A068418 -id:A068418 - cp's OEIS frontend

A069737 Integer quotient defining A068418 is 3.

1320, 3696, 5520, 13800, 19872, 22560, 23688, 87000, 89964, 366720, 785808, 1235052, 1470720, 8690112, 45986616, 46862592, 78584640, 142588224, 231718968, 235668528, 377425920, 1598249952, 9518204448, 12026499072, 40964894160, 65368504980

Offset: 1

Labos Elemer, Apr 05 2002

nonn

			40th term in A068418 is m=142588224, sum-of-proper-divisors[m]=294860736, cototient[m]=98286912, quotient=3=294860736/98286912,

Cf. A068418, A068414, A069714.

(Sum of proper divisors[m])/cototient[m]=3, where m is a term of A068418.

More terms from Labos Elemer, Apr 22 2002

a(22)-a(26) from Donovan Johnson, May 27 2011

A069719 Integer quotients arising in A068418: sum of proper divisors is divided by cototient of terms of A068418.

Original entry on oeis.org

2, 2, 2, 2, 3, 2, 2, 2, 3, 3, 3, 2, 3, 3, 3, 2, 2, 3, 3, 2, 2, 3, 3, 4, 2, 3, 2, 2, 3, 4, 2, 2, 3, 3, 3, 2, 2, 3, 4, 3, 2, 3, 3, 2, 3, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 4, 2, 3, 3, 2, 2, 2, 4, 2, 2, 2, 3, 3

Offset: 1

Labos Elemer, Apr 05 2002

			n=39: m=A068418(39)=130141440, sigma[m]-m=405550080, m-Phi[m]=101387520, quotient=4=a(39).

Cf. A068418, A001065, A000203, A051953, A068414, A069714, A069737.

a(n) = A001065(A068418(n)) - A051953(A068418(n)).

a(47)-a(68) from Donovan Johnson, May 27 2011

A055940 Counterbalanced numbers: Composite numbers k such that phi(k)/(sigma(k)-k) is an integer.

Original entry on oeis.org

133, 403, 583, 713, 817, 2077, 2623, 2923, 4453, 4717, 5311, 5773, 7093, 7747, 9313, 11023, 11581, 11653, 12877, 14353, 15553, 19303, 20803, 21409, 21733, 21971, 24307, 31169, 35033, 39283, 39337, 43873, 46297, 46357, 50573, 50879, 53863

Offset: 1

Robert G. Wilson v, Jul 22 2000

nonn

Banks and Luca (2007) showed that the number of terms <= x, N(x) <= x * exp(-((1/3)*(log(8))^(1/3) + o(1))*(log(x))^(1/3)*(log(log(x)))^(1/3)) as x -> infinity, and that under Dickson's conjecture this sequence is infinite, since for each positive integer m, if p = 5m + 1 and q = 20m + 13 are primes, then p*q is a term. - Amiram Eldar, Apr 13 2020

			k = 133 = 7*19: phi(133)=108, sigma(133)-133 = 1+7+19 = 27, q = 4.

Donovan Johnson, Table of n, a(n) for n = 1..10000
William D. Banks and Florian Luca, When the sum of aliquots divides the totient, Proceedings of the Edinburgh Mathematical Society, Vol. 50, No. 3 (2007), pp. 563-569.
Douglas E. Iannucci, On the Equation sigma(n) = n + phi(n), Journal of Integer Sequences, Vol. 20 (2017), Article 17.6.2.

Cf. A000010, A001065, A068418, A020492, A070037.

Do[s=EulerPhi[n]/(DivisorSigma[1, n]-n); If[ !PrimeQ[n]&&IntegerQ[s], Print[n]], {n, 2, 1000000}]
Select[Range[54000],CompositeQ[#]&&IntegerQ[EulerPhi[#]/(DivisorSigma[ 1,#]-#)]&] (* Harvey P. Dale, Nov 16 2021 *)

is(n)=!isprime(n) && n>1 && eulerphi(n)%(sigma(n)-n)==0 \\ Charles R Greathouse IV, Jan 02 2014

A068414 Numbers k such that sigma(k) = 3k - 2*phi(k).

Original entry on oeis.org

1, 12, 56, 260, 992, 1976, 2156, 2754, 16256, 25232, 41072, 133984, 145888, 1100864, 1270208, 1439552, 2237888, 4729664, 67100672, 75398912, 171627376, 277060144, 473089984, 538178048, 558585344, 629225984, 1192258048, 1863840112, 2181070592, 4534854656

Offset: 1

Benoit Cloitre, Mar 03 2002

nonn

If 2^p-1 is prime (a Mersenne prime) and n = 2^p*(2^p-1) then n is in the sequence because 3*n-2*phi(n) = 3*2^p*(2^p-1)-2^p*(2^p-2) = 2^p*(2^(p+1)-1) = sigma(2^p-1)*sigma(2^p) = sigma(2^p*(2^p-1)) = sigma(n). - Farideh Firoozbakht, Dec 31 2005

Cf. A000010, A000203, A068418, A069719, A069737.

Select[Range[10^6], DivisorSigma[1, #] == 3*# - 2*EulerPhi[#] &] (* Amiram Eldar, May 14 2022 *)

for(n=1,500000, if(sigma(n)==3*n-2*eulerphi(n),print1(n,",")))

More terms (complete up to 50000000). - Rick L. Shepherd, Mar 28 2002
More terms from Labos Elemer, Apr 03 2002
a(24)-a(30) from Donovan Johnson, Feb 08 2012

A070159 Numbers k such that phi(k)/(sigma(k)-k) is an integer.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 133, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269

Offset: 1

Labos Elemer, Apr 26 2002

nonn

This sequence consists of all primes p (for which the given ratio equals (p-1)/1, see A000040) and of composites listed in A055940 (see examples).

