A088830 -id:A088830 - cp's OEIS frontend

A055234 Smallest x such that sigma(x) = n*phi(x), or -1 if no such x exists.

1, 3, 2, 14, 56, 6, 12, 42, 30, 168, 2580, 210, 630, 420, 840, 20790, 416640, 9240, 291060, 83160, 120120, 5165160, 1719277560, 43825320, 26860680, 277560360, 1304863560, 569729160, 587133466920, 16522145640, 33044291280, 563462139240, 1140028049160, 9015394227840, 1255683068640, 65361608151840

Offset: 1

Jud McCranie, Jun 21 2000

nonn

Conjecture: For each n, a(n) > 0. - Farideh Firoozbakht, Sep 12 2004
a(33) > 10^12. - Donovan Johnson, Mar 06 2012
a(34) <= 9015394227840, a(35) <= 1255683068640. - Giovanni Resta, May 08 2017
Terms after a(36) are > 10^14. a(37) <= 4771397395084320, a(38) <= 2418379501618080, a(39) <= 413956851628320, a(40) <= 1241870554884960, and a(42) <= 50916692750283360. - Jud McCranie, Sep 13 2017
a(38) = 299761858075680, a(39) = 413956851628320. a(37), a(40), and higher terms are > 4.2*10^14. - Jud McCranie, Nov 27 2017
a(37), a(40), and higher terms are > 6.0 x 10^14. - Jud McCranie, Dec 27 2017

			sigma(14) = 24 = 4*phi(14), so a(4) = 14.
n = 21: a(21) = 120120 = 2*2*2*3*5*7*11*13, sigma(120120) = 483840 = n*phi(120120), phi(120120) = 23040.

Cf. A000010, A000203, A020492, A023897, A088830.

a[n_]:=(For[m=1,DivisorSigma[1,m]!=n EulerPhi[m],m++ ];m);Do[Print[a[n]], {n,31}] (* Farideh Firoozbakht, Oct 31 2008 *)

a(n) = {k = 1; while(sigma(k) != n*eulerphi(k), k++); k;} \\ Michel Marcus, Sep 01 2014

from math import prod
from itertools import count
from sympy import factorint
def A055234(n):
    for m in count(1):
        f = factorint(m)
        if n*m*prod((p-1)**2 for p in f)==prod(p**(e+2)-p for p,e in f.items()):
            return m # Chai Wah Wu, Aug 12 2024

a(n) = Min{x : A000203(x)/A000010(x) = n} = Min{x : A023897(x) = n}

More terms from Farideh Firoozbakht, Sep 12 2004
a(32) from Donovan Johnson, Mar 06 2012
a(33) from Giovanni Resta, May 08 2017
a(34)-a(36) from Jud McCranie, Sep 10 2017

A088832 Numbers k whose abundance is 4: sigma(k) - 2*k = 4.

Original entry on oeis.org

12, 70, 88, 1888, 4030, 5830, 32128, 521728, 1848964, 8378368, 34359083008, 66072609790, 549753192448, 259708613909470, 2251799645913088, 9223372026117357568

Offset: 1

Labos Elemer, Oct 28 2003

If 2^m-5 is prime (A059608) then n=2^(m-1)*(2^m-5) is in the sequence. 12, 88, 1888, 32128, 521728, 8378368 & 34359083008 are such terms. See comments in A088831. - Farideh Firoozbakht, Feb 15 2008
Any term x of this sequence can be combined with any term y of A125246 to satisfy the property (sigma(x)+sigma(y))/(x+y) = 2, which is a necessary (but not sufficient) condition for two numbers to be amicable. - Timothy L. Tiffin, Sep 13 2016
Also contains 865268370658615254581248 = 2^23 * 16823249 * 6131278669. - Max Alekseyev, May 29 2025

			Abundances of terms in A045769: {-5,4,4,4,4,4,4,4,4,4} so A045769(1)=9 is not here.

Subsequence of A045769.

Cf. A033880, A045768, A088830, A059608, A125246 (deficiency 4).

Do[If[DivisorSigma[1,n]==2n+4,Print[n]],{n,650000000}] (* Farideh Firoozbakht, Feb 15 2008 *)

is(n)=sigma(n)==2*n+4 \\ Charles R Greathouse IV, Feb 21 2017

Solutions to sigma(x)-2*x=4.

One more terms from Farideh Firoozbakht, Feb 15 2008
a(11)-a(12) from Donovan Johnson, Dec 23 2008
a(13) from Donovan Johnson, Dec 08 2011
a(14)-a(15) from Hiroaki Yamanouchi, Aug 23 2018
a(16) from Max Alekseyev, May 29 2025

A088829 Even numbers with even abundance.

