A070265 -id:A070265 - cp's OEIS frontend

A097054 Nonsquare perfect powers.

8, 27, 32, 125, 128, 216, 243, 343, 512, 1000, 1331, 1728, 2048, 2187, 2197, 2744, 3125, 3375, 4913, 5832, 6859, 7776, 8000, 8192, 9261, 10648, 12167, 13824, 16807, 17576, 19683, 21952, 24389, 27000, 29791, 32768, 35937, 39304, 42875, 50653

Offset: 1

Hugo Pfoertner, Jul 21 2004

Terms of A001597 that are not in A000290.

All terms of this sequence are also in A070265 (odd powers), but omitting those odd powers that are also a square (e.g. 64=4^3=8^2).

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Perfect Power.
Eric Weisstein's World of Mathematics, Odd Power.

Cf. A001597 (perfect powers), A000290 (the squares), A008683, A070265 (odd powers), A097055, A097056, A239870, A239728, A093771.

import Data.Map (singleton, findMin, deleteMin, insert)
a097054 n = a097054_list !! (n-1)
a097054_list = f 9 (3, 2) (singleton 4 (2, 2)) where
   f zz (bz, be) m
    | xx < zz && even be =
                f zz (bz, be+1) (insert (bx*xx) (bx, be+1) $ deleteMin m)
    | xx < zz = xx :
                f zz (bz, be+1) (insert (bx*xx) (bx, be+1) $ deleteMin m)
    | xx > zz = f (zz+2*bz+1) (bz+1, 2) (insert (bz*zz) (bz, 3) m)
    | otherwise = f (zz + 2 * bz + 1) (bz + 1, 2) m
    where (xx, (bx, be)) = findMin m
-- Reinhard Zumkeller, Mar 28 2014

# uses code of A001597
for n from 4 do
    if not issqr(n) and isA001597(n) then
        printf("%d,\n",n);
    end if;
end do: # R. J. Mathar, Jan 13 2021

nn = 50653; Select[Union[Flatten[Table[n^i, {i, Prime[Range[2, PrimePi[Log[2, nn]]]]}, {n, 2, nn^(1/i)}]]], ! IntegerQ[Sqrt[#]] &] (* T. D. Noe, Apr 19 2011 *)

is(n)=ispower(n)%2 \\ Charles R Greathouse IV, Aug 28 2016

list(lim)=my(v=List()); forprime(e=3,logint(lim\=1,2), for(b=2,sqrtnint(lim,e), if(!issquare(b), listput(v,b^e)))); Set(v) \\ Charles R Greathouse IV, Jan 09 2023

from sympy import mobius, integer_nthroot
def A097054(n):
    def f(x): return int(n-1+x+sum(mobius(k)*(integer_nthroot(x,k)[0]-1) for k in range(3,x.bit_length())))
    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 14 2024

A052409(a(n)) is odd. - Reinhard Zumkeller, Mar 28 2014

Sum_{n>=1} 1/a(n) = 1 - zeta(2) + Sum_{k>=2} mu(k)*(1-zeta(k)) = 0.2295303015... - Amiram Eldar, Dec 21 2020

A126032 Numbers of the form b^m/2 for even b and odd m > 2.

Original entry on oeis.org

4, 16, 32, 64, 108, 256, 500, 512, 864, 1024, 1372, 2048, 2916, 3888, 4000, 4096, 5324, 6912, 8192, 8788, 10976, 13500, 16384, 19652, 23328, 27436, 32000, 37044, 42592, 48668, 50000, 55296, 62500, 65536, 70304, 78732, 87808, 97556, 108000, 119164, 124416

