A006093 -id:A006093 - cp's OEIS frontend

A061286 Smallest integer for which the number of divisors is the n-th prime.

2, 4, 16, 64, 1024, 4096, 65536, 262144, 4194304, 268435456, 1073741824, 68719476736, 1099511627776, 4398046511104, 70368744177664, 4503599627370496, 288230376151711744, 1152921504606846976

Offset: 1

Labos Elemer, May 22 2001

Seems to be the same as "Even numbers with prime number of divisors" - Jason Earls, Jul 04 2001

Except for the first term, smallest number == 1 (mod prime(n)) having n divisors (by Fermat's little theorem). - Amarnath Murthy and Meenakshi Srikanth (menakan_s(AT)yahoo.com), Jun 20 2003

Karl V. Keller, Jr., Table of n, a(n) for n = 1..460
Index to divisibility sequences

Cf. A000005, A005179, A003680, A061283, A061286, A006093, A005097, A006254, A119523, A196202.

Table[2^(p-1),{p,Table[Prime[n],{n,1,18}]}] (* Geoffrey Critzer, May 26 2013 *)

forstep(n=2,100000000,2,x=numdiv(n); if(isprime(x),print(n)))

a(n)=2^(prime(n)-1) \\ Charles R Greathouse IV, Apr 08 2012

from sympy import isprime, divisor_count as tau
[2] + [2**(2*n) for n in range(1, 33) if isprime(tau(2**(2*n)))] # Karl V. Keller, Jr., Jul 10 2020

a(n) = 2^(prime(n)-1) = 2^A006093(n).
a(n) = A005179(prime(n)). - R. J. Mathar, Aug 09 2019
Sum_{n>=1} 1/a(n) = A119523. - Amiram Eldar, Aug 11 2020

A068361 Numbers n such that the number of squarefree numbers between prime(n) and prime(n+1) = prime(n+1)-prime(n)-1.

Original entry on oeis.org

1, 3, 10, 13, 26, 33, 60, 89, 104, 113, 116, 142, 148, 201, 209, 212, 234, 265, 268, 288, 313, 320, 332, 343, 353, 384, 398, 408, 477, 484, 498, 542, 545, 551, 577, 581, 601, 625, 636, 671, 719, 723, 726, 745, 794, 805, 815, 862, 864, 884, 944, 964, 995, 1054

Offset: 1

Benoit Cloitre, Feb 28 2002

nonn

Also numbers k such that all numbers from prime(k) to prime(k+1) are squarefree. All such primes are twins, so this is a subset of A029707. The other twin primes are A061368. - Gus Wiseman, Dec 11 2024

Amiram Eldar, Table of n, a(n) for n = 1..10000

A subset of A029707 (lesser index of twin primes).
Prime index of each (prime) term of A061351.
Positions of zeros in A061399.
For perfect power instead of squarefree we have A377436, zeros of A377432.
Positions of zeros in A377784.
The rest of the twin primes are at A378620, indices of A061368.
A000040 lists the primes, differences A001223, (run-lengths A333254, A373821).
A005117 lists the squarefree numbers, differences A076259.
A006562 finds balanced primes.
A013929 lists the nonsquarefree numbers, differences A078147.
A014574 is the intersection of A006093 and A008864.
A038664 locates the first prime gap of size 2n.
A046933 counts composite numbers between primes.
A061398 counts squarefree numbers between primes, zeros A068360.
A120327 gives the least nonsquarefree number >= n.
Cf. A000720, A007674, A013928, A057627, A070321, A071403, A072284, A073247, A112925, A122535, A155752, A224363, A240473, A251092.

Select[Range[100],And@@SquareFreeQ/@Range[Prime[#],Prime[#+1]]&] (* Gus Wiseman, Dec 11 2024 *)

isok(n) = for (k=prime(n)+1, prime(n+1)-1, if (!issquarefree(k), return (0))); 1; \\ Michel Marcus, Apr 29 2016

n such that A061398(n) = prime(n+1)-prime(n)-1.

prime(a(n)) = A061351(n). - Gus Wiseman, Dec 11 2024

A377436 Numbers k such that there is no perfect-power x in the range prime(k) < x < prime(k+1).

