Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%I #36 Apr 10 2019 13:56:25
%S 2,5,8,17,32,37,101,125,128,197,257,401,512,577,677,1297,1601,2048,
%T 2917,3125,3137,4357,4913,5477,7057,8101,8192,8837,12101,13457,14401,
%U 15377,15877,16901,17957,21317,22501,24337,25601,28901,30977,32401
%N Odd powers of primes of the form q = x^2 + 1 (A002496).
%C A002496 is a subset; the odd power exponent is 1.
%C From _Bernard Schott_, Mar 16 2019: (Start)
%C The terms of this sequence are exactly the integers with only one prime factor and whose Euler's totient is square, so this sequence is a subsequence of A039770. The primitive terms of this sequence are the primes of the form q = x^2 + 1, which are exactly in A002496.
%C Additionally, the terms of this sequence also have a square cototient, so this sequence is a subsequence of A063752 and A054754.
%C If q prime = x^2 + 1, phi(q) = x^2, phi(q^(2k+1)) = (x*q^k)^2, and cototient(q) = 1^2, cototient(q^(2k+1)) = (q^k)^2. (End)
%H David A. Corneth, <a href="/A054755/b054755.txt">Table of n, a(n) for n = 1..18864</a> (terms <= 10^11)
%H Bernard Schott, <a href="/A306908/a306908.pdf">Subfamilies and subsequences</a>
%F A000010(a(n)) = (q^(2k))*(q-1) and A051953(a(n)) = q^(2k), where q = 1 + x^2 and is prime.
%e a(20) = 3125 = 5^5, q = 5 = 4^2+1 and Phi(3125) = 2500 = 50^2, cototient(3125) = 3125 - Phi(3125) = 625 = 25^2.
%t Select[Range[10^5], And[PrimeNu@ # == 1, IntegerQ@ Sqrt@ EulerPhi@ #] &] (* _Michael De Vlieger_, Mar 31 2019 *)
%o (PARI) isok(m) = (omega(m)==1) && issquare(eulerphi(m)); \\ _Michel Marcus_, Mar 16 2019
%o (PARI) upto(n) = {my(res = List([2]), q); forstep(i = 2, sqrtint(n), 2, if(isprime(i^2 + 1), listput(res, i^2 + 1) ) ); q = #res; forstep(i = 3, logint(n, 2), 2, for(j = 1, q, c = res[j]^i; if(c <= n, listput(res, c) , next(2) ) ) ); listsort(res); res } \\ _David A. Corneth_, Mar 17 2019
%Y Cf. A000010, A051953, A039770, A063752, A054754, A334745 (with 2 distinct prime factors), A306908 (with 3 distinct prime factors).
%Y Subsequences: A002496 (primitive primes: m^2+1), A004171 (2^(2k+1)), A013710 (5^(2k+1)), A013722 (17^(2k+1)), A262786 (37^(2k+1)).
%K nonn
%O 1,1
%A _Labos Elemer_, Apr 25 2000