A066230 -id:A066230 - cp's OEIS frontend

A066218 Numbers k such that sigma(k) = Sum_{j|k, j

198, 608, 11322, 20826, 56608, 3055150, 565344850, 579667086, 907521650, 8582999958, 13876688358, 19244570848, 195485816050, 255701999358, 1038635009650, 1410759512050, 3308222326688, 6293446033554, 12859914783762, 15343909268584, 18359652610976, 19142664182226, 41584649258178, 45090324794034, 56293124233554

Offset: 1

Joseph L. Pe, Dec 17 2001

I propose this generalization of perfect numbers: for an arithmetical function f, the "f-perfect numbers" are the n such that f(n) = sum of f(k) where k ranges over proper divisors of n. The usual perfect numbers are i-perfect numbers, where i is the identity function. This sequence lists the sigma-perfect numbers. It is not hard to see that the EulerPhi-perfect numbers are the powers of 2 and the d-perfect numbers are the squares of primes (d(n) denotes the number of divisors of n).
Problems: Find an expression generating sigma-perfect numbers. Are there infinitely many of these? Find other interesting sets generated by other f's. 3.
a(17) > 2*10^12. - Giovanni Resta, Jun 20 2013
Numbers k such that A296075(k) = 0. - Amiram Eldar, Apr 16 2024
No more terms < 10^14. - Jud McCranie, Nov 28 2024

			Proper divisors of 198 = {1, 2, 3, 6, 9, 11, 18, 22, 33, 66, 99}; sum of their sigma values = 1 + 3 + 4 + 12 + 13 + 12 + 39 + 36 + 48 + 144 + 156 = 468 = sigma(198).

Joseph L. Pe, On a Generalization of Perfect Numbers, J. Rec. Math., 31(3) (2002-2003), 168-172.
Giovanni Resta, 34 numbers > 3*10^12 which belong to the sequence.

Cf. A211779, A000203, A066229, A066230, A296075.

f[ x_ ] := DivisorSigma[ 1, x ]; Select[ Range[ 1, 10^5 ], 2 * f[ # ] == Apply[ Plus, Map[ f, Divisors[ # ] ] ] & ]

is(n)=sumdiv(n,d,sigma(d))==2*sigma(n) \\ Charles R Greathouse IV, Mar 09 2014

Integer n = p1^k1 * p2^k2 * ... * pm^km is in this sequence if and only if g(p1^k1)*g(p2^k2)*...*g(pm^km)=2, where g(p^k) = (p^(k+2)-(k+2)*p+k+1)/(p^(k+1)-1)/(p-1) for prime p and integer k>=1. - Max Alekseyev, Oct 23 2008

More terms from Naohiro Nomoto, May 07 2002
a(7)-a(8) from Farideh Firoozbakht, Sep 18 2006
a(9)-a(13) from Donovan Johnson, Jun 25 2012
a(14)-a(16) from Giovanni Resta, Jun 20 2013
a(17)-a(25) from Jud McCranie, Nov 28 2024

A066229 f-perfect numbers, where f(m) = m + 1.

Original entry on oeis.org

4, 10, 44, 2336, 8896, 34432, 449295, 549775212544, 2251801457852416, 9007202677293056, 9223372167851278336, 20055918935605248255

Offset: 1

Joseph L. Pe, Dec 18 2001

f-perfect numbers are defined in A066218.
From Farideh Firoozbakht, Sep 18 2006: (Start)
n is in the sequence iff sigma(n) = 2*n - d(n) + 2, where d(n) is number of positive divisors of n.
If 2^(i+1) + 2*i - 1 is prime then n = 2^i*(2^(i+1) + 2*i - 1) is in the sequence because sigma(n) + d(n) - 2 = (2^(i+1) - 1)*(2^(i+1) + 2*i) + 2*(i+1) - 2 = 2^(2*i+2) + 2*i*2^(i+1) - 2^(i+1) = 2^(i+1)*(2^(i+1) + 2*i - 1) = 2*n, so sigma(n) = 2*n - d(n) + 2.
Hence if i is in {1, 2, 5, 6, 7, 19, 25, 26, 31, 38, 62, 80, 97, 110, 126, 133, 137, 409, 469, 685, 758, 1004, 1025, 1385, 2077, 2646, 2969, 3438, 7806, 8683, ...} then 2^i*(2^(i+1) + 2*i - 1) is in the sequence. 10, 44, 2336, 8896, 34432, 549775212544, 2251801457852416, 9007202677293056, 9223372167851278336, 151115727472444489859072, ... are such terms. (End)
Also, numbers n such that the number of nontrivial proper subgroups of a dihedral group D_{2n} is the same as its order. - Ivan Neretin, Jun 21 2016, after Dietrich Burde, see MSE link

			f(10) = 11 = 2 + 3 + 6 = f(1) + f(2) + f(5), hence 10 is a term of the sequence.

Joseph L. Pe, On a Generalization of Perfect Numbers, J. Rec. Math., 31(3) (2002-2003), 168-172.
Math StackExchange, Discussion on math.SE.