Original entry on oeis.org

1, 3, 5, 7, 8, 10, 12, 13, 14, 16, 17, 19, 20, 21, 23, 24, 26, 27, 28, 29, 32, 33, 35, 36, 37, 38, 40, 41, 42, 43, 45, 46, 49, 50, 51, 52, 55, 56, 57, 58, 59, 60, 62, 63, 64, 65, 67, 69, 70, 71, 73, 74, 75, 76, 77, 79, 80, 81, 82, 83, 84, 86, 87, 88, 89, 90

Offset: 1

Gus Wiseman, Nov 02 2024

nonn

Perfect-powers (A001597) are numbers with a proper integer root, complement A007916.

			Primes 8 and 9 are 19 and 23, and the interval (20,21,22) contains no prime-powers, so 8 is in the sequence.

For powers of 2 instead of primes see A377467, A013597, A014210, A014234, A244508.
For squarefree instead of perfect-power we have A068360, see A061398, A377430, A377431.
For just squares (instead of all perfect-powers) we have A221056, primes A224363.
For prime-powers (instead of perfect-powers) we have A377286.
These are the positions of 0 in A377432.
For one instead of none we have A377434, for prime-powers A377287.
For two instead of none we have A377466, for prime-powers A377288, primes A053706.
A000015 gives the least prime-power >= n.
A000040 lists the primes, differences A001223.
A000961 lists the powers of primes, differences A057820.
A046933 counts the interval from A008864(n) to A006093(n+1).
A065514 gives the nearest prime-power before prime(n)-1, difference A377289.
A080101 and A366833 count prime-powers between primes, see A377057, A053607, A304521.
A081676 gives the nearest perfect-power up to n.
A246655 lists the prime-powers not including 1, complement A361102.
A377468 gives the nearest perfect-power after n.
Cf. A001597, A002808, A024619, A053707, A064113, A065890, A075526, A276781, A345531, A376597, A377054, A377282.

perpowQ[n_]:=n==1||GCD@@FactorInteger[n][[All,2]]>1;
Select[Range[100],Length[Select[Range[Prime[#]+1, Prime[#+1]-1],perpowQ]]==0&]

A005278 Noncototients: numbers k such that x - phi(x) = k has no solution.

Original entry on oeis.org

10, 26, 34, 50, 52, 58, 86, 100, 116, 122, 130, 134, 146, 154, 170, 172, 186, 202, 206, 218, 222, 232, 244, 260, 266, 268, 274, 290, 292, 298, 310, 326, 340, 344, 346, 362, 366, 372, 386, 394, 404, 412, 436, 466, 470, 474, 482, 490, 518, 520

Offset: 1

N. J. A. Sloane

Browkin & Schinzel show that this sequence is infinite. - Labos Elemer, Dec 21 1999
If the strong Goldbach conjecture (every even number > 6 is the sum of at least 2 distinct primes p and q) is true, the sequence contains only even values, since p*q - phi(p*q) = p+q-1 and then every odd number can be expressed as x-phi(x). - Benoit Cloitre, Mar 03 2002
Browkin & Schinzel and Hee-sung Yang (Myerson link, problem 012.17d) ask if this sequence has a positive lower density. - Charles R Greathouse IV, Nov 04 2013
From Amiram Eldar, Feb 13 2021: (Start)
Sierpiński (1959) asked if this sequence is infinite.
Erdős (1973) asked if this sequence has a positive lower density.
Browkin and Schinzel (1995) proved that 509203*2^k is a term for all k>=1.
Flammenkamp and Luca (2000) proved that 509203 can be replaced with any other term of A263958 (and found 6 more terms of A263958).
Banks and Luca (2004) proved that the relative density of primes p within the sequence of primes such that 2*p is noncototient is 1. (End)

