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%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
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