A066218 -id:A066218 - cp's OEIS frontend

A066367 The floor(prime(n)/n)-perfect numbers, where prime(n) denotes the n-th prime and f-perfect numbers for an arithmetical function f are defined in A066218.

5, 7, 11, 49, 169

Offset: 1

Joseph L. Pe, Dec 21 2001

There do not seem to be any more terms. There are no terms between 170 and 10^5.

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

f[x_] := Floor[Prime[x] / x]; Select[ Range[2, 10^3], 2 * f[ # ] == Apply[ Plus, Map[ f, Divisors[ # ] ] ] & ]

A066731 f-triperfect numbers, where f(n) = sigma(n) and f-triperfect numbers are defined similarly to f-perfect numbers in A066218.

Original entry on oeis.org

4680, 60192, 179424, 1737000, 2578968, 3441888, 5604192, 6008184, 6331104, 302459850, 320457888, 477229032, 589459104, 731925000, 766073448, 928765600

Offset: 1

Joseph L. Pe, Jan 15 2002

nonn

n is f-triperfect iff sum_{k divides n} f(k) = 3*f(n). f-triperfect numbers appear to be rare for many f. For the usual f(n) = n, there are at least 6 f-triperfect numbers known < 3.2 x 10^10, the first being 120. However, there are no f-triperfect numbers < 10^6 for f(n) = n+1 and f(n) = n-1 (if one ignores the trivial n = 1).

			Divisors of 4680 = {1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 13, 15, 18, 20, 24, 26, 30, 36, 39, 40, 45, 52, 60, 65, 72, 78, 90, 104, 117, 120, 130, 156, 180, 195, 234, 260, 312, 360, 390, 468, 520, 585, 780, 936, 1170, 1560, 2340, 4680}; f applied to these yield {1, 3, 4, 7, 6, 12, 15, 13, 18, 28, 14, 24, 39, 42, 60, 42, 72, 91, 56, 90, 78, 98, 168, 84, 195, 168, 234, 210, 182, 360, 252, 392, 546, 336, 546, 588, 840, 1170, 1008, 1274, 1260, 1092, 2352, 2730, 3276, 5040, 7644, 16380}, which sum to 49140 = 3 * 16380 = 3 * f(4680). Hence 4680 is a term of the sequence.

Wells, D. Curious and Interesting Numbers, Revised Edition. Penguin Books, 1997. (See the entry on "120".)

Cf. A066218.

f[x_] := DivisorSigma[1, x]; Do[If[ Apply[ Plus, Map[ f, Divisors[ n ] ] ] == 3*f[n], Print[n]], {n, 1, 10^7}]

a(5)-a(16) from Giovanni Resta, Jul 13 2016

A296075 Sum of deficiencies of divisors of n.

Original entry on oeis.org

1, 2, 3, 3, 5, 4, 7, 4, 8, 8, 11, 1, 13, 12, 13, 5, 17, 6, 19, 7, 19, 20, 23, -10, 24, 24, 22, 13, 29, 4, 31, 6, 31, 32, 33, -16, 37, 36, 37, -2, 41, 12, 43, 25, 30, 44, 47, -37, 48, 34, 49, 31, 53, 8, 53, 6, 55, 56, 59, -49, 61, 60, 46, 7, 63, 28, 67, 43, 67, 36, 71, -78, 73, 72, 58, 49, 75, 36, 79, -27, 63, 80, 83, -47, 83

Offset: 1

Antti Karttunen, Dec 04 2017

a(n)=0 for n in A066218. Are 1 and 12 the only solutions to a(n)=1? - Robert Israel, Dec 04 2017

			For n = 6, whose divisors are 1, 2, 3, 6, their deficiencies are 1, 1, 2, 0, thus a(6) = 1 + 1 + 2 + 0 = 4.
For n = 24, whose divisors are 1, 2, 3, 4, 6, 8, 12, 24, their deficiencies are 1, 1, 2, 1, 0, 1, -4, -12, thus a(24) = 1 + 1 + 2 + 1 + 0 + 1 + -4 + -12 = -10.

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A000203, A033879, A066218, A296074.

Cf. also A007429, A187793, A187794, A187795.

f:= n -> add(2*t-numtheory:-sigma(t), t=numtheory:-divisors(n)):
map(f, [$1..100]); # Robert Israel, Dec 04 2017

f1[p_, e_] := (p^(e+1)-1)/(p-1); f2[p_, e_] := (p*(p^(e+1)-1) - (p-1)*(e+1))/(p-1)^2; a[1] = 1; a[n_] := Module[{f = FactorInteger[n]}, 2 * Times @@ f1 @@@ f - Times @@ f2 @@@ f]; Array[a, 100] (* Amiram Eldar, Dec 04 2023 *)

A033879(n) = ((2*n)-sigma(n));
A296075(n) = sumdiv(n,d,A033879(d));

a(n) = Sum_{d|n} A033879(d).
a(n) = A296074(n) + A033879(n).
If m and n are coprime, a(m*n) = 2*a(m)*A000203(n)+2*a(n)*A000203(m)-a(m)*a(n)-2*A000203(m)*A000203(n). - Robert Israel, Dec 04 2017
a(n) = 2*A000203(n) - A007429(n). - Ridouane Oudra, Jul 29 2019
Sum_{k=1..n} a(k) ~ (Pi^2/6 - Pi^4/72) * n^2. - Amiram Eldar, Dec 04 2023

A221219 Numbers k such that sigma(k) divides Sum_{d|k} sigma(d).

