A062972 -id:A062972

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A070159

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

+10
3

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

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

LINKS

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.

FORMULA

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

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

EXAMPLE

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).

MATHEMATICA

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 *)

PROG

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

CROSSREFS

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

KEYWORD

nonn

AUTHOR

Labos Elemer, Apr 26 2002

EXTENSIONS

Edited by M. F. Hasler, Dec 11 2007

STATUS

approved

A070160

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

+10
3

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

Euler phi value divided by Chowla function gives integer.

LINKS

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

FORMULA

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

EXAMPLE

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.

MATHEMATICA

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

CROSSREFS

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

Union of A000040 (primes) and A070161.

KEYWORD

nonn

AUTHOR

Labos Elemer, Apr 26 2002

STATUS

approved

A070161

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

+10
3

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

LINKS

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

FORMULA

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

EXAMPLE

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

MATHEMATICA

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

CROSSREFS

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

KEYWORD

nonn

AUTHOR

Labos Elemer, Apr 26 2002

STATUS

approved

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