Richard K. Guy, Unsolved Problems in Number Theory, 3rd ed., Springer, 2004, section B36, pp. 138-142.
Wacław Sierpiński, Number Theory, Part II, PWN Warszawa, 1959 (in Polish).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Donovan Johnson, Table of n, a(n) for n = 1..10000 (first 963 terms from T. D. Noe)
William D. Banks and Florian Luca, Noncototients and Nonaliquots, arXiv:math/0409231 [math.NT], 2004.
J. Browkin and A. Schinzel, On integers not of the form n-phi(n), Colloq. Math., Vol. 68 (1995), pp. 55-58.
Paul Erdős, Über die Zahlen der form sigma(n)-n und n-phi(n), (in German), Elem. Math., Vol. 28 (1973), pp. 83-86; alternative link.
Achim Flammenkamp and Florian Luca, Infinite families of noncototients, Colloq. Math., Vol. 86 (2000), pp. 37-41.
Aleksander Grytczuk and Barbara Medryk, On a result of Flammenkamp-Luca concerning noncototient sequence, Tsukuba Journal of Mathematics, Vol. 29, No. 2 (2005), pp. 533-538.
Gerry Myerson, Western Number Theory Problems, Dec 17 & 19 2012.
David Ng, Introduction to Noncototients, 2017.
Carl Pomerance and Hee-Sung Yang, On untouchable numbers and related problems, 2012.
Carl Pomerance and Hee-Sung Yang, Variant of a theorem of Erdős on the sum-of-proper-divisors function, Math. Comp., Vol. 83, No. 288 (2014), pp. 1903-1913; alternative link.
Eric Weisstein's World of Mathematics, Noncototient.

Cf. A006093, A126887, A263958. Complement of A051953.

Cf. A063740 (number of k such that cototient(k) = n).

nmax = 520; cototientQ[n_?EvenQ] := (x = n; While[test = x - EulerPhi[x] == n ; Not[test || x > 2*nmax], x++]; test); cototientQ[n_?OddQ] = True; Select[Range[nmax], !cototientQ[#]&] (* Jean-François Alcover, Jul 20 2011 *)

lista(nn)=v = vecsort(vector(nn^2, n, n - eulerphi(n)), ,8); for (n=1, nn, if (! vecsearch(v, n), print1(n, ", "))); \\ Michel Marcus, Oct 03 2016

{ k | A063740(k) = 0 }. - M. F. Hasler, Jan 11 2018

More terms from Jud McCranie, Jan 01 1997

A088362 Numbers k such that k^4096 + 1 is prime (a generalized Fermat prime).

Original entry on oeis.org

1, 1534, 7316, 17582, 18224, 28234, 34954, 41336, 48824, 51558, 51914, 57394, 61686, 62060, 89762, 96632, 98242, 100540, 101578, 109696, 110540, 114690, 125440, 125442, 127596, 138068, 144362, 154908, 157310, 161822, 161900, 166224

Offset: 1

Jeppe Stig Nielsen, Sep 27 2003

nonn

Jeppe Stig Nielsen, Table of n, a(n) for n = 1..10000
Yves Gallot, Generalized Fermat Prime Search.
Jeppe Stig Nielsen, All terms up to 2*10^9 (from a distributed search)
Jeppe Stig Nielsen, Generalized Fermat Primes sorted by base.
'stream', GFN-12 Final Statistic.
Eric Weisstein's World of Mathematics, Generalized Fermat Number.

Cf. A006093, A005574, A000068, A006314, A006313, A006315, A006316, A056994, A056995, A057465, A057002, A088361, A226528, A226529, A226530, A251597, A253854, A244150, A243959, A321323.

isA088362(n)=isprime(n^4096+1) \\ Michael B. Porter, Apr 22 2010

A072670 Number of ways to write n as i*j + i + j, 0 < i <= j.

