A019434 -id:A019434 - cp's OEIS frontend

A102634 Numbers k such that 2^k + 13 is prime.

2, 4, 8, 20, 38, 64, 80, 292, 1132, 4108, 19934, 125278, 175628, 282184

Offset: 1

Lei Zhou, Jan 20 2005

If k is odd, then 2^k + 13 is divisible by 3. - Robert G. Wilson v, Jan 24 2005
a(15) > 5*10^5. - Robert Price, Aug 15 2015
For k in this sequence, the number 2^(k-1)*(2^k+13) has deficiency 14, cf. A141550. - M. F. Hasler, Jul 18 2016

			2^2+13 = 17 is prime.
2^4+13 = 29 is prime.
2^3+13 = 21 is not prime.

H. Lifchitz and R. Lifchitz (Editors), PRP Top Records, of the form 2^n+13.
Lei Zhou, Between 2^n and primes [broken link].

Cf. A019434 (2^k+1), A057732 (2^k+3), A059242 (2^k+5), A057195 (2^k+7), A057196 (2^k+9), A102633 (2^k+11), A102634 (this), A057197 (2^k+15), A057200 (2^k+17), A057221 (2^k+19), A057201 (2^k+21), A057203 (2^k+23).

Do[m = n; If[PrimeQ[2^n + 13], Print[n]], {n, 2, 19125, 2}] (* Robert G. Wilson v, Jan 24 2005 *)

first(m)=my(v=vector(m),r=1);for(i=1,m,while(!isprime(2^r + 13),r++);v[i]=r;r++);v; \\ Anders Hellström, Aug 15 2015

a(n) = 2*A253772(n). - Elmo R. Oliveira, Nov 12 2023

a(10) from Robert G. Wilson v, Jan 24 2005

a(11)-a(14) from Robert Price, Aug 15 2015

A125045 Odd primes generated recursively: a(1) = 3, a(n) = Min {p is prime; p divides Q+2}, where Q is the product of previous terms in the sequence.

Original entry on oeis.org

3, 5, 17, 257, 65537, 641, 7, 318811, 19, 1747, 12791, 73, 90679, 67, 59, 113, 13, 41, 47, 151, 131, 1301297155768795368671, 20921, 1514878040967313829436066877903, 5514151389810781513, 283, 1063, 3027041, 29, 24040758847310589568111822987, 154351, 89

Offset: 1

Nick Hobson, Nov 18 2006

nonn

The first five terms comprise the known Fermat primes: A019434.

			a(7) = 7 is the smallest prime divisor of 3 * 5 * 17 * 257 * 65537 * 641 + 2 = 2753074036097 = 7 * 11 * 37 * 966329953.

Sean A. Irvine, Table of n, a(n) for n = 1..64

Cf. A000945, A019434, A057204-A057208, A051308-A051335, A124984-A124993, A125037-A125045.

a={3}; q=1;
For[n=2,n<=20,n++,
    q=q*Last[a];
    AppendTo[a,Min[FactorInteger[q+2][[All,1]]]];
    ];
a (* Robert Price, Jul 16 2015 *)

A057201 Numbers k such that 2^k + 21 is prime.

Original entry on oeis.org

1, 3, 4, 5, 7, 8, 11, 15, 16, 19, 44, 48, 51, 52, 61, 163, 196, 456, 492, 911, 997, 1616, 1631, 1647, 1803, 1899, 3112, 3584, 3956, 6848, 7023, 9535, 16657, 27035, 33843, 36551, 38859, 81485, 107287, 131383, 139476, 158497, 210061, 216752, 339168, 341355, 376731, 1173095

Offset: 1

Robert G. Wilson v, Sep 16 2000

nonn

a(48) > 5*10^5. - Robert Price, Sep 17 2015

			k = 15, 2^15 + 21 = 32789 is prime.
k = 16, 2^16 + 21 = 65557 is prime.

Henri Lifchitz and Renaud Lifchitz, Search for 2^n+21, PRP Top Records.

Cf. A094076.

Cf. A019434 (primes 2^k+1), A057732 (2^k+3), A059242 (2^k+5), A057195 (2^k+7), A057196 (2^k+9), A102633 (2^k+11), A102634 (2^k+13), A057197 (2^k+15), A057200 (2^k+17), A057221 (2^k+19), this sequence (2^k+21), A057203 (2^k+23).

