A066230 f-perfect numbers, where f(m) = m - 1.

1, 12, 196, 368, 1696, 30848, 437745, 2075648, 8341504, 33452032, 34355150848, 562949131337728, 2305842943715442688, 590295809173294678016

Offset: 1

Author

Joseph L. Pe, Dec 18 2001

Keywords

• nonn
• more

Comments

f-perfect numbers are defined in A066218.

Equivalently, let g(n) = sigma(n)-n-d(n)+2, where d(n) is the number of divisors of n and sigma(n) is their sum. Then n is in the sequence if g(n)=n.

If 2^k - 2*k + 1 is prime (i.e. k in A301744), then 2^(k-1)*(2^k - 2*k + 1) is a term. The only known terms not of this form are 1, 196, and 437745. - Lambert Klasen (lambert.klasen(AT)gmx.net), Jul 31 2005; updated by Max Alekseyev, Jul 30 2025

If 2^(i + 1)-(2i + 1) is prime then n = 2^i*(2^(i + 1)-(2i + 1)) is in the sequence because sigma(n)-d(n) + 2 = (2^(i + 1)-1)*(2^(i + 1)-2i)-2(i + 1) + 2 = 2^(i + 1)*(2^(i + 1)-(2i + 1)) = 2n, so sigma(n)-n-d(n) + 2 = n. - Farideh Firoozbakht, Sep 18 2006

Examples

			f(12) = 11 = 0 + 1 + 2 + 3 + 5 = f(1) + f(2) + f(3) + f(4) + f(6), hence 12 is a term of the sequence.
		

Links

• J. Pe, On a Generalization of Perfect Numbers, J. Rec. Math., 31(3) (2002-2003), 168-172.

Crossrefs

Cf. A066218, A066511, A066229, A301744.

Programs

Mathematica

g[ n_ ] := DivisorSigma[ 1, n ]-n-DivisorSigma[ 0, n ]+2; For[ n=1, True, n++, If[ g[ n ]==n, Print[ n ] ] ]

Extensions

Edited by Dean Hickerson, Jan 10 2002.

More terms from Jason Earls, May 14 2002

2 more terms from Farideh Firoozbakht, Sep 18 2006

a(11) from Donovan Johnson, Jun 25 2012

a(12)-a(14) from Max Alekseyev, Jul 11 2025