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 1-7 of 7 results.

A015762 Numbers n such that phi(n) | sigma_4(n).

Original entry on oeis.org

1, 2, 3, 6, 249, 498
Offset: 1

Views

Author

Robert G. Wilson v

Keywords

Comments

sigma_4(n) is the sum of the 4th powers of the divisors of n (A001159).
sigma_{8j+4}(x)/phi(x) is an integer for j=0..500, x=1,2,3,6,249,498, and this is conjectured to hold for possible larger terms of A015762 and all j. Compare with comments to A015759, A091285, A015770. - Labos Elemer, May 27 2004
For any odd n in this sequence, 2n is also in the sequence, since phi(2n) = phi(n) and sigma_4(2n) = 17 sigma_4(n). More generally, if gcd(m,n) = 1 and m and n both are in this sequence, then mn is also in the sequence. No odd prime > 3 can be in the sequence, since if p = 2r + 1, then sigma_4(p) = 8r(2r^3 + 4r^2 + 3r + 1) + 2 is divisible by phi(p) = 2r only for r = 1. The term a(5) = 3*83 is the only odd semiprime term with a factor < 10^5. - M. F. Hasler, Aug 21 2017
a(7) > 3*10^11, if it exists. - Giovanni Resta, Aug 23 2017

Crossrefs

Cf. A020492, A015759, A015761, A015763, A015764, A015765, A015766, A015767, A015768, A015769, A015770, A015771, A015773, A015774, A094470.

Programs

  • Magma
    [n: n in [1..1000]| DivisorSigma(4, n) mod EulerPhi(n) eq 0]; // Vincenzo Librandi, Aug 22 2017
  • Mathematica
    Select[Range[500],Divisible[DivisorSigma[4,#],EulerPhi[#]]&] (* Harvey P. Dale, Dec 16 2012 *)
  • PARI
    select( is(n)=sigma(n,4)%eulerphi(n)==0, [1..10^4])  \\ M. F. Hasler, Aug 21 2017

A015770 Numbers k such that phi(k) divides sigma_12(k).

Original entry on oeis.org

1, 2, 3, 6, 249, 498, 118578, 99295058, 297885174, 4005374907
Offset: 1

Views

Author

Robert G. Wilson v

Keywords

Comments

sigma_12(n) = A013960(n) is the sum of the 12th powers of the divisors of n.
sigma(24j+12,x)/phi(x) is an integer for j in the range 0, ..., 500 for x = 1, 2, 3, 6, 249, 498, 118578 and supposed to hold for possible larger terms of A015770 and all j. Compare with comments to A015759, A091285, A015762. - Labos Elemer, May 27 2004
a(11) > 5*10^9. - Giovanni Resta, Aug 22 2017
All known terms of A015762 (and also of this sequence) are squarefree. In that case, sigma_12(x)/sigma_4(x) = Product_{primes p|x} (p^8 - p^4 + 1) is an integer, so x is also in this sequence. - M. F. Hasler, Aug 22 2017

Crossrefs

Cf. A020492, A015759, A015761, A015762, A015763, A015764, A015765, A015766, A015767, A015768, A015769, A015771, A015773, A015774, A094470.

Programs

  • Mathematica
    Select[Range[1200000],Divisible[DivisorSigma[12,#],EulerPhi[#]]&] (* Harvey P. Dale, Dec 04 2015 *)

Extensions

Corrected by Harvey P. Dale, Dec 04 2015
Offset corrected by and a(8)-a(10) from Giovanni Resta, Aug 22 2017

A015766 Numbers k such that phi(k) | sigma_8(k).

Original entry on oeis.org

1, 2, 3, 6, 19689, 39378
Offset: 1

Views

Author

Robert G. Wilson v

Keywords

Comments

sigma_8(n) is the sum of the 8th powers of the divisors of n.
sigma(16j+8,x)/phi(x) is an integer for j = 0, ..., 500 and 6 actual terms of this sequence. Compare to A015759, A015762, A015770 and A091285. - Labos Elemer, May 27 2004
No additional terms up to 5 million. - Harvey P. Dale, Jan 31 2016

Crossrefs

Cf. A015759, A015762, A015770, A091285.

Programs

  • Mathematica
    Select[Range[40000],Divisible[DivisorSigma[8,#],EulerPhi[#]]&] (* Harvey P. Dale, Jan 31 2016 *)

A094468 Numbers k such that sum of 9th powers of divisors of k is divisible by the square of Euler-phi of k.