[n: n in [0..1000] | IsPrime(2^n+21)]; // Vincenzo Librandi, Aug 28 2015

Do[ If[ PrimeQ[ 2^n + 21 ], Print[ n ] ], {n, 1, 4000} ]
Select[Range[10000], PrimeQ[2^# + 21] &] (* Vincenzo Librandi, Aug 28 2015 *)

is(n)=isprime(2^n+21) \\ Charles R Greathouse IV, Feb 17 2017

a(30)-a(47) from Robert Price, Dec 06 2013

a(48) found by Stefano Morozzi, added by Elmo R. Oliveira, Nov 25 2023

A057203 Numbers k such that 2^k + 23 is prime.

Original entry on oeis.org

3, 7, 39, 79, 359, 451, 1031, 1039, 11311, 30227, 47599, 55731, 307099, 351831, 418851

Offset: 1

Robert G. Wilson v, Sep 16 2000

a(16) > 5*10^5. - Robert Price, Sep 06 2015

All terms are odd. - Elmo R. Oliveira, Dec 01 2023

			For k = 39, 2^39 + 23 = 549755813911 is prime.

Henri Lifchitz and Renaud Lifchitz, Search for 2^n+23, PRP Top Records.

Cf. A094076.

Cf. A019434 (primes 2^k+1), A057732 (2^k+3), A059242 (2^k+5), A057195 (2^k+7), A057196(2^k+9), A102633 (2^k+11), A102634 (2^k+13), A057197 (2^k+15), A057200 (2^k+17), A057221 (2^k+19), A057201 (2^k+21), this sequence (2^k+23).

Do[ If[ PrimeQ[2^n + 23], Print[ n ]], {n, 1, 5000} ]

is(n)=isprime(2^n+23) \\ Charles R Greathouse IV, Feb 17 2017

a(9)-a(15) from Robert Price, Sep 06 2015

A057221 Numbers k such that 2^k + 19 is prime.

Original entry on oeis.org

2, 6, 30, 162, 654, 714, 1370, 1662, 1722, 2810, 77142, 156254, 432974, 1092242, 1245230

Offset: 1

Robert G. Wilson v, Sep 17 2000

a(14) > 5*10^5. - Robert Price, Aug 27 2015
All terms are even. - Robert Israel, Aug 28 2015
For numbers k in this sequence, the number 2^(k-1)*(2^k+19) has deficiency 20 (see A223607). - M. F. Hasler, Jul 18 2016

Henri Lifchitz and Renaud Lifchitz (Editors), Search for 2^n+19, PRP Top Records.

Cf. A223607, A253774.

Cf. A019434 (primes 2^k+1), A057732 (2^k+3), A059242 (2^k+5), A057195 (2^k+7), A057196(2^k+9), A102633 (2^k+11), A102634 (2^k+13), A057197 (2^k+15), A057200 (2^k+17), this sequence (2^k+19), A057201 (2^k+21), A057203 (2^k+23).

[n: n in [0..1000] | IsPrime(2^n+19)]; // Vincenzo Librandi, Aug 28 2015

select(n -> isprime(2^n+19), [seq(2*i,i=1..10000)]); # Robert Israel, Aug 28 2015

Do[ If[ PrimeQ[ 2^n + 19 ], Print[ n ] ], {n, 1, 15000} ]
Select[Range[10000], PrimeQ[2^# + 19] &] (* Vincenzo Librandi, Aug 28 2015 *)

for(n=1,oo,ispseudoprime(2^n+19)&&print1(n",")) \\ M. F. Hasler, Jul 18 2016

a(n) = 2*A253774(n). - Joerg Arndt, Aug 28 2015

a(11)-a(13) from Robert Price, Aug 27 2015
Edited by M. F. Hasler, Jul 18 2016
a(14)-a(15) found by Stefano Morozzi, added by Elmo R. Oliveira, Nov 19 2023

A001494 Numbers k such that phi(k) = phi(k+2).

