A057732 -id:A057732 - cp's OEIS frontend

A089437 Numbers k such that 4^k + 3 is prime.

1, 2, 3, 6, 8, 9, 14, 15, 42, 114, 195, 392, 555, 852, 1004, 1185, 2001, 2030, 2031, 2276, 8610, 8967, 10362, 11366, 15927, 16514, 17877, 19122, 19898, 27728, 29156, 61275, 102981, 117663, 181560, 239922, 342789, 371226, 717200, 937776, 970406, 1102722

Offset: 1

Herman H. Rosenfeld (herm3(AT)pacbell.net), Dec 28 2003

The halved even terms of A057732. - R. J. Mathar, Feb 26 2008

			4^15 + 3 = 1073741827 a prime number.

Cf. A057732, A228026 (the actual primes).

is(n)=ispseudoprime(4^n+3) \\ Charles R Greathouse IV, Feb 20 2017

More terms from Ray Chandler, Dec 30 2003
More terms from R. J. Mathar, Feb 26 2008
a(33)-a(37) derived from A057732 by Robert Price, Apr 26 2014
a(38)-a(42) derived from A057732 by Amiram Eldar, Nov 16 2023

A135535 Primes of the form 4^k - 3.

Original entry on oeis.org

13, 61, 1021, 4093, 16381, 1048573, 4194301, 16777213, 19807040628566084398385987581, 83076749736557242056487941267521533, 5316911983139663491615228241121378301, 1427247692705959881058285969449495136382746621, 23945242826029513411849172299223580994042798784118781, 118571099379011784113736688648896417641748464297615937576404566024103044751294461, 139984046386112763159840142535527767382602843577165595931249318810236991948760059086304843329475444733

Offset: 1

Daniele Corradetti (d.corradetti(AT)gmail.com), Feb 21 2008

nonn

Involved in the "New Mersenne Prime Conjecture" and in some generalizations of Mersenne primes.

Subsequence of A050415. - Elmo R. Oliveira, Nov 28 2023

			16381 is a term because 4^7 - 3 = 16381 is prime.

Daniel Minoli, Voice over MPLS, McGraw-Hill, New York, NY, 2002, ISBN 0-07-140615-8 (pp. 114-134).

P. T. Bateman, J. L. Selfridge and S. S. Wagstaff, Jr., The New Mersenne Conjecture, Amer. Math. Monthly 96, 125-128, 1989.
D. Minoli and Robert Bear, Hyperperfect Numbers, Pi Mu Epsilon Journal, Fall 1975, pp. 153-157. [Daniel Minoli (daniel.minoli(AT)ses.com), Aug 26 2009]
Daniel Minoli and W. Nakamine, Mersenne Numbers Rooted On 3 For Number Theoretic Transforms, 1980 IEEE International Conf. on Acoust., Speech and Signal Processing. [Daniel Minoli (daniel.minoli(AT)ses.com), Aug 26 2009]
Paul Tannery, Questions 659 et 660, L'Intermédiaire des mathématiciens, Tome II (1895) p. 317.
Eric Weisstein's World of Mathematics, New Mersenne Prime Conjecture

Cf. A000040, A050415, A057732, A057733, A059266.

Do[If[PrimeQ[4^n - 3], Print[4^n - 3]], {n, 100}] (* Robert G. Wilson v, Feb 29 2008 *)
Select[4^Range[200]-3,PrimeQ] (* Harvey P. Dale, Jul 11 2022 *)

a(n) = 4^A059266(n) - 3. - Ryan Propper, Feb 26 2008

More terms from R. J. Mathar, Robert G. Wilson v and Ryan Propper, Feb 26 2008

A217354 Numbers n such that 8^n + 3 is prime.

Original entry on oeis.org

1, 2, 4, 5, 6, 10, 28, 76, 130, 370, 568, 713, 789, 790, 1334, 1354, 1849, 2913, 5729, 5740, 5978, 6908, 10618, 11918, 12748, 13449, 40850, 68654, 78442, 121040, 159948, 228526

Offset: 1

Vincenzo Librandi, Oct 02 2012

nonn

3*A217354 is a subsequence of A057732. - Bruno Berselli, Oct 02 2012

Cf. A057732, A089437, A217353.

Mathematica
```
Select[Range[5000], PrimeQ[8^# + 3] &]
```

is(n)=ispseudoprime(8^n+3) \\ Charles R Greathouse IV, Feb 17 2017

a(19)-a(32) are obtained from A057732; by Bruno Berselli, Oct 02 2012

A257273 a(n) = 2^(n-1)*(2^n+3).

Original entry on oeis.org

2, 5, 14, 44, 152, 560, 2144, 8384, 33152, 131840, 525824, 2100224, 8394752, 33566720, 134242304, 536920064, 2147581952, 8590131200, 34360131584, 137439739904, 549757386752, 2199026401280, 8796099313664, 35184384671744, 140737513521152, 562950003752960, 2251799914348544, 9007199456067584

Offset: 0

M. F. Hasler, Apr 27 2015

a(n) is in A125246 <=> n is in A057732 <=> A062709(n) is in A057733.

These are also the row sum of the triangle A146769: For n>=1, a(n-1) is the sum of row n of A146769.

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (6,-8).

Cf. A000079, A007582, A028403, A256873, A256871, A257272.

Cf. A062709, A057732, A057733.