Offset: 1

Alexander Adamchuk, Feb 28 2007

The old definition was: Numbers n such that A123669(n) = -1, or no generalized Fermat prime exists of the form (2n)^(2^k) + 1. But that sequence is probably missing a lot of terms, such as {6, 9, 11, 19, 21, 25, 26, 29, 30, 31, ...}, where no generalized Fermat prime has been found yet, and it seems unlikely any exist. Currently it can only be proved that none exist if n is of form b^m/2 for even b and odd m > 1. The listed terms are the first numbers of this form: 4 = 2^3/2, 16 = 2^5/2, 32 = 4^3/2, 64 = 2^7/2, 108 = 6^3/2, 256 = 2^9/2 = 8^3/2, 500 = 10^3/2. - Jens Kruse Andersen, Jul 24 2014

The even terms of A070265, divided by two. - Jeppe Stig Nielsen, Jul 02 2017

Jeppe Stig Nielsen, Generalized Fermat Primes sorted by base.
Eric Weisstein's World of Mathematics, Generalized Fermat Number.

Cf. A123669 = Smallest generalized Fermat prime of the form (2n)^(2^k) + 1, where k>0.

Cf. A070265.

Module[{nn = 2^17, a = {}, n}, Do[If[b > nn, Break[], Do[If[Set[n, b^m/2] > nn, Break[], AppendTo[a, n]], {m, 3, Infinity, 2}]], {b, 2, Infinity, 2}]; Union@ a] (* Michael De Vlieger, Jul 04 2017 *)

isOK(n)=ip=ispower(2*n);ip&&bitand(ip,ip-1) \\ Jeppe Stig Nielsen, Jul 02 2017

Definition changed by N. J. A. Sloane, Jul 26 2014 following the advice of Jens Kruse Andersen.

Terms after a(7) from Jeppe Stig Nielsen, Jul 02 2017

A228101 a(n) is the least k such that (2n)^(2^k) + 1 is a prime; a(n) = -1 if a prime (2n)^(2^k) + 1 is unknown, or = -2 if impossible.

Original entry on oeis.org

0, 0, 0, -2, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0, 0, -2, 2, 0, -1, 0, 0, 4, 0, 2, -1, 0, 1, 1, 0, 0, -1, -2, 0, -1, 0, 0, 1, 4, 0, 2, 0, 1, -1, 0, 1, -1, 1, 0, -1, 0, 0, -1, 0, 0, 1, 0, 5, 1, 2, 1, -1, 1, 0, -2, 0, 2, 1, 0, 0, 2, 2, -1, 1, 0, 0, 3, 2, 0, 4, 1, 0, 2, 0

Offset: 1

Yves Gallot (galloty(AT)wanadoo.fr) and Robert G. Wilson v, Aug 14 2013

A prime number of the form b^(2^k) + 1 is called a generalized Fermat prime to base b.
See the hyperlink for more information and links.
The impossibility case, a(n) = -2, occurs exactly if 2n is a member of A070265. Or equivalently, n is in A126032. - Jeppe Stig Nielsen, Jul 02 2017

Robert G. Wilson v, Table of n, a(n) for n = 1..500
Lucile and Yves Gallot, Welcome to the Generalized Fermat Prime Search!

Cf. A056993, A079706, A084712, A070265, A126032.

