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.

A110173 Least k such that phi(n) = phi(k) + phi(n-k) for 0

Original entry on oeis.org

0, 0, 1, 2, 0, 0, 0, 4, 4, 4, 0, 6, 0, 4, 5, 8, 0, 6, 0, 6, 5, 6, 0, 6, 6, 4, 11, 6, 0, 0, 0, 16, 6, 8, 10, 12, 0, 4, 13, 12, 0, 12, 0, 6, 7, 8, 0, 12, 0, 10, 16, 6, 0, 6, 26, 12, 19, 26, 0, 30, 0, 4, 12, 32, 24, 24, 0, 6, 23, 28, 0, 18, 0, 10, 12, 8, 24, 12, 0, 24, 0, 8, 0, 24, 8, 4, 6, 12, 0, 30
Offset: 1

Views

Author

T. D. Noe, Jul 15 2005

Keywords

Comments

Sequence A110174 gives the number of solutions 0A110175.

Links

Crossrefs

Cf. A066426 (least k such that phi(n)+phi(k)=phi(n+k)), A110174.
Cf. also A110176.

Programs

  • Mathematica
    a[n_] := Select[Range[n-1], EulerPhi[n]==EulerPhi[n-# ]+EulerPhi[ # ]&]; Table[s=a[n]; If[Length[s]==0, 0, First[s]], {n, 150}]
  • PARI
    A110173(n) = { my(ph=eulerphi(n)); for(k=1,n-1,if(ph == (eulerphi(k)+eulerphi(n-k)), return(k))); (0); }; \\ Antti Karttunen, Feb 20 2023

A091531 Primes p such that k = 2p is the smallest positive solution to the equation phi(p+k) = phi(p) + phi(k), where phi is Euler's totient function.

Original entry on oeis.org

7, 23, 31, 43, 59, 67, 71, 73, 101, 103, 107, 127, 131, 137, 139, 179, 199, 211, 223, 227, 239, 269, 281, 283, 307, 311, 331, 347, 359, 367, 379, 383, 431, 439, 463, 467, 479, 487, 491, 503, 523, 547, 563, 571, 607, 619, 631, 643, 659, 661, 683, 691, 719, 727
Offset: 1

Views

Author

T. D. Noe, Jan 19 2004

Keywords

Comments

Note that for all primes p > 3, phi(3p) = phi(p) + phi(2p).

Crossrefs

Cf. A066426 (least k such that phi(n+k)=phi(n)+phi(k)).

Programs

  • Mathematica
    lst={}; Do[p=Prime[n]; k=1; While[EulerPhi[p+k]!=EulerPhi[p]+EulerPhi[k], k++ ]; If[k==2p, AppendTo[lst, p]], {n, 3, 200}]; lst

A110172 Conjectured numbers j such that phi(j) + phi(k) = phi(j+k) has no solution k, where phi is Euler's totient function.

Original entry on oeis.org

3, 15, 21, 39, 45, 57, 69, 105, 147, 165, 177, 195, 213, 273, 285, 315, 345, 393, 399, 465, 489, 525, 585, 615, 633, 645, 651, 681, 717, 777, 807, 813, 843, 855, 879, 885, 903, 915, 933, 939, 1005, 1035, 1041, 1065, 1095, 1149, 1263, 1281, 1293, 1317, 1395
Offset: 1

Views

Author

T. D. Noe, Jul 15 2005

Keywords

Comments

All k < 10^8 have been checked. All of these numbers are multiples of 3.
The observation above is true for every term. Substituting k=j into phi(j) + phi(k) = phi(j+k) gives phi(j) + phi(j) = phi(j+j), i.e., 2*phi(j) = phi(2j), which is true for every positive even number j; thus k=j yields a solution for every positive even number j. Substituting k=2j into phi(j) + phi(k) = phi(j+k) gives phi(j) + phi(2j) = phi(j+2j), i.e., phi(j) + phi(2j) = phi(3j); since phi(j) = phi(2j) for every odd number j, this is equivalent (for odd j) to phi(j) + phi(j) = phi(3j), i.e., 2*phi(j) = phi(3j), which holds for every odd j that is not a multiple of 3; thus, k=2j yields a solution for every odd j that is not a multiple of 3. Consequently, every term of the sequence is an odd multiple of 3. - Flávio V. Fernandes, May 10 2022

Crossrefs

Cf. A066426 (least k such that phi(n) + phi(k) = phi(n+k)).
Cf. A306771.
Showing 1-3 of 3 results.

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