A301859 Abundant numbers whose abundance is a perfect number.

48, 2002, 2632, 4540, 5170, 6952, 8925, 29056, 32445, 32980, 88330, 133042, 174856, 189472, 280228, 442365, 518368, 566752, 892552, 1266952, 2030368, 2052256, 2218450, 3959752, 4120672, 4558936, 5568448, 9071752, 15921112, 38551936, 65969536, 70114936, 88149352, 97364848

Offset: 1

Author

Waldemar Puszkarz, Mar 27 2018

Keywords

• nonn

Comments

There are 34 terms up to 10^8. The abundance of odd terms (only 3 terms) is 6 (see also A087167). The abundance of even terms is 28, 496, 8128, and 33550336 (for 97364848). There exist deficient numbers whose abundance is a perfect number in absolute terms, e.g., 7, 29, 62.

Examples

			48 is a term as it is abundant and its abundance, sigma(48)-2*48 = 28, is the second perfect number.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..68

Crossrefs

Cf. A005101 (abundant numbers), A033880 (abundance), A000396 (perfect numbers), A087167, A088834, A088012, A077374 (sequences related to the odd terms of this sequence).

Programs

Mathematica

Select[Range[10^8], PerfectNumberQ[DivisorSigma[1,# ]-2#]&]

PARI

for(n=1,10^8, a=sigma(n)-2*n; a>0&&sigma(a)==2*a&&print1(n ","))