f[b_?EvenQ] := f[b] = Block[{k = 0}, While[! PrimeQ[b^(2^k) + 1], k++]; k];
lst = {38, 50, 62, 68, 86, 92, 98, 104, 122, 144, 168, 182, 186, 200, 202, 212, 214, 218, 244, 246, 252, 258, 286, 294, 298, 302, 304, 308, 322, 324, 338, 344, 354, 356, 362, 368, 380, 390, 394, 398, 402, 404, 410, 416, 422, 424, 446, 450, 454, 458, 468, 480, 482, 484, 500, 514, 518, 524, 528, 530, 534, 538, 552, 558, 564, 572, 574, 578, 580, 590, 602, 604, 608, 620, 622, 626, 632, 638, 648, 650, 662, 666, 668, 670, 678, 684, 692, 694, 698, 706, 712, 720, 722, 724, 734, 744, 746, 752, 754, 762, 766, 770, 792, 794, 802, 806, 812, 814, 818, 836, 840, 842, 844, 848, 854, 868, 870, 872, 878, 888, 896, 902, 904, 908, 922, 924, 926, 932, 938, 942, 944, 948, 954, 958, 964, 968, 974, 978, 980, 988, 994, 998}; (f[#] = -1) & /@ lst;
lst = {8, 32, 64, 128, 216, 512, 1000}; (f[#] = -2) & /@ lst; Table[ f[b], {b, 2, 1000, 2}]
(* Second program: *)
Module[{r = 83, nn = 12, s = {}, k}, Do[If[b > r, Break[], Do[If[Set[k, b^m/2] > r, Break[], AppendTo[s, k]], {m, 3, Infinity, 2}]], {b, 2, Infinity, 2}]; Table[If[MemberQ[s, n], -2, SelectFirst[Range[0, nn], PrimeQ[(2 n)^(2^#) + 1] &] /. x_ /; MissingQ@ x -> -1], {n, r}]] (* Michael De Vlieger, Jul 04 2017, Version 10.2 *)

Definition rewritten by Jeppe Stig Nielsen, Jul 02 2017

A253242 Least k>=0 such that n^(2^k)+1 is prime (for even n), or (n^(2^k)+1)/2 is prime (for odd n); -1 if no such k exists.

Original entry on oeis.org

0, 0, 0, 0, 0, 2, -1, 0, 0, 1, 0, 0, 1, 1, 0, 2, 0, 1, 1, 0, 0, 2, 1, 0, 1, -1, 0, 1, 0

Offset: 2

Eric Chen, Apr 19 2015

Least k such that the generalized Fermat number in base n (GFN(k,n)) is prime.
a(n) = -1 if n is in A070265 (perfect powers with an odd exponent).
a(n) is currently unknown for n = {31, 38, 50, 55, 62, 63, 67, 68, 77, 83, 86, 89, 91, 92, 97, 98, 99, 104, 107, 109, 122, 123, 127, 135, 137, ...}
Corresponding primes are {3, 2, 5, 3, 7, 1201, 0, 5, 11, 61, 13, 7, 197, 113, 17, 41761, 19, 181, 401, 11, 23, 139921, 577, 13, 677, 0, 29, 421, 31, ...}. (use 0 if a(n) = -1)
All 2 <= n <= 1500 and 0 <= k <= 14 are checked, the first occurrence of k (start with k = 0) in a(n) are {2, 11, 7, 43, 41, 75, 274, 234, 331, 1342, 824, ...}.

			a(7) = 2 since (7^(2^0)+1)/2 and (7^(2^1)+1)/2 are not primes, but (7^(2^2)+1)/2 = 1201 is prime.
a(14) = 1 since 14^(2^0)+1 is not prime, but 14^(2^1)+1 = 197 is prime.

Eric Chen, Table of n, a(n) for n = 2..1500 status (for the -1s, only a(n) for n in A070265 are proved, all other -1s are only conjectured)
Gary Barnes, List of generalized Fermat primes in even bases up to 1030
Eric Chen, List of generalized Fermat primes in bases up to 1000
Chris Caldwell, Generalized Fermat number
Richard Fischer, List of generalized Fermat primes in odd bases
Yves Gallot, Generalized Fermat prime search
Wilfrid Keller, Factorization of GFN(n,2)
Wilfrid Keller, Factorization of GFN(n,3)
Wilfrid Keller, Factorization of GFN(n,5)
Wilfrid Keller, Factorization of GFN(n,6)
Wilfrid Keller, Factorization of GFN(n,10)
Wilfrid Keller, Factorization of GFN(n,12)
Jeppe Stig Salling Nielsen, List of generalized Fermat primes in even bases up to 1000
MathWorld, Generalized Fermat number
OEIS wiki, Generalized Fermat number
Wikipedia, Generalized Fermat number

Cf. A079706, A228101, A084712, A123669, A058064, A057856, A130536, A080121, A077659, A122900.

