A330251 Numbers k such that phi(k) = phi(k+3), where phi (A000010) is Euler's totient function.

3, 5, 8720288051472, 9134280520365, 41544070492925, 42466684755492, 51363581614342, 68616494581632, 113312918293575, 210911076210835, 215517565688425, 294988451482725, 383617980270525, 432759876053505, 442863123838135, 532068058516992, 892813363927485, 923102743748185, 929531173876305

Offset: 1

Author

Michel Marcus and Giovanni Resta, Feb 29 2020

Keywords

• nonn

Comments

10^15 < a(20) <= 1089641067389872.

Also terms: 1248817919303952, 1332436545865422, 1394926716616125, 1868522795664525, 1950445682260072.

a(4) and a(9) appear in Kevin Ford's paper.

Links

• Kevin Ford, Solutions of phi(n)=phi(n+k) and sigma(n)=sigma(n+k), arXiv:2002.12155 [math.NT], 2020.
• S. W. Graham, J. J. Holt, and C. Pomerance, On the solutions to phi(n) = phi(n+k), Number Theory in Progress, K. Gyory, H. Iwaniec, and J. Urbanowicz, eds., vol. 2, de Gruyter, Berlin and New York, 1999, pp. 867-882.
• Mathematics StackExchange, Conjecture on the gap between integers having the same number of co-primes, Sep 25 2019.

Crossrefs

Cf. A000010, A007015.

Cf. A001274, A001494, A179186, A179187, A179188, A179189, A179202, A330429.

Cf. A276503, A276504, A217139.

Programs

Mathematica

Select[Range[100000], EulerPhi[#] == EulerPhi[# + 3] &] (* Alonso del Arte, Mar 01 2020 *)

PARI

isok(k) = eulerphi(k) == eulerphi(k+3); \\ Michel Marcus, Feb 29 2020