A246551 - OEIS

%I #55 Sep 24 2024 09:28:01

%S 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 79,83,89,97,101,103,107,109,113,125,127,128,131,137,139,149,151,157,

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

%C 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 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 Jens Kruse Andersen, <a href="/A246551/b246551.txt">Table of n, a(n) for n = 1..10000</a>

%t Take[Union[Flatten[Table[Prime[n]^(k + 1), {n, 100}, {k, 0, 14, 2}]]], 100] (* _Vincenzo Librandi_, Jan 10 2019 *)

%o (PARI) for(n=1, 10^4, my(e=isprimepower(n)); if(e%2==1, print1(n, ", ")))

%o (Magma) [n:n in [2..1000]| #PrimeDivisors(n) eq 1 and IsSquare(n-EulerPhi(n))]; // _Marius A. Burtea_, May 15 2019

%o (Python)

%o from sympy import primepi, integer_nthroot

%o def A246551(n):

%o     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     kmin, kmax = 1,2

%o     while f(kmax) >= kmax:

%o         kmax <<= 1

%o     while True:

%o         kmid = kmax+kmin>>1

%o         if f(kmid) < kmid:

%o             kmax = kmid

%o         else:

%o             kmin = kmid

%o         if kmax-kmin <= 1:

%o             break

%o     return kmax # _Chai Wah Wu_, Aug 13 2024

%Y Cf. A000961, A246547, A246549, A168363, A197680, subsequence of A171561.

%Y Cf. also A056798 (prime powers with even exponents >= 0).

%Y Subsequence of A063752.

%K nonn

%O 1,1

%A _Joerg Arndt_, Aug 29 2014