226             if isA196226(a) then
%p A196226                 return a;
%p A196226             end if;
%p A196226         end do:
%p A196226     end if;
%p A196226 end proc:
%p A196226 seq(A196226(n),n=1..100) ; # _R. J. Mathar_, Aug 24 2023
%o A196226 (PARI) lista(nn) = {for(n=1, nn, if ((sigma(n) % n) == (3 + n/2), print1(n, ", ")););} \\ _Michel Marcus_, Jul 12 2014
%Y A196226 Cf. A054024, A073582, A161344.
%K A196226 nonn
%O A196226 1,1
%A A196226 _John W. Layman_, Sep 29 2011