Table[k=0; While[p=If[EvenQ[n], (2n)^(2^k)+1, ((2n)^(2^k)+1)/2]; k<12 && !PrimeQ[p], k=k+1]; If[k==12, -1, k], {n, 2, 1500}]

f(n) = for(k=0, 11, if(ispseudoprime(n^(2^k)+1), return(k))); -1
g(n) = for(k=0, 11, if(ispseudoprime((n^(2^k)+1)/2), return(k))); -1
a(n) = if(n%2==0, f(n), g(n))

f(n,k)=if(n%2, (n^(2^k)+1)/2, n^(2^k)+1)
a(n)=if(ispower(-n), -1, my(k); while(!ispseudoprime(f(n,k)), k++); k) \\ Charles R Greathouse IV, Apr 20 2015

a(2n) = A228101(n) = log_2(A079706(n)).

a(A006093(n)) = 0, a(A076274(n)) = 0, a(A070265(n)) = -1.

A375888 Rectangular array: row n shows all k such that n is the number of rises in the trajectory of k in the Collatz problem.

Original entry on oeis.org

1, 2, 5, 4, 10, 3, 8, 20, 6, 17, 16, 21, 12, 34, 11, 32, 40, 13, 35, 22, 7, 64, 42, 24, 68, 23, 14, 9, 128, 80, 26, 69, 44, 15, 18, 25, 256, 84, 48, 70, 45, 28, 19, 49, 33, 512, 85, 52, 75, 46, 29, 36, 50, 65, 43, 1024, 160, 53, 136, 88, 30, 37, 51, 66, 86, 57

Offset: 0

Clark Kimberling, Sep 11 2024

Assuming that the Collatz conjecture (also known as the 3x+1 conjecture) is true, this is a permutation of the positive integers; viz., every positive integer occurs exactly once. Conjecture: every row contains a pair of consecutive integers.

			Corner:
   1     2     4     8    16    32    64   128   256   512  1024
   5    10    20    21    40    42    80    84    85   160   168
   3     6    12    13    24    26    48    52    53    96   104
  17    34    35    68    69    70    75   136   138   140   141
  11    22    23    44    45    46    88    90    92    93   176
   7    14    15    28    29    30    56    58    60    61   112
   9    18    19    36    37    38    72    74    76    77    81
6 is in row 2 because the trajectory, (6, 3, 10, 5, 16, 4, 2, 1), has exactly 2 rises: 3 to 10, and 5 to 16.

Index entries for sequences related to 3x+1 (or Collatz) problem

Cf. A000027, A000079 (row 1), A092893 (column 1), A006667, A070265, A078719.

Cf. A354236.

t = Table[Count[Differences[NestWhileList[If[EvenQ[#], #/2, 3 # + 1] &, n, # > 1 &]], ? Positive], {n, 2048}]; (* after _Harvey P. Dale, A006667 *)
r[n_] := Flatten[Position[t, n - 1]];
Column[Table[r[n], {n, 1, 21}]] (* array *)
u = Table[r[k][[n + 1 - k]], {n, 1, 12}, {k, 1, n}]
Flatten[u] (* sequence *)

Transpose of the array in A354236.

cp's OEIS Frontend

A097054 Nonsquare perfect powers.

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A126032 Numbers of the form b^m/2 for even b and odd m > 2.

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A228101 a(n) is the least k such that (2n)^(2^k) + 1 is a prime; a(n) = -1 if a prime (2n)^(2^k) + 1 is unknown, or = -2 if impossible.

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A253242 Least k>=0 such that n^(2^k)+1 is prime (for even n), or (n^(2^k)+1)/2 is prime (for odd n); -1 if no such k exists.

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A375888 Rectangular array: row n shows all k such that n is the number of rises in the trajectory of k in the Collatz problem.

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