Up to 10^7, there is no element of this sequence having more than 2 prime factors. - M. F. Hasler, Dec 11 2007

			The prime p=47 is in this sequence since phi[p]/(sigma[p]-p) = p-1 is an integer, as is the case for any other prime.
The composite n=403=13*31 is in this sequence, since the ratio phi(n)/(sigma[n]-n) =360/(1+13+31)=8 is an integer.
The first few composites in this sequence are 133,403,583,713,... (A055940).

Amiram Eldar, Table of n, a(n) for n = 1..10000
Douglas E. Iannucci, On the Equation sigma(n) = n + phi(n), Journal of Integer Sequences, Vol. 20 (2017), Article 17.6.2.

Cf. A000010, A000040, A001065, A000203, A055940, A070037, A020492, A068418, A062972.

Do[s=EulerPhi[n]/(DivisorSigma[1, n]-n); If[IntegerQ[s], Print[n]], {n, 2, 1000}]
Select[Range[2,300],IntegerQ[EulerPhi[#]/(DivisorSigma[1,#]-#)]&] (* Harvey P. Dale, Dec 25 2019 *)

for(n=2,999,eulerphi(n)%(sigma(n)-n) || print1(n",")) \\ M. F. Hasler, Dec 11 2007

{ a(k) } = { n in N | A000010(n)/A001065(n) is an integer }.

{ a(k) } = { A000040(k) } union { A055940(k) }.

Edited by M. F. Hasler, Dec 11 2007

A070160 Nonprime numbers k such that phi(k)/(sigma(k) - k - 1) is an integer.

Original entry on oeis.org

4, 9, 15, 25, 35, 49, 95, 119, 121, 143, 169, 209, 287, 289, 319, 323, 361, 377, 527, 529, 559, 779, 841, 899, 903, 923, 961, 989, 1007, 1189, 1199, 1343, 1349, 1369, 1681, 1763, 1849, 1919, 2159, 2209, 2507, 2759, 2809, 2911, 3239, 3481, 3599, 3721

Offset: 1

Labos Elemer, Apr 26 2002

nonn

Euler phi value divided by Chowla function gives integer.

			In A062972, n=15: q = 8/8 = 1; n=101: q = 100/1 = 100. While integer quotient chowla(n)/phi(n) gives only 5 nonprime solutions below 20000000 (see A070037), here, the integer reciprocals, q = phi(n)/chowla(n) obtained with squared primes and with other composites. If n=p^2, q = p(p-1)/p = p-1. So for squared primes, the quotients give A006093.

Donovan Johnson, Table of n, a(n) for n = 1..10000

Cf. A000010, A001065, A000203, A020492, A068418, A062972, A055940, A070159, A070037.

Union of A000040 (primes) and A070161.

Do[s=EulerPhi[n]/(DivisorSigma[1, n]-n-1); If[IntegerQ[s], Print[n]], {n, 2, 100000}]

{k : A000010(k)/A048050(k) is an integer}.

A070161 Nonprime numbers n such that q=phi(n)/(sigma(n)-n-1) is an integer and n is not a prime square.

Original entry on oeis.org

15, 35, 95, 119, 143, 209, 287, 319, 323, 377, 527, 559, 779, 899, 903, 923, 989, 1007, 1189, 1199, 1343, 1349, 1763, 1919, 2159, 2507, 2759, 2911, 3239, 3599, 3827, 4031, 4607, 5183, 5207, 5249, 5459, 5543, 6439, 6887, 7067, 7279, 7739, 8159, 8639, 9179

Offset: 1

Labos Elemer, Apr 26 2002

nonn

			n=35: phi(35)=24, sigma(35)=1+5+7+35=48, chowla(35)=12, quotient=2

Donovan Johnson, Table of n, a(n) for n = 1..10000

Cf. A000010, A001065, A000203, A048050, A055940, A070159, A070160, A070037 A020492, A068418, A062972.

Do[s=EulerPhi[n]/(DivisorSigma[1, n]-n-1); If[ !PrimeQ[n]&&!PrimeQ[Sqrt[n]]&&IntegerQ[s], Print[n]], {n, 2, 100000}]

q=A000010(n)/A048050(n) and n is not in A001248.

A069714 Least integer k such that (sigma(k)-k)/(k-phi(k)) = n.

Original entry on oeis.org

2, 12, 1320, 851760

Offset: 1

Labos Elemer, Apr 02 2002

10^11 < a(5) <= 124176229632000. - Donovan Johnson, Oct 22 2011

Cf. A068414, A068418, A001065, A051953, A000010, A000203.

a(n) = Min{x: A001065(x) / A051953(x) = n}.

cp's OEIS Frontend

A069737 Integer quotient defining A068418 is 3.

Views

Author

Keywords

Examples

Crossrefs

Formula

Extensions

A069719 Integer quotients arising in A068418: sum of proper divisors is divided by cototient of terms of A068418.

Views

Author

Keywords

Examples

Crossrefs

Formula

Extensions

A055940 Counterbalanced numbers: Composite numbers k such that phi(k)/(sigma(k)-k) is an integer.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A068414 Numbers k such that sigma(k) = 3k - 2*phi(k).

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

PARI

Extensions

A070159 Numbers k such that phi(k)/(sigma(k)-k) is an integer.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A070160 Nonprime numbers k such that phi(k)/(sigma(k) - k - 1) is an integer.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A070161 Nonprime numbers n such that q=phi(n)/(sigma(n)-n-1) is an integer and n is not a prime square.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A069714 Least integer k such that (sigma(k)-k)/(k-phi(k)) = n.

Views

Author

Keywords

Comments

Crossrefs

Formula