Original entry on oeis.org

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

Offset: 0

Reinhard Zumkeller, Jun 30 2002

nonn

a(n) is the number of partitions of n+1 with summands in arithmetic progression having common difference 2. For example a(29)=3 because there are 3 partitions of 30 that are in arithmetic progressions: 2+4+6+8+10, 8+10+12 and 14+16. - N-E. Fahssi, Feb 01 2008
From Daniel Forgues, Sep 20 2011: (Start)
a(n) is the number of nontrivial factorizations of n+1, in two factors.
a(n) is the number of ways to write n+1 as i*j + i + j + 1 = (i+1)(j+1), 0 < i <= j. (End)
a(n) is the number of ways to write n+1 as i*j, 1 < i <= j. - Arkadiusz Wesolowski, Nov 18 2012
For a generalization, see comment in A260804. - Vladimir Shevelev, Aug 04 2015
Number of partitions of n into 3 parts whose largest part is equal to the product of the other two. - Wesley Ivan Hurt, Jan 04 2022

			a(11)=2: 11 = 1*5 + 1 + 5 = 2*3 + 2 + 3.
From _Daniel Forgues_, Sep 20 2011 (Start)
Number of nontrivial factorizations of n+1 in two factors:
  0 for the unit 1 and prime numbers
  1 for a square: n^2 = n*n
  1 for 6 (2*3), 10 (2*5), 14 (2*7), 15 (3*5)
  1 for a cube: n^3 = n*n^2
  2 for 12 (2*6, 3*4), for 18 (2*9, 3*6) (End)

Robert Israel, Table of n, a(n) for n = 0..10000
Joseph W. Andrushkiw, Roman I. Andrushkiw and Clifton E. Corzatt, Representations of Positive Integers as Sums of Arithmetic Progressions, Mathematics Magazine, Vol. 49, No. 5 (Nov., 1976), pp. 245-248.
M. A. Nyblom and C. Evans, On the enumeration of partitions with summands in arithmetic progression, Australian Journal of Combinatorics, Vol. 28 (2003), pp. 149-159.
Vladimir Shevelev, Representation of positive integers by the form x1...xk+x1+...+xk, arXiv:1508.03970 [math.NT], 2015.

Cf. A001620, A067432, A066938, A072668, A006093, A072671, A161840, A260803, A260804.

0, seq(ceil(numtheory:-tau(n+1)/2)-1, n=1..100); # Robert Israel, Aug 04 2015

p2[n_] := 1/2 (Length[Divisors[n]] - 2 + ((-1)^(Length[Divisors[n]] + 1) + 1)/2); Table[p2[n + 1], {n, 0, 104}] (* N-E. Fahssi, Feb 01 2008 *)
Table[Ceiling[DivisorSigma[0, n + 1]/2] - 1, {n, 0, 104}] (* Arkadiusz Wesolowski, Nov 18 2012 *)

is_ok(k,i,j)=0=i&&k===i*j+i+j;
first(m)=my(v=vector(m,z,0));for(l=1,m,for(j=1,l,for(i=1,j,if(is_ok(l,i,j),v[l]++))));concat([0],v); /* Anders Hellström, Aug 04 2015 */

a(n)=(numdiv(n+1)+issquare(n+1))/2-1 \\ Charles R Greathouse IV, Jul 14 2017

a(n) = A038548(n+1) - 1.
From N-E. Fahssi, Feb 01 2008: (Start)
a(n) = p2(n+1), where p2(n) = (1/2)*(d(n) - 2 + ((-1)^(d(n)+1)+1)/2); d(n) is the number of divisors of n: A000005.
G.f.: Sum_{n>=1} a(n) x^n = 1/x Sum_{k>=2} x^(k^2)/(1-x^k). (End)
lim_{n->infinity} a(A002110(n)-1) = infinity. - Vladimir Shevelev, Aug 04 2015
a(n) = A161840(n+1)/2. - Omar E. Pol, Feb 27 2019
Sum_{k=1..n} a(k) ~ n * (log(n) + 2*gamma - 3) / 2, where gamma is Euler's constant (A001620). - Amiram Eldar, Jan 14 2024

A088361 Numbers n such that n^2048 + 1 is prime (a generalized Fermat prime).