Original entry on oeis.org

4, 7, 8, 10, 26, 32, 70, 74, 122, 146, 308, 314, 386, 512, 554, 572, 626, 635, 728, 794, 842, 910, 914, 1015, 1082, 1226, 1322, 1330, 1346, 1466, 1514, 1608, 1754, 1994, 2132, 2170, 2186, 2306, 2402, 2426, 2474, 2590, 2642, 2695, 2762, 2906, 3242, 3314

Offset: 1

N. J. A. Sloane

If p and 2p-1 are odd primes then 2*(2p-1) is a solution of the equation. Other terms (7,8,32,70,...) are not of this form.
There are 506764111 terms under 10^12. - Jud McCranie, Feb 13 2012
If 2^(2^m) + 1 is a Fermat prime in A019434, so, m = 0, 1, 2, 3, 4, then k = 2^(2^m + 1) is a term; this subsequence consists of {4, 8, 32, 512, 131072} and, in this case, phi(k) = phi(k+2) = 2^(2^m). - Bernard Schott, Apr 22 2022

D. M. Burton, Elementary Number Theory, section 7-2.
R. K. Guy, Unsolved Problems Number Theory, Sect. B36.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Jud McCranie, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
Kevin Ford, Solutions of phi(n)=phi(n+k) and sigma(n)=sigma(n+k), arXiv:2002.12155 [math.NT], 2020.
M. F. Hasler, Table of n, a(n) for n = 1..17286. (Terms up to 10^7.)
V. L. Klee, Jr., Some remarks on Euler's totient function, Amer. Math. Monthly, 54 (1947), 332.
Leo Moser, Some equations involving Euler's totient function, Amer. Math. Monthly, 56 (1949), 22-23.

Cf. A000010, A001274, A007015, A179186, A179187, A179188, A179189, A179202, A217139.

import Data.List (elemIndices)
a001494 n = a001494_list !! (n-1)
a001494_list = map (+ 1) $ elemIndices 0 $
               zipWith (-) (drop 2 a000010_list) a000010_list
-- Reinhard Zumkeller, Feb 08 2013

[n: n in [1..4000] | EulerPhi(n) eq EulerPhi(n+2)]; // Vincenzo Librandi, Sep 07 2016

Select[Range[3500], EulerPhi[#]==EulerPhi[#+2]&] (* Harvey P. Dale, Apr 24 2011 *)
Flatten[Position[Partition[EulerPhi[Range[3500]],3,1],?(#[[1]]==#[[3]]&),{1},Heads->False]] (* This program is more efficient than the first program above because it only has to calculate phi of each number once. *) (* _Harvey P. Dale, Aug 20 2014 *)
SequencePosition[EulerPhi[Range[4300]],{x_,,x}][[All,1]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Sep 04 2020 *)

op=[0,c=0]; for( n=1,1e7,if( op[bittest(n,0)+1]+0==op[bittest(n,0)+1]=eulerphi(n), write("b001494.txt",c++," "n-2))) \\ M. F. Hasler, Jan 05 2011

A000010(a(n)) = A000010(a(n) + 2). - Reinhard Zumkeller, Feb 08 2013

More terms from Jud McCranie, Dec 24 1999

A064896 Numbers of the form (2^(m*r)-1)/(2^r-1) for positive integers m, r.

Original entry on oeis.org

1, 3, 5, 7, 9, 15, 17, 21, 31, 33, 63, 65, 73, 85, 127, 129, 255, 257, 273, 341, 511, 513, 585, 1023, 1025, 1057, 1365, 2047, 2049, 4095, 4097, 4161, 4369, 4681, 5461, 8191, 8193, 16383, 16385, 16513, 21845, 32767, 32769, 33825, 37449, 65535, 65537

Offset: 1

Marc LeBrun, Oct 11 2001

Binary expansion of n consists of single 1's diluted by (possibly empty) equal-sized blocks of 0's.
According to Stolarsky's Theorem 2.1, all numbers in this sequence are sturdy numbers; this sequence is a subsequence of A125121. - T. D. Noe, Jul 21 2008
These are the numbers k > 0 for which k + 2^m = k*2^n + 1 has a solution m,n > 0. For k > 1, these are numbers k such that (k - 2^x)*2^y + 1 = k has a solution in positive integers x,y. In other words, (k - 1)/(k - 2^x) = 2^y for some x,y > 0. If t = (2^m - 1)/(2^n - 1) is a term of this sequence (i.e. if and only if n|m), then t' = t + 2^m = t*2^n + 1 is also a term. Primes in this sequence (A245730) include: all Mersenne primes (A000668), all Fermat primes (A019434), and other primes (73, 262657, 4432676798593, ...). - Thomas Ordowski, Feb 14 2024

			73 is included because it is 1001001 in binary, whose 1's are diluted by blocks of two 0's.

