A382506 a(n) is the smallest k such that sigma(n) + k is a perfect number.

5, 3, 2, 21, 0, 16, 20, 13, 15, 10, 16, 0, 14, 4, 4, 465, 10, 457, 8, 454, 464, 460, 4, 436, 465, 454, 456, 440, 466, 424, 464, 433, 448, 442, 448, 405, 458, 436, 440, 406, 454, 400, 452, 412, 418, 424, 448, 372, 439, 403, 424, 398, 442, 376, 424, 376, 416, 406, 436, 328, 434, 400

Offset: 1

Author

Leo Hennig, Mar 29 2025

Keywords

• nonn

Examples

			sigma(1) = 1, 1 + 5 = 6, k = 5.
sigma(6) = 12, 12 + 16 = 28, k = 16.
sigma(180) = 546, 546 + 7582 = 8128, k = 7582.
As sigma(3000) = 9360 and the smallest perfect number at least as large as 9360 is 2^12 * (2^13 - 1) = 33550336 we have a(3000) = 33550336 - sigma(3000) = 33540976. - _David A. Corneth_, Apr 10 2025
		

Links

• David A. Corneth, Table of n, a(n) for n = 1..10000 (first 3000 terms from Michel Marcus)
• Michael De Vlieger, Log log scatterplot of a(n), n = 1..10000.

Crossrefs

Cf. A000203, A000396, A081357, A146542, A382929.

Programs

Mathematica

Do[k=0;s=DivisorSigma[1,n];While[DivisorSigma[1,s+k]!=2*(s+k),k++];a[n]=k,{n,62}];Array[a,62] (* James C. McMahon, Apr 10 2025 *)

PARI

a(n) = my(s=sigma(n),k=0); while (sigma(s+k) != 2*(s+k), k++); k; \\ Michel Marcus, Mar 30 2025

PARI

a(n) = {my(s = sigma(n));
    forprime(p = 2, oo,
        my(c = 2^p-1);
        if(isprime(c) && binomial(c+1, 2) >= s,
           return(binomial(c+1, 2) - s)))
} \\ David A. Corneth, Apr 10 2025

Formula

a(A081357(n)) = 0 and a(A146542(n)) = 0.