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 1-2 of 2 results.

A079537 a(n) = phi(2*n+1)*d(2*n+1) - sigma(2*n+1).

Original entry on oeis.org

0, 0, 2, 4, 5, 8, 10, 8, 14, 16, 16, 20, 29, 32, 26, 28, 32, 48, 34, 40, 38, 40, 66, 44, 69, 56, 50, 88, 64, 56, 58, 112, 108, 64, 80, 68, 70, 116, 144, 76, 149, 80, 148, 104, 86, 176, 112, 168, 94, 204, 98, 100, 192, 104, 106, 136, 110, 208, 250, 240, 197, 152, 244, 124, 160
Offset: 0

Views

Author

N. J. A. Sloane, Jan 23 2003

Keywords

Comments

It is known that a(n) >= 0.

References

  • D. S. Mitrinovic et al., Handbook of Number Theory, Kluwer, p. 10.

Links

Crossrefs

Cf. A000010, A000005, A000203, A079536.

Programs

  • Magma
    [EulerPhi(2*n+1)*DivisorSigma(0,2*n+1) - DivisorSigma(1,2*n+1): n in [0..80]]; // G. C. Greubel, Jan 15 2019
  • Mathematica
    Table[EulerPhi[2*n+1]*DivisorSigma[0, 2*n+1] - DivisorSigma[1, 2*n+1], {n, 0, 80}] (* G. C. Greubel, Jan 15 2019 *)
  • PARI
    vector(80, n,  n--; eulerphi(2*n+1)*sigma(2*n+1,0) - sigma(2*n+1,1)) \\ G. C. Greubel, Jan 15 2019
  • Sage
    [euler_phi(2*n+1)*sigma(2*n+1, 0) - sigma(2*n+1,1) for n in (0..80)] # G. C. Greubel, Jan 15 2019

A055650 Numbers k such that k | phi(k)*d(k) - sigma(k), where phi=A000010, d=A000005 and sigma=A000203.

Original entry on oeis.org

1, 3, 14, 42, 76, 376, 3608, 163712, 163944, 196128, 277688, 491136, 833064, 849120, 905814, 911008, 1080328, 1653520, 1847898, 1935128, 2733024, 3145216, 3240984, 4586240, 4734736, 4960560, 5805384, 13758720, 16582752, 25244956, 34961040, 38521440, 48177990, 56240352
Offset: 1

Views

Author

Robert G. Wilson v, Jun 06 2000

Keywords

Comments

From Farideh Firoozbakht, Mar 17 2007: (Start)
I. If p is an odd prime then m = 2^k*p is in the sequence iff p = (k+3)*2^k - 1. For example, 14, 76, 376, 163712, 3145216, 1073733632, 1443108749312 and 67185481812096157153425363042304 are such terms. The numbers k such that (k+3)*2^k - 1 is prime up to 10000 are 1, 2, 3, 7, 9, 13, 18, 50, 210, 301, 349, 1160, 1796, 2677 and 8823. Thus 2^8823*(8826*2^8823-1) is the largest such term that I have found.
II. If m is in the sequence and 3 | phi(m)*d(m) - sigma(m) but 3 doesn't divide m then 3*m is in the sequence. Thus 1, 14, 163712, 277688, 911008, 1080328, 1653520, 1935128 and 4586240 are such terms and 2^2677*(2680*2^2677-1) is the largest such term that I have found. (End)

References

  • Inspired by David Wells, Curious and Interesting Numbers (Revised), Penguin Books.

Links

Crossrefs

Cf. A000005, A000010, A000203, A079536.

Programs

  • Mathematica
    Do[If[Mod[EulerPhi[n]*DivisorSigma[0, n]-DivisorSigma[1, n], n]==0, Print[n]], {n, 1, 1.05*10^7}]
Select[Range[6000000],Divisible[EulerPhi[#]DivisorSigma[0,#]- DivisorSigma[ 1,#], #]&] (* Harvey P. Dale, Mar 10 2012 *)
  • PARI
    isok(k) = {my(f=factor(k)); (eulerphi(f)*numdiv(f)-sigma(f))%k == 0; } \\ Jinyuan Wang, Mar 17 2020

Extensions

More terms from Jinyuan Wang, Mar 17 2020
Showing 1-2 of 2 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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