A070037 -id:A070037 - cp's OEIS frontend

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

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

A062972 Numbers k such that the Chowla function of k is divisible by phi(k).

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 11, 13, 15, 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, 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

Jason Earls, Jul 24 2001

nonn

Chowla's function (A048050) = sum of divisors of n except 1 and n.

Sequence contains all primes; see A070037 for nonprime terms. - Charles R Greathouse IV, Apr 14 2010

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

Cf. A000010, A048050, A070037.

chowla[1] = 0; chowla[n_] := DivisorSigma[1, n] - n - 1; Select[Range[270], Divisible[chowla[#], EulerPhi[#]] &] (* Amiram Eldar, Dec 01 2019 *)

j=[]; for(n=1,600, if(Mod(sigma(n)-n-1,eulerphi(n)) == 0,j=concat(j,n))); j

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.

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

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A062972 Numbers k such that the Chowla function of k is divisible by phi(k).

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

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

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

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