A171561 -id:A171561 - cp's OEIS frontend

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

Joerg Arndt, Aug 29 2014

nonn

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

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

Cf. A000961, A246547, A246549, A168363, A197680, subsequence of A171561.
Cf. also A056798 (prime powers with even exponents >= 0).
Subsequence of A063752.

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

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

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

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

A268391 Numbers of the form p^A001317(k) where p is prime and k >= 0.

Original entry on oeis.org

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, 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

Antti Karttunen, Feb 10 2016

nonn

Sequence A268392 sorted into ascending order.

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A001221, A001317, A050376, A067029, A268384, A268392.
Cf. A000040 (subsequence).
Subsequence of A000961 and A246551.
Differs from A171561 and A246551 for the first time at n=36, which here a(36) = 131.

A304291 Composite numbers k such that for all primes p dividing k, p-1 divides k-1 and p+1 divides k+1.

Original entry on oeis.org

8, 27, 32, 125, 128, 243, 343, 512, 1331, 2048, 2187, 2197, 3125, 4913, 6859, 8192, 12167, 16807, 19683, 24389, 29791, 32768, 50653, 68921, 74431, 78125, 79507, 103823, 131072, 148877, 161051, 177147, 205379, 226981, 300763, 357911, 371293, 389017, 493039, 524288

Offset: 1

Paolo P. Lava, May 17 2018

nonn

Intersection of A080062 and A056729.
Mainly odd powers of a prime: A056824 is a subset of this sequence.
If the additional limitations p-2|n-2 and p+2|n+2 should be added, only 243, 19683, 78125, 1594323 would be terms of the sequence for n <= 10^7.
Terms that are not perfect powers are 31*7^4, 31^3*7^4, 71*11^6, .... - Altug Alkan, May 17 2018
It appears that this is the intersection of A002808 and A171561. - Michel Marcus, May 19 2018
From Robert Israel, May 25 2018: (Start)
  If i is odd and 4|j, then 31^i*7^j is a member.
  If i is odd and 6|j, then 71^i*11^j is a member.
  If i is odd and 12|j, then 17^i*5^j is a member.
  If i is odd and 36|j, then 53^i*5^j is a member.
  If i == 9 (mod 18) and 6|j, then 13^i*37^j is a member.
  If i == 9 (mod 18) and 12|j, then 29^i*53^j is a member.
  If i == 18 (mod 36), j == 3 (mod 6) and k == 2 (mod 4), then 5^i*17^j*53^k is a member.
(End)
Composite numbers k such that for all primes p dividing k, p+1 divides k-1 and p-1 divides k+1 are the union of 2^2j and 3^2j, with j>0. - Paolo P. Lava, May 16 2019

			Prime factors of 74431 are 7 and 31 and (74431-1)/(7-1) = 12405, (74431-1)/(31-1) = 2481, (74431+1)/(7+1) = 9304, (74431+1)/(31+1) = 2326.

Giovanni Resta, Table of n, a(n) for n = 1..1000 (first 192 terms from Robert Israel)

Cf. A056729, A056824, A080062.

sol:=[]; m:=1; p:=[]; for u in [1..600000] do if not IsPrime(u) then p:=PrimeDivisors(u);  s:=0; for i in [1..#p] do if IsIntegral((u-1)/(p[i]-1)) and  IsIntegral((u+1)/(p[i]+1)) then  s:=s+1; end if; if s eq #p then sol[m]:=u; m:=m+1; end if; end for; end if; end for; sol; // Marius A. Burtea, May 16 2019

with(numtheory): P:=proc(q) local a,b,k,n,ok;
for n from 2 to q do if not isprime(n) then a:=factorset(n); ok:=1;
for k from 1 to nops(a) do if frac((n-1)/(a[k]-1))>0 or frac((n+1)/(a[k]+1))>0 then ok:=0; break; fi; od;
if ok=1 then print(n); fi; fi; od; end: P(10^6);

Select[Range[4, 2^19], Function[k, And[CompositeQ@ k, AllTrue[FactorInteger[k][[All, 1]], And[Mod[k - 1, # - 1] == 0, Mod[k + 1, # + 1] == 0] &]]]] (* Michael De Vlieger, May 22 2018 *)

lista(nn) = {forcomposite(c=1, nn, my(f = factor(c)); ok = 1; for (k=1, #f~, my(p = f[k,1]); if (((c-1) % (p-1)) || ((c+1) % (p+1)), ok = 0; break);); if (ok, print1(c, ", ")););} \\ Michel Marcus, May 19 2018

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

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A268391 Numbers of the form p^A001317(k) where p is prime and k >= 0.

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A304291 Composite numbers k such that for all primes p dividing k, p-1 divides k-1 and p+1 divides k+1.

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