Original entry on oeis.org

1, 2, 3, 6, 14, 42, 3810, 26670, 34162, 41256, 48546, 87096, 102486, 131934, 210482, 288792, 315723, 318990, 430122, 529848, 609672, 631446, 979830, 1023366, 1203960, 1473374, 1683126, 1920699, 2210061, 2241934, 2506086, 2549610
Offset: 1

Views

Author

Labos Elemer, May 19 2004

Keywords

Links

Crossrefs

Cf. A015763, A000010, A091285, A013957.

Programs

  • Mathematica
    Select[Range[2600000],Divisible[Total[Divisors[#]^9],(EulerPhi[#])^2]&]  (* Harvey P. Dale, Mar 04 2011 *)
  • PARI
    for(n=1,10000000,if(Mod(sigma(n,9),eulerphi(n)^2)==0,print1(n,","))) \\ C. Ronaldo

Formula

A013957(k)/A000010(k)^2 is an integer.

Extensions

More terms from C. Ronaldo (aga_new_ac(AT)hotmail.com), Jan 18 2005

A091286 Numbers n such that sigma_3(n) is divisible by square of cototient of n, while n is not a prime number.

Original entry on oeis.org

15, 85, 259, 295, 391, 589, 799, 1111, 1717, 3193, 4171, 4369, 12361, 17473, 23533, 25429, 28243, 31351, 34477, 36181, 41407, 47989, 51143, 52537, 58103, 65641, 68377, 69541, 69919, 70453, 72619, 81121, 83131, 83767, 85069, 91759
Offset: 1

Views

Author

Labos Elemer, Feb 03 2004

Keywords

Examples

			n=15: cototient(15) = 7, sigma_3(15) = 3528 = 72 * 49.

Links

Crossrefs

Cf. A051953, A001158, A091285.

Programs

  • Mathematica
    Do[s=DivisorSigma[3, n]/(n-EulerPhi[n])^2; If[IntegerQ[s]&&!PrimeQ[n], Print[n]], {n, 1, 100000}]
  • PARI
    isok(n) = (n!=1) && !isprime(n) && !(sigma(n, 3)%(n-eulerphi(n))^2); \\ Michel Marcus, Aug 13 2019

A093038 Numbers k such that sum of 5th powers of divisors of k is divisible by the square of Euler-phi of k.

Original entry on oeis.org

1, 2, 3, 6, 322, 966, 473732, 1420296, 6655780894
Offset: 1

Views

Author

Labos Elemer, May 19 2004

Keywords

Comments

No more terms through 10^9. - Ryan Propper, Jan 09 2007

Crossrefs

Cf. A000010, A015763, A091285.

Programs

  • Mathematica
    Do[ If[ Mod[ DivisorSigma[5, n], EulerPhi[n]^2] == 0, Print[n]], {n, 10^7}] (* Robert G. Wilson v, May 23 2004 *)
  • PARI
    isok(k) = (sigma(k, 5) % eulerphi(k)^2) == 0; \\ Michel Marcus, Mar 07 2020

Formula

A001160(k)/A000010(k)^2 is an integer.

Extensions

a(9) from Amiram Eldar, Mar 07 2020

A094469 Numbers k such that sum of 11th powers of divisors of k is divisible by the square of Euler-phi of k.

Original entry on oeis.org

1, 2, 3, 6, 1645, 3290, 4935, 9870, 3831674, 11495022, 346014339, 692028678
Offset: 1

Views

Author

Labos Elemer, May 19 2004

Keywords

Crossrefs

Cf. A000010, A013959, A015763, A091285.

Programs

  • Mathematica
    Do[ If[ Mod[ DivisorSigma[11, n], EulerPhi[n]^2] == 0, Print[n]], {n, 10^7}] (* Robert G. Wilson v, May 23 2004 *)
  • PARI
    isok(k) = (sigma(k, 11) % eulerphi(k)^2) == 0; \\ Michel Marcus, Mar 07 2020

Formula

A013959(k)/A000010(k)^2 is an integer.

Extensions

a(9) from Robert G. Wilson v, May 23 2004
a(10) from Labos Elemer, May 26 2004
a(6) corrected and a(11)-a(12) added by Amiram Eldar, Mar 07 2020
Showing 1-7 of 7 results.

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

Content is available under The OEIS End-User License Agreement.