A382483 a(n) = smallest number k such that at least one of sigma(n) - k and sigma(n) + k is a perfect number.

5, 3, 2, 1, 0, 6, 2, 9, 7, 10, 6, 0, 8, 4, 4, 3, 10, 11, 8, 14, 4, 8, 4, 32, 3, 14, 12, 28, 2, 44, 4, 35, 20, 26, 20, 63, 10, 32, 28, 62, 14, 68, 16, 56, 50, 44, 20, 96, 29, 65, 44, 70, 26, 92, 44, 92, 52, 62, 32, 140, 34, 68, 76, 99, 56, 116, 40, 98, 68, 116, 44, 167, 46, 86, 96, 112, 68, 140

Offset: 1

Author

Leo Hennig, Mar 27 2025

Keywords

• nonn
• easy

Examples

			sigma(6) = 12, the nearest perfect number is 6, thus a(6) = 12 - 6 = 6.
sigma(26) = 42, the nearest perfect number is 28, thus a(26) = 42 - 28 = 14.
		

Crossrefs

Cf. A000396, A081357, A146542, A382506.

Programs

Maple

isA000396 := proc(n::integer)
    if n < 6 then
        false ;
    elif numtheory[sigma](n) = 2*n then
        true;
    else
        false;
    end if;
end proc:
A382483 := proc(n)
    local k ;
    for k from 0 do
        if isA000396(numtheory[sigma](n)-k) or isA000396(numtheory[sigma](n)+k)  then
            return k;
        end if;
    end do:
end proc:
seq(A382483(n),n=1..50) ; # R. J. Mathar, Apr 01 2025

PARI

isp(x) = if (x>0, sigma(x) == 2*x);
a(n) = my(k=0, s=sigma(n)); while (!(isp(s-k) || isp(s+k)), k++); k; \\ Michel Marcus, Apr 01 2025

Formula

a(A081357(k)) = 0.

a(A146542(k)) = 0.

a(A000396(k)) = A000396(k).