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.
%I A051488 #34 Jul 02 2025 16:01:58 %S A051488 30,60,66,120,132,138,174,210,240,246,264,276,318,330,348,420,480,492, %T A051488 498,510,528,534,552,630,636,660,678,690,696,786,840,870,910,960,984, %U A051488 996,1020,1038,1056,1068,1074,1104,1122,1146,1260,1272,1320,1330,1356 %N A051488 Numbers k such that phi(k) < phi(k - phi(k)). %C A051488 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 %C A051488 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 %D A051488 Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section B42, p. 150. %D A051488 József Sándor, Dragoslav S. Mitrinovic and Borislav Crstici, Handbook of Number Theory I, Springer Science & Business Media, 2005, Chapter III, p. 209. %H A051488 Amiram Eldar, <a href="/A051488/b051488.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..1000 from T. D. Noe) %H A051488 Paul Erdős, <a href="https://books.google.com/books?id=KBD87GWw9mMC&lpg=PA381&pg=PA505">Problem P. 294</a>, Canad. Math. Bull., Vol. 23, No. 4 (1980), p. 505. %H A051488 Aleksander Grytczuk, Florian Luca and Marek Wojtowicz, <a href="https://www.researchgate.net/profile/Marek-Wojtowicz-2/publication/266273753_A_conjecture_of_Erdos_concerning_inequalities_for_the_Euler_totient_function">A conjecture of Erdős concerning inequalities for the Euler totient function</a>, Publ. Math. Debrecen, Vol. 59, No. 1-2, (2001), pp. 9-16. %t A051488 Select[Range[1360], EulerPhi[ # ] < EulerPhi[ # - EulerPhi[ # ]] &] (* _Farideh Firoozbakht_, Jun 19 2005 *) %o A051488 (Haskell) %o A051488 a051488 n = a051488_list !! (n-1) %o A051488 a051488_list = [x | x <- [2..], let t = a000010 x, t < a000010 (x - t)] %o A051488 -- _Reinhard Zumkeller_, Apr 12 2014 %Y A051488 Cf. A051487, A005384. %Y A051488 Cf. A000010, A051953. %K A051488 nonn,nice,easy %O A051488 1,1 %A A051488 _R. K. Guy_ %E A051488 More terms from _James Sellers_