A051488 Numbers k such that phi(k) < phi(k - phi(k)).

30, 60, 66, 120, 132, 138, 174, 210, 240, 246, 264, 276, 318, 330, 348, 420, 480, 492, 498, 510, 528, 534, 552, 630, 636, 660, 678, 690, 696, 786, 840, 870, 910, 960, 984, 996, 1020, 1038, 1056, 1068, 1074, 1104, 1122, 1146, 1260, 1272, 1320, 1330, 1356

Offset: 1

Author

R. K. Guy

Keywords

• nonn
• nice
• easy

Comments

If p is a Sophie Germain prime greater than 3 and n is a natural number then 2^n*3*p is in the sequence. That is because if m = 2^n*3*p then phi(m) = 2^n*(p-1) and phi(m - phi(m)) = phi(2^n*3*p - 2^n*(p-1)) = phi(2^n*(2p+1)) = 2^n*p so phi(m) < phi(m-phi(m)) and m is in the sequence. - Farideh Firoozbakht, Jun 19 2005

Erdős (1980) proposed the problem to prove that this sequence is infinite and has an asymptotic density 0. Grytczuk et al. (2001) proved that this sequence is infinite with an upper asymptotic density < 0.45637. - Amiram Eldar, May 22 2021

References

• Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section B42, p. 150.
• József Sándor, Dragoslav S. Mitrinovic and Borislav Crstici, Handbook of Number Theory I, Springer Science & Business Media, 2005, Chapter III, p. 209.

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)
• Paul Erdős, Problem P. 294, Canad. Math. Bull., Vol. 23, No. 4 (1980), p. 505.
• Aleksander Grytczuk, Florian Luca and Marek Wojtowicz, A conjecture of Erdős concerning inequalities for the Euler totient function, Publ. Math. Debrecen, Vol. 59, No. 1-2, (2001), pp. 9-16.

Crossrefs

Cf. A051487, A005384.

Cf. A000010, A051953.

Programs

Haskell

a051488 n = a051488_list !! (n-1)
a051488_list = [x | x <- [2..], let t = a000010 x, t < a000010 (x - t)]
-- Reinhard Zumkeller, Apr 12 2014

Mathematica

Select[Range[1360], EulerPhi[ # ] < EulerPhi[ # - EulerPhi[ # ]] &] (* Farideh Firoozbakht, Jun 19 2005 *)

Extensions

More terms from James Sellers