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 1-4 of 4 results.

A110176 Least k such that sigma(n) = sigma(k) + sigma(n-k) for 0

Original entry on oeis.org

0, 0, 1, 0, 0, 0, 0, 2, 4, 2, 0, 0, 0, 0, 5, 0, 0, 0, 0, 2, 7, 0, 0, 0, 0, 0, 0, 0, 0, 6, 0, 4, 11, 0, 0, 0, 0, 0, 13, 10, 0, 0, 0, 0, 0, 0, 0, 0, 8, 0, 17, 0, 0, 0, 12, 14, 19, 0, 0, 0, 0, 17, 19, 0, 0, 0, 0, 0, 14, 14, 0, 0, 0, 0, 25, 0, 0, 0, 0, 0, 0, 0, 0, 0, 38, 0, 22, 22, 0, 18, 0, 30, 31, 19, 0, 12
Offset: 1

Views

Author

T. D. Noe, Jul 15 2005

Keywords

Comments

Sequence A110177 gives the number of solutions 0A110178.

Links

Crossrefs

Cf. A066435 (least k such that sigma(n)+sigma(k)=sigma(n+k)), A110177.
Cf. also A110173.

Programs

  • Mathematica
    a[n_] := Select[Range[n-1], DivisorSigma[1, n]==DivisorSigma[1, n-# ]+DivisorSigma[1, # ]&]; Table[s=a[n]; If[Length[s]==0, 0, First[s]], {n, 150}]
  • PARI
    A110176(n) = { my(x=sigma(n)); for(k=1,n-1,if(x == (sigma(k)+sigma(n-k)), return(k))); (0); }; \\ Antti Karttunen, Feb 20 2023

A066426 Conjectured values for a(n) = least natural number k such that phi(n+k) = phi(n) + phi(k), if k exists; otherwise 0.

Original entry on oeis.org

2, 1, 0, 4, 4, 4, 14, 6, 6, 4, 16, 6, 14, 6, 0, 5, 8, 6, 6, 8, 0, 4, 46, 12, 10, 8, 6, 12, 26, 12, 62, 6, 12, 4, 16, 12, 28, 6, 0, 10, 24, 24, 86, 8, 0, 6, 38, 6, 62, 25, 12, 16, 24, 18, 32, 24, 0, 4, 118, 24, 80, 6, 12, 10, 28, 12, 134, 8, 0, 35, 142, 24, 146, 8, 30, 12, 8, 24, 46, 20, 6
Offset: 1

Views

Author

Joseph L. Pe, Dec 27 2001

Keywords

Comments

It would be nice to remove the word "Conjectured" from the description. - N. J. A. Sloane
The values of a(3), a(15) and a(21) listed above, namely 0, are conjectural. There is no natural number k < 10^6 satisfying the "homomorphic condition" phi(n+k) = phi(n) + phi(k) for n = 3, 15, 21.
The terms for which there is no solution k < 10^6 are n = 3, 15, 21, 39, 45, 57, 69, 105, 147, 165, 177, 195, 213, 273, 285,..., which satisfy n=3 (mod 6). - T. D. Noe, Jan 20 2004
All n < 2000 and k < 10^8 have been tested. Sequence A110172 gives the n for which there is no solution k < 10^8. For n=1 (mod 3) or n=2 (mod 3), it appears that the least solution k satisfies k<=2n. For n=0 (mod 3), the least k, if it exists, can be greater than 2n. - T. D. Noe, Jul 15 2005

References

  • R. K. Guy, Unsolved Problems in Number Theory, 3rd Ed., New York, Springer-Verlag, 2004, Section B36.

Crossrefs

Cf. A000010.
Cf. A091531 (primes p such that k=2p is the smallest solution to phi(p+k) = phi(p) + phi(k)).
Cf. A110173 (least k such that phi(n) = phi(k) + phi(n-k) for 0 < k < n).

Programs

  • Mathematica
    a[ n_ ] := Min[ Select[ Range[ 1, 10^6 ], EulerPhi[ 1, n + # ] == EulerPhi[ 1, n ] + EulerPhi[ 1, # ] & ] ]; Table[ a[ i ], {i, 1, 21} ]

Extensions

More terms from T. D. Noe, Jan 20 2004

A110174 Number of solutions 0

Original entry on oeis.org

0, 0, 2, 1, 0, 0, 0, 1, 2, 2, 0, 1, 0, 4, 4, 1, 0, 2, 0, 5, 4, 4, 0, 3, 4, 4, 4, 7, 0, 0, 0, 1, 6, 2, 2, 3, 0, 6, 2, 7, 0, 2, 0, 9, 6, 4, 0, 3, 0, 4, 6, 9, 0, 4, 2, 9, 4, 2, 0, 1, 0, 4, 4, 1, 4, 4, 0, 9, 4, 4, 0, 5, 0, 4, 8, 9, 2, 6, 0, 7, 0, 2, 0, 3, 4, 4, 8, 9, 0, 2, 0, 11, 8, 4, 0, 3, 0, 2, 6, 9, 0, 6, 0
Offset: 1

Views

Author

T. D. Noe, Jul 15 2005

Keywords

Links

Crossrefs

Cf. A110173 (least k such that phi(n) = phi(k) + phi(n-k)).
Cf. also A110177.

Programs

  • Mathematica
    a[n_] := Select[Range[n-1], EulerPhi[n]==EulerPhi[n-# ]+EulerPhi[ # ]&]; Table[Length[a[n]], {n, 150}]
  • PARI
    A110174(n) = { my(ph=eulerphi(n)); sum(k=1,n-1,(ph == (eulerphi(k)+eulerphi(n-k)))); }; \\ Antti Karttunen, Feb 20 2023

A110175 Composite numbers n such that the equation phi(n)=phi(k)+phi(n-k) has no solution, where phi is Euler's totient function.

Original entry on oeis.org

6, 30, 49, 81, 91, 95, 115, 121, 155, 187, 205, 210, 221, 243, 254, 259, 287, 298, 299, 329, 341, 355, 361, 377, 403, 415, 437, 451, 469, 473, 502, 533, 551, 559, 565, 611, 625, 629, 649, 655, 662, 667, 674, 679, 685, 703, 713, 731, 737, 746, 767, 779, 781
Offset: 1

Views

Author

T. D. Noe, Jul 15 2005

Keywords

Comments

The only prime for which this equation has a solution is 3.

Crossrefs

Cf. A110173 (least k such that phi(n)=phi(k)+phi(n-k)).

Programs

  • Mathematica
    a[n_] := Select[Range[n-1], EulerPhi[n]==EulerPhi[n-# ]+EulerPhi[ # ]&]; t=Table[Length[a[n]], {n, 1000}]; Complement[Flatten[Position[t, 0]], Prime[Range[PrimePi[1000]]]]
Showing 1-4 of 4 results.

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

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