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

8, 10, 14, 22, 26, 34, 38, 46, 58, 62, 74, 82, 86, 94, 106, 118, 122, 134, 142, 146, 158, 166, 178, 194, 202, 206, 214, 218, 226, 254, 262, 274, 278, 298, 302, 314, 326, 334, 346, 358, 362, 382, 386, 394, 398, 422, 446, 454, 458, 466, 478, 482, 502, 514

Offset: 1

Author

John W. Layman, Sep 29 2011

Keywords

• nonn

Comments

This sequence appears to be identical to A073582 with its first term omitted and to A161344 with its first two terms omitted.

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.)

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..

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

The terms in this sequence which are not even semiprimes are 8, 690, 12978, 176946, ... - R. J. Mathar, Aug 24 2023

Crossrefs

Cf. A054024, A073582, A161344.

Programs

Maple

isA196226 := proc(n)
    sigmar := modp(numtheory[sigma](n),n) ;
    if sigmar = 3+n/2 then
        true;
    else
        false;
    end if;
end proc:
A196226 := proc(n)
     option remember;
     if n =1 then
        8;
    else
        for a from procname(n-1)+1 do
            if isA196226(a) then
                return a;
            end if;
        end do:
    end if;
end proc:
seq(A196226(n),n=1..100) ; # R. J. Mathar, Aug 24 2023

PARI

lista(nn) = {for(n=1, nn, if ((sigma(n) % n) == (3 + n/2), print1(n, ", ")););} \\ Michel Marcus, Jul 12 2014