Original entry on oeis.org

1, 150, 2558, 4650, 4772, 11272, 13236, 15048, 23302, 26946, 29504, 31614, 33308, 35054, 36702, 37062, 39020, 39056, 43738, 44174, 45654, 46502, 47348, 49190, 49204, 49544, 54514, 57210, 59770, 61184, 66894, 68194, 70574, 72446, 82642

Offset: 1

Jeppe Stig Nielsen, Sep 27 2003

nonn

T. D. Noe, Table of n, a(n) for n = 1..1000 (from Yves Gallot)
Yves Gallot, Generalized Fermat Prime Search.
Jeppe Stig Nielsen, Generalized Fermat Primes sorted by base.
Eric Weisstein's World of Mathematics, Generalized Fermat Number.

Cf. A006093, A005574, A000068, A006314, A006313, A006315, A006316, A056994, A056995, A057465, A057002, A088362, A226528, A226529, A226530, A251597, A253854, A244150, A243959, A321323.

isA088361(n)=isprime(n^2048+1) \\ Michael B. Porter, Apr 23 2010

A377287 Numbers k such that there is exactly one prime-power between prime(k)+1 and prime(k+1)-1.

Original entry on oeis.org

2, 6, 11, 15, 18, 22, 31, 39, 53, 54, 61, 68, 72, 97, 99, 114, 129, 146, 162, 172, 217, 219, 263, 283, 309, 329, 357, 409, 445, 487, 519, 564, 609, 656, 675, 705, 811, 847, 882, 886, 1000, 1028, 1163, 1252, 1294, 1381, 1423, 1457, 1523, 1715, 1821, 1877, 1900

Offset: 1

Gus Wiseman, Oct 25 2024

nonn

			Primes 18 and 19 are 61 and 67, and the interval (62, 63, 64, 65, 66) contains only the one prime-power 64, so 18 is in the sequence.

The interval from A008864(n) to A006093(n+1) has A046933(n) elements.
For powers of 2 instead of primes see A013597, A014210, A014234, A244508, A304521.
The nearest prime-power before prime(n)-1 is A065514, difference A377289.
The nearest prime-power after prime(n)+1 is A345531, difference A377281.
These are the positions of 1 in A080101, or 2 in A366833.
For at least one prime-power we have A377057, primes A053607.
For no prime-powers we have A377286.
For two prime-powers we have A377288, primes A053706.
For squarefree instead of prime-power see A377430, A061398, A377431, A068360.
A000015 gives the least prime-power >= n.
A000040 lists the primes, differences A001223.
A000961 lists the powers of primes, differences A057820.
A031218 gives the greatest prime-power <= n.
A246655 lists the prime-powers not including 1, complement A361102.
Cf. A001597, A002808, A024619, A053707, A064113, A065890, A075526, A095195, A224363, A276781, A376597, A377282.

Select[Range[100],Length[Select[Range[Prime[#]+1,Prime[#+1]-1],PrimePowerQ]]==1&]

from itertools import count, islice
from sympy import factorint, nextprime
def A377287_gen(): # generator of terms
    p, q, k = 2, 3, 1
    for k in count(1):
        if sum(1 for i in range(p+1,q) if len(factorint(i))<=1)==1:
            yield k
        p, q = q, nextprime(q)
A377287_list = list(islice(A377287_gen(),53)) # Chai Wah Wu, Oct 28 2024

A377057 Numbers k such that there is at least one prime-power between prime(k)+1 and prime(k+1)-1.

Original entry on oeis.org

2, 4, 6, 9, 11, 15, 18, 22, 30, 31, 39, 53, 54, 61, 68, 72, 97, 99, 114, 129, 146, 162, 172, 217, 219, 263, 283, 309, 327, 329, 357, 409, 445, 487, 519, 564, 609, 656, 675, 705, 811, 847, 882, 886, 1000, 1028, 1163, 1252, 1294, 1381, 1423, 1457

Offset: 1

Gus Wiseman, Oct 25 2024

nonn

			Primes 18 and 19 are 61 and 67, and the interval (62, 63, 64, 65, 66) contains the prime-power 64, so 18 is in the sequence.