T. D. Noe, Table of n, a(n) for n=1..1000
T. Chinburg and M. Henriksen, Sums of k-th powers in the ring of polynomials with integer coefficients, Acta Arithmetica, 29 (1976), 227-250.
K. B. Stolarsky, Integers whose multiples have anomalous digital frequencies, Acta Arithmetica, 38 (1980), 117-128.

Cf. A064894, A056538.
Cf. A076270 (k=3), A076275 (k=4), A076284 (k=5), A076285 (k=6), A076286 (k=7), A076287 (k=8), A076288 (k=9), A076289 (k=10).
Primes in this sequence: A245730.

f := proc(p) local m,r,t1; t1 := {}; for m from 1 to 10 do for r from 1 to 10 do t1 := {op(t1), (p^(m*r)-1)/(p^r-1)}; od: od: sort(convert(t1,list)); end; f(2); # very crude!
# Alternative:
N:= 10^6: # to get all terms <= N
A:= sort(convert({1,seq(seq((2^(m*r)-1)/(2^r-1),m=2..1/r*ilog2(N*(2^r-1)+1)),r=1..ilog2(N-1))},list)); # Robert Israel, Jun 12 2015

lista(nn) = {v = [1]; x = (2^nn-1); for (m=2, nn, r = 1; while ((y = (2^(m*r)-1)/(2^r-1)) <=x, v = Set(concat(v, y)); r++);); v;} \\ Michel Marcus, Jun 12 2015

A064894(a(n)) = A056538(n).

A334100 Square array where the row n lists all numbers k for which A329697(k) = n, read by falling antidiagonals.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 6, 9, 19, 16, 10, 11, 21, 43, 32, 12, 13, 23, 47, 127, 64, 17, 14, 27, 49, 129, 283, 128, 20, 15, 29, 57, 133, 301, 659, 256, 24, 18, 31, 59, 139, 329, 817, 1319, 512, 34, 22, 33, 63, 141, 343, 827, 1699, 3957, 1024, 40, 25, 35, 67, 147, 347, 839, 1787, 4079, 9227, 2048, 48, 26, 37, 69, 161, 361, 849, 1849, 4613, 9233, 21599

Offset: 1

Antti Karttunen, Apr 14 2020

Array is read by descending antidiagonals with (n,k) = (0,1), (0,2), (1,1), (0,3), (1,2), (2,1), ... where A(n,k) is the k-th solution x to A329697(x) = n. The row indexing (n) starts from 0, and column indexing (k) from 1.
Any odd prime that appears on row n is 1+{some term on row n-1}.
The e-th powers of the terms on row n form a subset of terms on row (e*n). More generally, a product of terms that occur on rows i_1, i_2, ..., i_k can be found at row (i_1 + i_2 + ... + i_k), because A329697 is completely additive.
The binary weight (A000120) of any term on row n is at most 2^n.

			The top left corner of the array:
  n\k |    1     2     3     4     5     6     7     8     9    10
------+----------------------------------------------------------------
   0  |    1,    2,    4,    8,   16,   32,   64,  128,  256,  512, ...
   1  |    3,    5,    6,   10,   12,   17,   20,   24,   34,   40, ...
   2  |    7,    9,   11,   13,   14,   15,   18,   22,   25,   26, ...
   3  |   19,   21,   23,   27,   29,   31,   33,   35,   37,   38, ...
   4  |   43,   47,   49,   57,   59,   63,   67,   69,   71,   77, ...
   5  |  127,  129,  133,  139,  141,  147,  161,  163,  171,  173, ...
   6  |  283,  301,  329,  343,  347,  361,  379,  381,  383,  387, ...
   7  |  659,  817,  827,  839,  849,  863,  883,  889,  893,  903, ...
   8  | 1319, 1699, 1787, 1849, 1977, 1979, 1981, 2021, 2039, 2083, ...
   9  | 3957, 4079, 4613, 4903, 5097, 5179, 5361, 5377, 5399, 5419, ...
etc.
Note that the row 9 is the first one which begins with composite, as 3957 = 3*1319. The next such rows are row 15 and row 22. See A334099.

