A092506 -id:A092506 - cp's OEIS frontend

A238735 Number of prime pairs {2^n + (2k + 1), (2k + 1)*2^n + 1}, k < n.

1, 2, 1, 2, 0, 3, 2, 2, 0, 1, 0, 2, 0, 0, 0, 3, 0, 1, 0, 3, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Ilya Lopatin and Juri-Stepan Gerasimov, Mar 04 2014

nonn

If k = 0, then the two numbers in the "prime pair" are actually the same number, 2^n + 1 (which is either 2 or a Fermat prime; see A019434, A092506).

			a(1) = 1 because 2^1+(2*0+1)=3 and (2*0+1)*2^1+1=3 is prime pair for k=0,
a(2) = 2 because 2^2+(2*0+1)=5 and (2*0+1)*2^2+1=5 is prime pair for k=0, 2^2+(2*1+1)=7 and (2*1+1)*2^2+1=13 is prime pair for k=1,
a(3) = 1 because 2^3+(2*2+1)=13 and (2*2+1)*2^3+1=41 is prime pair for k=2.

Antti Karttunen, Table of n, a(n) for n = 1..631

Cf. A019434, A238554.

a[n_] := Length@Select[Range[0, n-1], PrimeQ[2^n + (2*# + 1)] && PrimeQ[(2*# + 1)*2^n + 1] &]; Array[a, 100] (* Giovanni Resta, Mar 04 2014 *)

a(n)=sum(k=0,n-1,isprime(2^n+2*k+1)&&isprime((2*k+1)<Charles R Greathouse IV, Mar 06 2014

a(47)-a(87) from Giovanni Resta, Mar 04 2014

A268210 Primes p of the form 2^k + 1 such that p - k is a prime q (for k >= 0).

Original entry on oeis.org

2, 3, 5, 17, 65537

Offset: 1

Jaroslav Krizek, Jan 28 2016

nonn

Intersection of A092506 and A268209.
Sequence is not the same as A004249 because A004249(5) is a composite number.
Corresponding values of numbers k: 0, 1, 2, 4, 16; corresponding values of primes q: 2, 2, 3, 13, 65521.
4 out of 5 known Fermat primes from A019434 (3, 5, 17, 65537) are terms.

			Prime 17 = 2^4 + 1 is a term because 17 - 4 = 13 (prime).
257 = 2^8 + 1 is not a term because 257 - 8 = 249 (composite number).

Cf. A004249, A019434, A092506, A100361, A100362, A268209.

[2^k + 1: k in [0..60] | IsPrime(2^k + 1) and IsPrime(2^k - k + 1)];

2^# + 1 &@ Select[Range[0, 600], PrimeQ[2^# + 1] && PrimeQ[2^# - # + 1] &] (* Michael De Vlieger, Jan 29 2016 *)

A269019 a(n) = 2^n + 2*(-1)^n - 1.

Original entry on oeis.org

2, -1, 5, 5, 17, 29, 65, 125, 257, 509, 1025, 2045, 4097, 8189, 16385, 32765, 65537, 131069, 262145, 524285, 1048577, 2097149, 4194305, 8388605, 16777217, 33554429, 67108865, 134217725, 268435457, 536870909, 1073741825, 2147483645, 4294967297, 8589934589

Offset: 0

Jaroslav Krizek, Feb 17 2016

sign

Fermat numbers > 3 from A000215 are terms.
Prime terms are in A269018.
Union of A052539 and A267921.

			For n = 6; a(n) = 2^n + 2*(-1)^n - 1 = 2^6 + 2*(-1)^6 - 1 = 65.

Index entries for linear recurrences with constant coefficients, signature (2,1,-2).

Cf. A019434, A052539, A092506, A176680, A267921, A269018.

Magma
```
[2^n + 2*(-1)^n - 1: n in [0..300]]
```

Table [2^n + 2 (-1)^n - 1, {n, 0, 80}] (* or *) CoefficientList[Series[(2 - 5 x + 5 x^2) / ((1 - 2 x) (1 - x^2)), {x, 0, 33}], x] (* Vincenzo Librandi, Feb 18 2016 *)
LinearRecurrence[{2,1,-2},{2,-1,5},40] (* Harvey P. Dale, Feb 25 2022 *)

G.f.: (2-5*x+5*x^2)/((1-2*x)*(1-x^2)). - Vincenzo Librandi, Feb 18 2016

A274915 Powers of odd non-Fermat primes.

Original entry on oeis.org

1, 7, 11, 13, 19, 23, 29, 31, 37, 41, 43, 47, 49, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 121, 127, 131, 137, 139, 149, 151, 157, 163, 167, 169, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 263, 269, 271, 277, 281, 283, 293, 307, 311

Offset: 1

Juri-Stepan Gerasimov, Nov 11 2016

n is in the sequence if n = p^m where p is in A138889 and m >= 0. - Robert Israel, Sep 15 2017
The difference between two divisors of n is never a power of 2. The first number with this property that is not in the sequence is 91. - Robert Israel, Sep 15 2017
Subsequence of A061345.

