A066511 -id:A066511 - cp's OEIS frontend

A066573 Pairs (x, y), x < y, of f-amicable numbers where f(k) = floor(|k*sin(k)|) sorted by increasing y, then increasing x; f-amicable numbers are defined in A066511.

1, 3, 22, 223, 283, 355, 22, 421, 389, 1065, 365, 1508, 2130, 3079, 1065, 69203, 51872, 127539, 83282, 128604, 152628, 252271, 191963, 295294, 130252, 459590, 717615, 1401314, 2840, 7189717, 1258370, 10269235, 2130, 11671711, 11519862, 19177306, 17002972, 21316045

Offset: 1

Joseph L. Pe, Jan 07 2002

nonn

			Proper divisors of 22 are {1,2,11}; f applied to these = {0, 1, 10}, which sum to 11 = f(223). Proper divisors of 223 are {1}; f applied to these = {0}, which sum to 0 = f(22). Hence (22,223) is an f-amicable pair.

Sean A. Irvine, Java program (github)
J. Pe, On a Generalization of Perfect Numbers, J. Rec. Math., 31(3) (2002-2003), 168-172.

Cf. A066511, A066218.

f[x_] := Floor[Abs[x*Sin[x]]]; d[x_] := Apply[ Plus, Map[ f, Divisors[ x] ] ] - f[ x]; m = Table[{x, y}, {x, 1, 1000}, {y, 1, 1000}]; Do[a = m[[i, j]]; If[ (a[[1]] < a[[2]]) && (f[a[[1]]] == d[a[[2]]]) && (f[a[[2]]] == d[a[[1]]]), Print[{i, j}]], {j, 1, 1000}, {i, 1, 1000}]

More terms and entry revised by Sean A. Irvine, Oct 29 2023

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

Original entry on oeis.org

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

Offset: 1

Joseph L. Pe, Dec 18 2001

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

			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.

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

Cf. A066218, A066511, A066229, A301744.

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

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

cp's OEIS Frontend

A066573 Pairs (x, y), x < y, of f-amicable numbers where f(k) = floor(|k*sin(k)|) sorted by increasing y, then increasing x; f-amicable numbers are defined in A066511.

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