Original entry on oeis.org

6, 10, 12, 14, 20, 22, 24, 26, 28, 30, 34, 38, 40, 42, 44, 46, 48, 52, 54, 56, 58, 60, 62, 66, 68, 70, 74, 76, 78, 80, 82, 84, 86, 88, 90, 92, 94, 96, 102, 104, 106, 108, 110, 112, 114, 116, 118, 120, 122, 124, 126, 130, 132, 134, 136, 138, 140, 142, 146, 148, 150, 152

Offset: 1

Labos Elemer, Oct 28 2003

nonn

Odd numbers with odd abundance are in A016754. Odd numbers with even abundance are in A088828. Even numbers with odd abundance are in A088827.

			Even perfect numbers are here.
n=10: sigma[10]=18, abundance=18-20=-2<0;
n=12: sigma[12]=28, abundance=28-24=+4>0.
sigma[n] is also even number; sign of abundance:<=>0.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A016754, A088827-A088828, A088830.

Do[s=DivisorSigma[1, n]-2*n; If[ !OddQ[s]&&!OddQ[n], Print[{n, s}]], {n, 1, 1000}]

A087979 a(n) = Min {x : sigma(x) = phi(nx), x is not a prime}, least nonprime solutions to sigma(x) = phi(nx).

Original entry on oeis.org

1, 1, 15, 14, 14, 6, 6, 42, 30, 42, 168, 210, 210, 420, 840, 20790, 20790, 9240, 9240, 83160, 120120, 3984120, 5165160, 43825320, 26860680, 43825320, 1304863560, 569729160, 569729160, 16522145640, 18176198040, 563462139240, 1140028049160, 3844800479520, 1255683068640, 65361608151840, 65361608151840, 65361608151840, 413956851628320, 1241870554884960, 1241870554884960

Offset: 1

Labos Elemer, Sep 29 2003

nonn

If x is prime then by necessity we have x = 3 and n = 4. Hence, except for a(4), this sequence is the same as A256527. - Max Alekseyev, Sep 29 2023

			n=4: a(4)=14, sigma(14) = 24 = phi(4*14) = phi(56).
n=5: a(5)=14, sigma(14) = 24 = phi(5*14) = phi(70).

Cf. A074891, A000203, A000010, A088830, A256527.

ds[x_, de_] := DivisorSigma[1, x]-EulerPhi[de*x] a[n_] := Block[{m=1, s=ds[m, n]}, While[(s !=0||PrimeQ[m])&&!Greater[m, 4000000], m++ ]; m]; Table[a[n], {n, 22}]

For n >= 5, a(n) = A256527(n). - Conjectured by Manfred Scheucher, May 28 2015; proved by Max Alekseyev, Sep 29 2023

More terms from David Wasserman, Jun 20 2005
a(26)-a(31) from Donovan Johnson, Feb 06 2010
a(32)-a(34) from Donovan Johnson confirmed, a(35) added by Giovanni Resta, May 24 2016
a(36)-a(41) from Max Alekseyev, Oct 10 2024

A088847 a(n) = sigma(A087979(n)) / phi(A087979(n)).

Original entry on oeis.org

1, 1, 3, 4, 4, 6, 6, 8, 9, 8, 10, 12, 12, 14, 15, 16, 16, 18, 18, 20, 21, 20, 22, 24, 25, 24, 27, 28, 28, 30, 30, 32, 33, 32, 35, 36, 36, 36, 39, 40, 40, 42, 42, 44, 45, 46, 46, 48, 49, 50, 51, 52, 52, 54, 55, 56, 57, 56, 58, 60, 60, 62, 63, 64, 65, 66, 66, 68, 69, 70, 70, 72, 72, 74, 75, 76, 77, 78, 79, 80, 81, 82, 82, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100

Offset: 1

Labos Elemer, Nov 17 2003

nonn

Note that A087979(n) is a balanced number (A020492), hence this sequence is well-defined. - Max Alekseyev, Oct 10 2024

For all n, a(n) <= n and a(n) divides A299822(n). - Max Alekseyev, Oct 11 2024

			While A088830 includes special balanced numbers, A087979 does not include per definition. Nevertheless, it seems that A087979 consists only of balanced numbers. This is provable at least for special cases.

Cf. A000010, A000203, A087979, A020492, A088830, A299822.

a(24)-a(35) from Amiram Eldar, Dec 05 2019 (calculated from the data at A087979)

Terms a(36) onward from Max Alekseyev, Oct 10 2024

cp's OEIS Frontend

A055234 Smallest x such that sigma(x) = n*phi(x), or -1 if no such x exists.

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A088832 Numbers k whose abundance is 4: sigma(k) - 2*k = 4.

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A088829 Even numbers with even abundance.

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A087979 a(n) = Min {x : sigma(x) = phi(n*x), x is not a prime}, least nonprime solutions to sigma(x) = phi(n*x).

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A088847 a(n) = sigma(A087979(n)) / phi(A087979(n)).

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A087979 a(n) = Min {x : sigma(x) = phi(nx), x is not a prime}, least nonprime solutions to sigma(x) = phi(nx).