			49 is in this sequence because 49 = 7^2 and 7 is not a Fermat prime.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A019434, A061345, A092506, A138889, A277994.

N:= 500: # to get all terms <= N
P:= select(isprime, {seq(i,i=7..N,2)}) minus {seq(2^i+1, i=1..ilog2(N))}:
sort(convert(map(p -> seq(p^k,k=0..floor(log[p](N))), P), list)); # Robert Israel, Sep 15 2017

A277994(a(n)) = 0.

Edited, new name, and corrected by Robert Israel, Sep 15 2017

A367560 Numbers k such that k^7*2^k + 1 is a prime.

Original entry on oeis.org

1, 3, 11, 51, 76, 123, 149, 274, 311, 328, 381, 639, 737, 898, 1156, 9017, 13200, 18348, 26388, 30081

Offset: 1

Juri-Stepan Gerasimov, Nov 22 2023

No further terms <= 100000. - Michael S. Branicky, Aug 28 2024

Numbers k such that k^m*2^k + 1 is a prime: 0, 1, 2, 4, 8, 16, .. (m = 0), A005849 (m = 1), A058780 (m = 2), A357612 (m = 3), A366422 (m = 4), A367421 (m = 5), A367287 (m = 6), this sequence (m = 7).

Cf. A092506.

[k: k in [1..2000] | IsPrime(k^7*2^k+1)];

a(18)-a(20) from Michael S. Branicky, Nov 22 2023

A385748 Numbers k such that A384247(k) divides k.

Original entry on oeis.org

1, 2, 6, 8, 12, 24, 32, 54, 96, 108, 128, 192, 216, 240, 384, 486, 512, 864, 972, 1536, 1728, 1944, 2048, 2160, 3072, 3456, 4374, 6000, 6144, 7776, 8192, 8748, 13824, 15552, 17496, 19440, 24576, 27648, 31104, 32768, 39366, 49152, 54000, 55296, 61440, 65280, 69984

Offset: 1

Amiram Eldar, Jul 08 2025

nonn

(2^(2^k)-1) * 2^(2^k) is a term for k = 0..5.
Apparently, the only prime factors of any term are 2 and the Fermat primes (A019434), i.e., A092506.
Apparently, except for n = 1, a(n) / A384247(a(n)) is either 2 or 3.

			  n | a(n) | a(n) / A384247(a(n))
  --+------+---------------------
  1 |    1 | 1 / 1 = 1
  2 |    2 | 2 / 1 = 2
  3 |    6 | 6 / 2 = 3
  4 |    8 | 8 / 4 = 2
  5 |   12 | 12 / 6 = 2

Amiram Eldar, Table of n, a(n) for n = 1..250

Cf. A019434, A092506, A384247.

Similar sequences: A007694, A298759, A319481, A335327, A373057.

f[p_, e_] := p^e*(1 - 1/p^(2^(IntegerExponent[e, 2]))); iphi[1] = 1; iphi[n_] := iphi[n] = Times @@ f @@@ FactorInteger[n]; q[n_] := Divisible[n, iphi[n]]; Select[Range[70000], q]

iphi(n) = {my(f = factor(n)); n * prod(i = 1, #f~, (1 - 1/f[i, 1]^(1 << valuation(f[i, 2], 2))));}
isok(k) = !( k % iphi(k));

A092733 Sum of two consecutive primes of the form 2^x+1 (including Fermat primes and 2).

Original entry on oeis.org

5, 8, 22, 274, 65794

Offset: 1

Jorge Coveiro, Apr 12 2004

			a(1) = [2^0+1] + [2^(2^0)+1]
a(2) = [2^(2^0)+1] + [2^(2^1)+1]
a(3) = [2^(2^1)+1] + [2^(2^2)+1]
a(4) = [2^(2^2)+1] + [2^(2^3)+1]
a(5) = [2^(2^3)+1] + [2^(2^4)+1]

Cf. A092506, A019434, A063897.

Total/@Partition[Select[2^Range[0,50]+1,PrimeQ],2,1] (* Harvey P. Dale, Jun 21 2021 *)

Offset corrected by Arkadiusz Wesolowski, Oct 22 2011

A186884 Numbers k such that 2^(k-1) == 1 + b*k (mod k^2), where b divides k - 2^p for some integer p >= 0 and 2^p <= b.