Original entry on oeis.org

1, 198, 608, 4680, 11322, 20826, 56608, 60192, 179424, 1737000, 2578968, 3055150, 3441888, 5604192, 6008184, 6331104, 302459850, 320457888, 477229032, 565344850, 579667086, 589459104, 731925000, 766073448, 907521650, 928765600, 3586977576, 3732082848, 6487717600

Offset: 1

Paolo P. Lava, Feb 22 2013

nonn

A066218 is a subsequence of this sequence.
Numbers k such that A000203(k) divides A007429(k). - Jaroslav Krizek, Dec 22 2018
Corresponding values of (Sum_{d|k} sigma(d)) / sigma(k)  for numbers k from this sequence: 1, 2, 2, 3, 2, 2, 2, 3, 3, 3, 3, 2, 3, 3, 3, 3, 3, 3, 3, 2, 2, 3, 3, 3, 2, 3, 3, 3, ... - Jaroslav Krizek, Dec 22 2018

			4680 is in the sequence because sigma(4680)=16380, its proper divisors are 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 13, 15, 18, 20, 24, 26, 30, 36, 39, 40, 45, 52, 60, 65, 72, 78, 90, 104, 117, 120, 130, 156, 180, 195, 234, 260, 312, 360, 390, 468, 520, 585, 780, 936, 1170, 1560, 2340 and the sum of their sigma values is 32760. Finally 32760/16380=2.

Cf. A000203, A066218, A224488, A322655, A322656, A319296.

[k: k in [1..1000000] | &+[SumOfDivisors(d): d in Divisors(k)] mod  SumOfDivisors(k) eq 0] // Jaroslav Krizek, Dec 22 2018

with(numtheory);
A221219:=proc(q) local a,b,j,n;
for n from 1 to q do a:=divisors(n); b:=add(sigma(a[j]),j=1..nops(a));
  if type(b/sigma(n),integer) then print(n); fi; od; end:
A221219(10^10);

f1[p_, e_] := (p*(p^(e + 1) - 1) - (p - 1)*(e + 1))/(p - 1)^2; f2[p_, e_] := (p^(e+1) - 1)/(p - 1); aQ[1] = True; aQ[n_] := Module[{f = FactorInteger[n]}, Divisible[Times @@ f1 @@@ f, Times @@ f2 @@@ f]]; Select[Range[10^5], aQ] (* Amiram Eldar, Dec 23 2018 *)

isok(n) = (sumdiv(n, d, sigma(d)) % sigma(n) == 0); \\ Michel Marcus, Dec 22 2018

a(10)-a(28) from Donovan Johnson, Apr 05 2013

1 prepended by Jaroslav Krizek, Dec 22 2018

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

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

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.

Original entry on oeis.org

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

A291882 a(n) is the least number k such that sigma(k+n) = Sum_{j=1..i} sigma(d_j), where d_j are the divisors of k.

Original entry on oeis.org

1, 2, 10, 1183, 4514, 1179, 38, 3325, 9, 126855, 290, 261, 18, 6, 1930, 104771947, 344, 58, 326, 117, 270754, 13875, 32, 45, 32, 74, 70, 38, 18, 21200761175, 206, 1179, 86, 16, 56, 357, 85, 18, 124, 39948225, 361, 171, 1118, 63, 122, 38, 30, 239267, 482, 1367247

Offset: 0

Paolo P. Lava, Sep 05 2017

nonn

a(123) > 10^11. - Giovanni Resta, Sep 15 2017

			Divisors of 1183 are 1, 7, 13, 91, 169 and 1183: sigma(1) + sigma(7) + sigma(13) + sigma(91) + sigma(169) + sigma(1183)= 1 + 8 + 14 + 112 + 183 + 1464 = 1782 = sigma(1183+3) and 1183 is the least number to have this property.

Paolo P. Lava, Terms from a(1) to a(300) ('?' for unknown values)

Cf. A000203, A066218.

with(numtheory): P:=proc(q) local a,b,j,k,n; for n from 0 to q do for k from 1 to q do
a:=divisors(k); b:=add(sigma(a[j]),j=1..nops(a));
if sigma(k+n)=b then print(k); break; fi; od; od; end: P(10^6);

a(n) = my(k = 1); while(sigma(k+n) != sumdiv(k, d, sigma(d)), k++); k; \\ Michel Marcus, Sep 19 2017

a(15), a(29), a(39), a(75), a(84), a(89), a(111) from Giovanni Resta, Sep 15 2017

A066365 f-perfect numbers, where f(m) = sigma(m)-m.

Original entry on oeis.org

1, 1134, 1476, 1530, 16600, 282555

Offset: 1

Joseph L. Pe, Dec 21 2001

f-perfect numbers for an arithmetical function f is defined in A066218.

Also, numbers m such that 3*sigma(m)-2*m = A007429(m). - Max Alekseyev, Jul 30 2025

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

Cf. A007429, A066218.

f[x_] := Abs[DivisorSigma[1, x] - x]; Select[ Range[2, 10^6], 2 * f[ # ] == Apply[ Plus, Map[ f, Divisors[ # ] ] ] & ]

Edited and a(1)=1 inserted by Max Alekseyev, Jul 30 2025

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A066367 The floor(prime(n)/n)-perfect numbers, where prime(n) denotes the n-th prime and f-perfect numbers for an arithmetical function f are defined in A066218.

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A066731 f-triperfect numbers, where f(n) = sigma(n) and f-triperfect numbers are defined similarly to f-perfect numbers in A066218.

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A296075 Sum of deficiencies of divisors of n.

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A221219 Numbers k such that sigma(k) divides Sum_{d|k} sigma(d).

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

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

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

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