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

1,1

Links

Paolo Xausa, Table of n, a(n) for n = 1..10000

Formula

a(A081357(k)) = 0.

a(A146542(k)) = 0.

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

Example

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.

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

Mathematica

A000396Q[k_] := A000396Q[k] = k > 0 && 2*k == DivisorSigma[1, k];
A382483[n_] := Module[{k = -1}, While[NoneTrue[# + {++k, -k}, A000396Q]] & [DivisorSigma[1, n]]; k];
Array[A382483, 100] (* Paolo Xausa, Oct 17 2025 *)

Prog

(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

Crossrefs

Cf. A000203, A000396, A081357, A146542, A382506.

Sequence in context: A350041 A317674 A201333 * A391752 A353179 A088324

Adjacent sequences: A382480 A382481 A382482 * A382484 A382485 A382486

Keyword

nonn,easy

Author

Leo Hennig, Mar 27 2025

Status

approved