Original entry on oeis.org

3, 5, 7, 11, 13, 17, 19, 29, 31, 37, 71, 127, 173, 199, 233, 251, 257, 379, 491, 613, 881, 2047, 2633, 2659, 3373, 3457, 5501, 5683, 8191, 11497, 13249, 15823, 16879, 18839, 22669, 24763, 25037, 26893, 30139, 45337, 48473, 56671, 58921, 65537, 70687, 74531, 74597, 77023, 79669, 87211, 92237, 102407, 131071, 133493, 181421, 184511, 237379, 250583, 254491, 281381

Offset: 1

Alzhekeyev Ascar M, Feb 28 2011

nonn

This sequence contains A186645 as a subsequence (corresponding to p=0).
All composites in this sequence are 2-pseudoprimes, A001567. This sequence contains all terms of A054723. Another composite term is 4294967297 = 2^32 + 1, which does not belong to A054723. In other words, all known composite terms have the form (2^x + 1) or (2^x - 1). Are there composites not of this form?
This sequence contains all the primes of the forms (2^x + 1) and (2^x - 1), i.e., subsequences A092506 and A000668.

Edited by Max Alekseyev, Mar 14 2011

a(25) and a(26) interchanged by Georg Fischer, Jul 08 2022

A219196 A subsequence of the denominators of the Bernoulli numbers: a(n) = A027642(A131577(n)).

Original entry on oeis.org

1, 2, 6, 30, 30, 510, 510, 510, 510, 131070, 131070, 131070, 131070, 131070, 131070, 131070, 131070, 8589934590, 8589934590, 8589934590, 8589934590, 8589934590, 8589934590, 8589934590, 8589934590, 8589934590, 8589934590, 8589934590, 8589934590, 8589934590, 8589934590

Offset: 0

Paul Curtz, Nov 14 2012

nonn

Conjecture: a(15) = a(16) = 131070, a(17) through a(32) = 8589934590.
Number of different terms: 1, 1, 1, 2, 4, ... = abs(A141531)?
Factorization of terms from 2:
         2 = 2
         6 = 2*3
        30 = 2*3*5
       510 = 2*3*5*17
    131070 = 2*3*5*17*257
8589934590 = 2*3*5*17*257*65537.
Note that all factors shown are 2 or Fermat numbers (see A092506, A019434, A000215).
Empirical: using the von Staudt-Clausen theorem, terms a(17) through a(4215) are all 8589934590. - Simon Plouffe, Sep 20 2015
Using the von Staudt-Clausen theorem, a(n) is the product of 2 and all Fermat primes <= 2^(n-1)+1: see A019434.  The only known Fermat primes are 3,5,17,257,65537; it is known that there are no others < 2^(2^33)+1, so that a(n) = 8589934590 for n <= 2^33 = 8589934592. - Robert Israel, Sep 21 2015

Eric Weisstein's World of Mathematics, von Staudt-Clausen Theorem

Cf. A092506, A019434, A000215.

a[n_] := a[n] = Times @@ Select[ Divisors[2^(n-1)] + 1, PrimeQ]; a[0] = 1; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Dec 07 2012 *)

a(n) = denominator(bernfrac(1<Michel Marcus, Aug 14 2013

Extended up to a(20) by Jean-François Alcover, Dec 07 2012

More terms from Michel Marcus, Sep 27 2015

A237251 Primes p such that p*2^(p-1)-1 is prime.

Original entry on oeis.org

2, 3, 5, 17, 257, 16487

Offset: 1

Gerasimov Sergey, Feb 05 2014

The fifth Fermat prime, 65537, is not in the sequence: 65537*2^65536-1 is composite (per PFGW). - Michael B. Porter, Feb 11 2014

Also 65537*2^65536-1 is divisible by 16267 and 2058772459. - Jeppe Stig Nielsen, Jan 04 2020

Cf. A019434, A092506, A230769, A236752.

isok(p) = isprime(p) && isprime(p*2^(p-1) - 1); \\ Michel Marcus, Feb 06 2014

a(5) from Ralf Stephan, Feb 03 2014

a(6) = A230769(26)+1 appended by Jeppe Stig Nielsen, Jan 04 2020

cp's OEIS Frontend

A238735 Number of prime pairs {2^n + (2k + 1), (2k + 1)*2^n + 1}, k < n.

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A268210 Primes p of the form 2^k + 1 such that p - k is a prime q (for k >= 0).

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Magma

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A269019 a(n) = 2^n + 2*(-1)^n - 1.

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Magma

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A274915 Powers of odd non-Fermat primes.

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Maple

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A367560 Numbers k such that k^7*2^k + 1 is a prime.

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Magma

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A385748 Numbers k such that A384247(k) divides k.

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A092733 Sum of two consecutive primes of the form 2^x+1 (including Fermat primes and 2).

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Mathematica

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A186884 Numbers k such that 2^(k-1) == 1 + b*k (mod k^2), where b divides k - 2^p for some integer p >= 0 and 2^p <= b.

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