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A246551 Prime powers p^e where p is a prime and e is odd.

%I A246551 #55 Sep 24 2024 09:28:01
%S A246551 2,3,5,7,8,11,13,17,19,23,27,29,31,32,37,41,43,47,53,59,61,67,71,73,
%T A246551 79,83,89,97,101,103,107,109,113,125,127,128,131,137,139,149,151,157,
%U A246551 163,167,173,179,181,191,193,197,199,211,223,227,229,233,239,241,243,251,257,263,269,271,277,281,283,293,307,311,313
%N A246551 Prime powers p^e where p is a prime and e is odd.
%C A246551 These are the integers with only one prime factor whose cototient is square, so this sequence is a subsequence of A063752. Indeed, cototient(p^(2k+1)) = (p^k)^2 and cototient(p) = 1 = 1^2. - _Bernard Schott_, Jan 08 2019
%C A246551 With 1 prepended, this sequence is the lexicographically earliest sequence of distinct numbers whose partial products are all numbers whose exponents in their prime power factorization are squares (A197680). - _Amiram Eldar_, Sep 24 2024
%H A246551 Jens Kruse Andersen, <a href="/A246551/b246551.txt">Table of n, a(n) for n = 1..10000</a>
%t A246551 Take[Union[Flatten[Table[Prime[n]^(k + 1), {n, 100}, {k, 0, 14, 2}]]], 100] (* _Vincenzo Librandi_, Jan 10 2019 *)
%o A246551 (PARI) for(n=1, 10^4, my(e=isprimepower(n)); if(e%2==1, print1(n, ", ")))
%o A246551 (Magma) [n:n in [2..1000]| #PrimeDivisors(n) eq 1 and IsSquare(n-EulerPhi(n))]; // _Marius A. Burtea_, May 15 2019
%o A246551 (Python)
%o A246551 from sympy import primepi, integer_nthroot
%o A246551 def A246551(n):
%o A246551     def f(x): return int(n-1+x-sum(primepi(integer_nthroot(x,k)[0])for k in range(1,x.bit_length(),2)))
%o A246551     kmin, kmax = 1,2
%o A246551     while f(kmax) >= kmax:
%o A246551         kmax <<= 1
%o A246551     while True:
%o A246551         kmid = kmax+kmin>>1
%o A246551         if f(kmid) < kmid:
%o A246551             kmax = kmid
%o A246551         else:
%o A246551             kmin = kmid
%o A246551         if kmax-kmin <= 1:
%o A246551             break
%o A246551     return kmax # _Chai Wah Wu_, Aug 13 2024
%Y A246551 Cf. A000961, A246547, A246549, A168363, A197680, subsequence of A171561.
%Y A246551 Cf. also A056798 (prime powers with even exponents >= 0).
%Y A246551 Subsequence of A063752.
%K A246551 nonn
%O A246551 1,1
%A A246551 _Joerg Arndt_, Aug 29 2014