The interval from A008864(n) to A006093(n+1) has A046933(n) elements.
For powers of 2 instead of primes see A013597, A014210, A014234, A244508, A304521.
The corresponding primes are A053607.
The nearest prime-power before prime(n)-1 is A065514, difference A377289.
These are the positions of positive terms in A080101, or terms >1 in A366833.
The nearest prime-power after prime(n)+1 is A345531, difference A377281.
For no prime-powers we have A377286.
For exactly one prime-power we have A377287.
For exactly two prime-powers we have A377288, primes A053706.
A000015 gives the least prime-power >= n.
A000040 lists the primes, differences A001223.
A000961 lists the powers of primes, differences A057820.
A031218 gives the greatest prime-power <= n.
A246655 lists the prime-powers not including 1, complement A361102.
Cf. A001597, A002808, A024619, A053707, A064113, A065890, A075526, A095195, A224363, A276781, A376597, A377051, A377282.

Select[Range[100], Length[Select[Range[Prime[#]+1,Prime[#+1]-1],PrimePowerQ]]>=1&]

from itertools import count, islice
from sympy import factorint, nextprime
def A377057_gen(): # generator of terms
    p, q, k = 2, 3, 1
    for k in count(1):
        if any(len(factorint(i))<=1 for i in range(p+1,q)):
            yield k
        p, q = q, nextprime(q)
A377057_list = list(islice(A377057_gen(),52)) # Chai Wah Wu, Oct 27 2024

prime(a(n)) = A053607(n).

A244376 Numbers k such that 1 + k + k^3 + k^5 + k^7 + k^9 + k^11 is prime.

Original entry on oeis.org

1, 2, 10, 40, 47, 55, 62, 121, 137, 152, 167, 201, 233, 278, 290, 293, 313, 333, 370, 382, 430, 452, 460, 506, 546, 555, 613, 625, 642, 675, 705, 711, 752, 767, 793, 797, 831, 835, 837, 872, 878, 891, 906, 917, 923, 978, 985, 1005, 1012, 1017, 1018, 1021

Offset: 1

Vincenzo Librandi, Jun 27 2014

Vincenzo Librandi, Table of n, a(n) for n = 1..1400

Cf. A127936.

Cf. numbers n such that 1+n+n^3 + ... + n^k, with k odd: A006093 (k=1), A049407 (k=3), A124154 (k=5), A124150 (k=7), A124163 (k=9), this sequence (k=11), A124164 (k=13), A244377 (k=15), A244378 (k=17), A124178 (k=19), A244379 (k=21), A124181 (k=23), A244380 (k=25), A124185 (k=27), A244383 (k=29), A124186 (k=31), A244384 (k=33), A124187 (k=35), A244385 (k=37), A124189 (k=39), A244386 (k=41), A124200 (k=43), A244387 (k=45), A124205 (k=47), A244388 (k=49), A124206 (k=51), A244389 (k=53), A124207 (k=55), A244390 (k=57), A124208 (k=59), A244391 (k=61), A124209 (k=63).

[n: n in [0..1500] | IsPrime(s) where s is 1+&+[n^i: i in [1..11 by 2]]];

Select[Range[4000], PrimeQ[Total[#^Range[1, 11, 2]] + 1] &]

isok(n) = isprime(1 + n + n^3 + n^5 + n^7 + n^9 + n^11); \\ Michel Marcus, Jun 27 2014

i,n = var('i,n')
[n for n in (1..2000) if is_prime(1+(n^(2*i+1)).sum(i,0,5))] # Bruno Berselli, Jun 27 2014

cp's OEIS Frontend

A061286 Smallest integer for which the number of divisors is the n-th prime.

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A068361 Numbers n such that the number of squarefree numbers between prime(n) and prime(n+1) = prime(n+1)-prime(n)-1.

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A377436 Numbers k such that there is no perfect-power x in the range prime(k) < x < prime(k+1).

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A005278 Noncototients: numbers k such that x - phi(x) = k has no solution.

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A088362 Numbers k such that k^4096 + 1 is prime (a generalized Fermat prime).

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A072670 Number of ways to write n as i*j + i + j, 0 < i <= j.

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A088361 Numbers n such that n^2048 + 1 is prime (a generalized Fermat prime).

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A377287 Numbers k such that there is exactly one prime-power between prime(k)+1 and prime(k+1)-1.

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