Index entries for sequences that are permutations of the natural numbers

Cf. A329697.
Cf. A334099 (the leftmost column).
Cf. A000079, A334101, A334102, A334103, A334104, A334105, A334106 for the rows 0-6.
Cf. A019434, A334092, A334093, A334094, A334095, A334096 for the primes on the rows 1-6.
Cf. also irregular triangle A334111.

Block[{nn = 16, s}, s = Values@ PositionIndex@ Array[-1 + Length@ NestWhileList[# - #/FactorInteger[#][[-1, 1]] &, #, # != 2^IntegerExponent[#, 2] &] &, 2^nn]; Table[s[[#, k]] &[m - k + 1], {m, nn - Ceiling[nn/4]}, {k, m, 1, -1}]] // Flatten (* Michael De Vlieger, Apr 30 2020 *)

up_to = 105; \\ up_to = 1081; \\ = binomial(46+1,2)
A329697(n) = if(!bitand(n,n-1),0,1+A329697(n-(n/vecmax(factor(n)[, 1]))));
memoA334100sq = Map();
A334100sq(n, k) = { my(v=0); if(!mapisdefined(memoA334100sq,[n,k-1],&v),if(1==k, v=0, v = A334100sq(n, k-1))); for(i=1+v,oo,if(A329697(i)==(n-1),mapput(memoA334100sq,[n,k],i); return(i))); };
A334100list(up_to) = { my(v = vector(up_to), i=0); for(a=1,oo, for(col=1,a, i++; if(i > up_to, return(v)); v[i] = A334100sq(col,(a-(col-1))))); (v); };
v334100 = A334100list(up_to);
A334100(n) = v334100[n];

A002586 Smallest prime factor of 2^n + 1.

Original entry on oeis.org

3, 5, 3, 17, 3, 5, 3, 257, 3, 5, 3, 17, 3, 5, 3, 65537, 3, 5, 3, 17, 3, 5, 3, 97, 3, 5, 3, 17, 3, 5, 3, 641, 3, 5, 3, 17, 3, 5, 3, 257, 3, 5, 3, 17, 3, 5, 3, 193, 3, 5, 3, 17, 3, 5, 3, 257, 3, 5, 3, 17, 3, 5, 3, 274177, 3, 5, 3, 17, 3, 5, 3, 97, 3, 5, 3, 17, 3, 5, 3, 65537, 3, 5, 3, 17, 3, 5

Offset: 1

N. J. A. Sloane

Conjecture: a(8+48*k) = 257 and a(40+48*k) = 257, where k is a nonnegative integer. - Thomas König, Feb 15 2017
Conjecture is true: 257 divides 2^(8+48*k)+1 and 2^(40+48*k)+1 but no prime < 257 ever does. Similarly, a(24+48*k) = 97. - Robert Israel, Feb 17 2017
From Robert Israel, Feb 17 2017: (Start)
If a(n) = p, there is some m such that a(n+m*j*n) = p for all j.
In particular, every member of the sequence occurs infinitely often.
a(k*n) <= a(n) for any odd k. (End)

			a(2^k) = 3, 5, 17, 257, 65537 is the k-th Fermat prime 2^(2^k) + 1 = A019434(k) for k = 0, 1, 2, 3, 4. - _Jonathan Sondow_, Nov 28 2012

J. Brillhart et al., Factorizations of b^n +- 1. Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 2nd edition, 1985; and later supplements.
M. Kraitchik, Recherches sur la Théorie des Nombres, Gauthiers-Villars, Paris, Vol. 1, 1924, Vol. 2, 1929, see Vol. 2, p. 85.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Max Alekseyev, Table of n, a(n) for n = 1..3967 (first 500 terms from T. D. Noe)
J. Brillhart et al., Factorizations of b^n +- 1, Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 3rd edition, 2002.
Thomas König, C program using gmp for testing the conjectures that a(8+k*48) = 257 and a(40+k*48) = 257
E. Lucas, Théorie des Fonctions Numériques Simplement Périodiques, I", Amer. J. Math., 1 (1878), 184-240, 289-321. See pages 239 and 240.
Edouard Lucas, The Theory of Simply Periodic Numerical Functions, Fibonacci Association, 1969. English translation of article "Théorie des Fonctions Numériques Simplement Périodiques, I", Amer. J. Math., 1 (1878), 184-240.
S. S. Wagstaff, Jr., The Cunningham Project