Cf. A066218, A066230, A083874.

Select[ Range[ 500000 ], DivisorSigma[ 1, # ] == 2# - DivisorSigma[ 0, # ] + 2 & ] (* Farideh Firoozbakht, Sep 18 2006 *)
f[x_] := x + 1; Select[ Range[ 1, 10^5], 2 * f[ # ] == Apply[ Plus, Map[ f, Divisors[ # ] ] ] & ]

isok(m) = sigma(m) == 2*m-numdiv(m)+2; \\ Michel Marcus, Mar 13 2020

a(8) from Giovanni Resta, Aug 21 2013

a(9)-a(11) from Farideh Firoozbakht confirmed and a(12) added by Max Alekseyev, Jul 30 2025

A066511 f-amicable numbers where f(n) = n-1.

Original entry on oeis.org

100, 110, 1806, 1872, 2404, 3742, 12488, 14378, 25130, 26696, 53418, 57448, 61962, 64938, 67528, 67624, 172362, 187624, 195114, 208072, 591882, 643624, 790758, 938948, 1361562, 1381624, 1803776, 1877682, 1892224, 2091770, 3335288, 3559402, 6585656, 8810794

Offset: 1

Joseph L. Pe, Jan 04 2002

nonn

f-amicable pairs are defined similarly to f-perfect numbers in A066218. That is, a, b is a f-amicable pair if f(a) = D(b) and f(b) = D(a), where D(n) = sum_{k divides n, k

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(g(n))=n but g(n) != n. (Sequence A066230 contains the solutions of g(n)=n.)

			Proper divisors of 100 = {1, 2, 4, 5, 10, 20, 25, 50}. f applied to these divisors = {0, 1, 3, 4, 9, 19, 24, 49}; their sum = 109. So D(100) = f(110). proper divisors of 110 = {1, 2, 5, 10, 11, 22, 55}. f applied to these divisors = {0, 1, 4, 9, 10, 21, 54}; their sum = 99. So D(110) = f(100). Therefore 100, 110 is an f-amicable pair.

Amiram Eldar, Table of n, a(n) for n = 1..162
J. Pe, On a Generalization of Perfect Numbers, J. Rec. Math., 31(3) (2002-2003), 168-172.

Cf. A066230, A066218.

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

Edited by Dean Hickerson, Jan 10 2002.

More terms from Amiram Eldar, Oct 02 2019

A317628 Take a number z of x digits and consider any concatenation z = concat(y_1, y_2, ..., y_i) where y_1, y_2, ..., y_i have the same number of digits. Then be g(z) the product of the sums y_1 + y_2 + ... + y_i for all those concatenations. Sequence lists numbers z such that g(g(z)) = z. (See example.)

Original entry on oeis.org

1782, 109000, 208000, 307000, 406000, 505000, 604000, 703000, 802000, 901000, 18802160, 33534424, 67452850, 71272872, 2496688768, 100000000000

Offset: 1

Paolo P. Lava, Aug 02 2018

The number of concatenations is d(x)-1, where d(x) is the number of divisors of x (z = concat(z) is excluded).
So far only fixed points of the transform g(z).
For n>1, the numbers equal to 10^(A066230(n)-1) belong to the sequence. - Giovanni Resta, Aug 04 2018

			z = 1782, x = 4. The concatenations are (1,7,8,2), (17,82) and g(z) = (1+7+8+2) * (17+82) = 1782.
z = 18802160, x = 8. The concatenations are (1,8,8,0,2,1,6,0), (18,80,21,60), (1880,2160) and g(z) = (1+8+8+0+2+1+6+0)*(18+80+21+60)*(1880+2160) = 18802160.

Cf. A000005, A317593, A066230.

with(numtheory): T:=proc(w) local i,k,x,y,z; z:=1; i:=sort([op(divisors(ilog10(w)+1))]); for k from 1 to nops(i)-1 do x:=0; y:=w; while y>0 do x:=x+(y mod 10^i[k]); y:=trunc(y/10^i[k]); od; z:=z*x; od; z; end: P:=proc(q) local c,d,j,n; for j from 1 to q do if not isprime(j+1) then for n from 10^j to 10^(j+1)-1 do c:=T(n); if c>0 then d:=T(c); fi; if d=n and d<>c then print(n); fi; od; fi; od; end: P(10^9); # Paolo P. Lava, Aug 02 2018

g[n_] := Block[{d = IntegerDigits[n], nd}, nd = Length@d; Times @@ (Plus @@ FromDigits /@ Partition[d, #] & /@ Most[Divisors@ nd])]; Select[Range[2, 10^6], g[g[#]] == # &] (* Giovanni Resta, Aug 04 2018 *)

a(12)-a(16) from Giovanni Resta, Aug 02 2018

Showing 1-4 of 4 results.

cp's OEIS Frontend

A066218 Numbers k such that sigma(k) = Sum_{j|k, j

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A066229 f-perfect numbers, where f(m) = m + 1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A066511 f-amicable numbers where f(n) = n-1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Extensions

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

Mathematica

Extensions