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

2, 3, 5, 7, 8, 11, 13, 17, 19, 23, 27, 29, 31, 32, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 125, 127, 128, 131, 137, 139, 149, 151, 157, 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

Offset: 1

Author

Joerg Arndt, Aug 29 2014

Keywords

• nonn

Comments

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

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

Links

• Jens Kruse Andersen, Table of n, a(n) for n = 1..10000

Crossrefs

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

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

Subsequence of A063752.

Programs

Magma

[n:n in [2..1000]| #PrimeDivisors(n) eq 1 and IsSquare(n-EulerPhi(n))]; // Marius A. Burtea, May 15 2019

Mathematica

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

PARI

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

Python

from sympy import primepi, integer_nthroot
def A246551(n):
    def f(x): return int(n-1+x-sum(primepi(integer_nthroot(x,k)[0])for k in range(1,x.bit_length(),2)))
    kmin, kmax = 1,2
    while f(kmax) >= kmax:
        kmax <<= 1
    while True:
        kmid = kmax+kmin>>1
        if f(kmid) < kmid:
            kmax = kmid
        else:
            kmin = kmid
        if kmax-kmin <= 1:
            break
    return kmax # Chai Wah Wu, Aug 13 2024