Cf. A000215, A001269, A002587, A019434, A050922, A093179.

f[n_] := FactorInteger[2^n + 1][[1, 1]]; Array[f, 100] (* Robert G. Wilson v, Nov 28 2012 *)
FactorInteger[#][[1,1]]&/@(2^Range[90]+1) (* Harvey P. Dale, Jul 25 2024 *)

a(n) = my(m=n%8); if(m, [3, 5, 3, 17, 3, 5, 3][m], factor(2^n+1)[1,1]); \\ Ruud H.G. van Tol, Feb 16 2024

from sympy import primefactors
smallest_primef = []
for n in range(1,87):
    y = (2 ** n) + 1
    smallest_primef.append(min(primefactors(y)))
print(smallest_primef) # Adrienne Leonardo, Dec 29 2024

a(n) = 3, 5, 3, 17, 3, 5, 3 for n == 1, 2, 3, 4, 5, 6, 7 (mod 8). (Proof. Let n = k*odd with k = 1, 2, or 4. As 2^k = 2, 4, 16 == -1 (mod 3, 5, 17), we get 2^n + 1 = 2^(k*odd) + 1 = (2^k)^odd + 1 == (-1)^odd + 1 == 0 (mod 3, 5, 17). Finally, 2^n + 1 !== 0 (mod p) for prime p < 3, 5, 17, respectively.) - Jonathan Sondow, Nov 28 2012

More terms from James Sellers, Jul 06 2000

Definition corrected by Jonathan Sondow, Nov 27 2012

A004729 Divisors of 2^32 - 1 (for a(1) to a(31), the 31 regular polygons with an odd number of sides constructible with ruler and compass).

Original entry on oeis.org

1, 3, 5, 15, 17, 51, 85, 255, 257, 771, 1285, 3855, 4369, 13107, 21845, 65535, 65537, 196611, 327685, 983055, 1114129, 3342387, 5570645, 16711935, 16843009, 50529027, 84215045, 252645135, 286331153, 858993459, 1431655765, 4294967295

Offset: 0

N. J. A. Sloane

The 32 divisors of the product of the 5 known Fermat primes.
The only known odd numbers whose totient is a power of 2. - Labos Elemer, Dec 06 2000
Equals first 32 members of A001317. Also, equals first 32 members of A053576. - Omar E. Pol, Dec 10 2008
Omitting the first term a(0)=1 gives A045544 (the number of sides of constructible odd-sided regular polygons.)

J. H. Conway and R. K. Guy, The Book of Numbers, Copernicus Press, New York, 1996; see p. 140.

Eric Weisstein's World of Mathematics, Regular Polygon, Sierpiński Sieve, Constructible Polygon
OEIS Wiki, Constructible odd-sided polygons
OEIS Wiki, Sierpinski's triangle
Index entries for sequences related to divisors of numbers

Essentially same as A045544.

Cf. A000010, A000215, A001317, A003401, A003527, A004169, A004729, A019434, A045544, A047999, A053576, A054432.

Mathematica
```
Divisors[2^32-1]
```
PARI
```
divisors(1<<32-1)
```

Edited by Daniel Forgues, Jun 17 2011

cp's OEIS Frontend

A102634 Numbers k such that 2^k + 13 is prime.

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A125045 Odd primes generated recursively: a(1) = 3, a(n) = Min {p is prime; p divides Q+2}, where Q is the product of previous terms in the sequence.

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A057201 Numbers k such that 2^k + 21 is prime.

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A057203 Numbers k such that 2^k + 23 is prime.

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A057221 Numbers k such that 2^k + 19 is prime.

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A001494 Numbers k such that phi(k) = phi(k+2).

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A064896 Numbers of the form (2^(m*r)-1)/(2^r-1) for positive integers m, r.

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