cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

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

Views

Author

Labos Elemer, Apr 26 2002

Keywords

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

Examples

			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).

Links

Crossrefs

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

Programs

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

Formula

{ a(k) } = { n in N | A000010(n)/A001065(n) is an integer }.
{ a(k) } = { A000040(k) } union { A055940(k) }.

Extensions

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

Views

Author

Labos Elemer, Apr 26 2002

Keywords

Comments

Euler phi value divided by Chowla function gives integer.

Examples

			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.

Links

Crossrefs

Cf. A000010, A001065, A000203, A020492, A068418, A062972, A055940, A070159, A070037.
Union of A000040 (primes) and A070161.

Programs

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

Formula

{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

Views

Author

Labos Elemer, Apr 26 2002

Keywords

Examples

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

Links

Crossrefs

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

Programs

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

Formula

q=A000010(n)/A048050(n) and n is not in A001248.
Showing 1-3 of 3 results.

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