[2^(n-1)*(2^n+3): n in [0..35]]; // Vincenzo Librandi, Apr 27 2015

Table[2^(n - 1) (2^n + 3), {n, 0, 30}] (* Bruno Berselli, Apr 27 2015 *)
CoefficientList[Series[(2 - 7 x)/((1 - 4 x) (1 - 2 x)), {x, 0, 30}], x] (* Vincenzo Librandi, Apr 27 2015 *)
LinearRecurrence[{6,-8},{2,5},30] (* Harvey P. Dale, Dec 21 2024 *)

PARI
```
a(n)=2^(n-1)*(2^n+3)
```

Vec((2-7*x)/((1-4*x)*(1-2*x)) + O(x^100)) \\ Colin Barker, Apr 27 2015

G.f.: (2-7*x)/((1-4*x)*(1-2*x)). - Vincenzo Librandi, Apr 27 2015

a(n) = 6*a(n-1)-8*a(n-2). - Colin Barker, Apr 27 2015

A057736 Primes p such that 2^p + 3 is prime.

Original entry on oeis.org

2, 3, 7, 67

Offset: 1

G. L. Honaker, Jr., Oct 29 2000

The next term p is greater than 100000, corresponding to a prime 2^p + 3 with more than 30000 digits. - Ryan Propper, Aug 24 2005

Next term > 2205444. - Joerg Arndt, Mar 07 2021

J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 67, p. 24, Ellipses, Paris 2008.

Cf. A057737 (the corresponding primes).

Subsequence of A057732.

Select[Prime[Range[20]],PrimeQ[2^#+3]&] (* Harvey P. Dale, Jun 06 2022 *)

from sympy import isprime, primerange
def afind(limit):
  for p in primerange(2, limit+1):
    if isprime(2**p + 3): print(p, end=", ")
afind(1000) # Michael S. Branicky, Mar 07 2021

A134916 Numbers n such that both 3^n+2 and 2^n+3 are primes.

Original entry on oeis.org

1, 2, 3, 4, 15

Offset: 1

Zak Seidov, Jan 29 2008

Intersection of A057732 and A051783. a(6)>1000.

Since this is the intersection of A057732 and A051783, a(6)>95504. - Dmitry Kamenetsky, Jul 29 2008

Cf. A051783, A057732.

Select[Range[20],AllTrue[{3^#+2,2^#+3},PrimeQ]&] (* Harvey P. Dale, Sep 02 2022 *)

for(n=0,1000,if(isprime(2^n+3)&&isprime(3^n+2),print(n)))

A234285 Positive odd numbers n such that sigma(m) - 2m is never equal to n, where sigma(.) is the sum of divisors function A000203. Conjectural.

Original entry on oeis.org

1, 5, 9, 11, 13, 15, 21, 23, 25, 27, 29, 33, 35, 37, 43, 45

Offset: 1

N. J. A. Sloane, Dec 28 2013

The terms shown here are conjectural, based on a search up to 10^20 made by Davis et al. (2013).
Comments from Farideh Firoozbakht, Jan 12 2014: (Start)
1. Mersenne primes are not in this sequence. Because if M=2^p-1 is prime then M=sigma(m)-2m, where m=2^(p-1)*(2^p-1)^2=(1/2)*(M+1)*M^2 (please see Proposition 2.1 of Firoozbakht-Hasler, 2010).
2. If M = 2^p - 1 is a Mersenne prime then M^2 + 3M + 1 = 4^p + 2^p - 1 is not in the sequence. Because M^2 + 3M + 1 = sigma(m) - 2m where m = M^3 + M^2 = 2^p(2^p-1)^2 (please see Proposition 2.5, op. cit.).
Examples:
p = 2, M = 3, 4^p + 2^p - 1 = 19, m = M^3 + M^2 = 2^p(2^p-1)^2 = 36; sigma(m) - 2m = 19
p = 3, M = 7, 4^p + 2^p - 1 = 71, m = M^3 + M^2 = 2^p(2^p-1)^2 = 392; sigma(m) - 2m = 71
3. Note that if r is an even number and if for a number k p = 2^k - r - 1 is an odd prime then r = sigma(m) - 2m where m = 2^(k-1)*p. Namely r is not in the sequence (see Theorem 1.1, op. cit.).
It seems that for each even number r, there exists at least one odd prime of the form 2^k - r - 1. This means there is no even term in the sequence.
Moreover I conjecture that, for each even number r, there exist infinitely many primes p of the form 2^k - r - 1, or equivalently, I conjecture that: For each odd number s, there exists infinitely many primes p of the form 2^k - s.
Special cases:
(i): s = 1, there exist infinitely many Mersenne primes.
(ii): s = -1, there exist infinitely many Fermat primes.
(iii): s = 3, sequence A050414 is infinite.
(iv): s = -3, sequence A057732 is infinite.
(v): s = -5, sequence A059242 is infinite.
and so on. (End)
Cohen (1983) showed that 203^2 is not a term since sigma(m) - 2*m = 203^2 has a solution m = 742^2. - Max Alekseyev, Aug 29 2025

Graeme L. Cohen, Generalised quasiperfect numbers, Ph.D. Dissertation, University of New South Wales, Sydney, 1982. Abstract in Bull. Australian Math. Soc., 27 (1983), 153-155.
Nichole Davis, Dominic Klyve and Nicole Kraght, On the difference between an integer and the sum of its proper divisors, Involve, Vol. 6 (2013), No. 4, 493-504; DOI: 10.2140/involve.2013.6.493
Farideh Firoozbakht and M. F. Hasler, Variations on Euclid's formula for perfect numbers, Journal of Integer Sequences 13 (2010), 18 pp. #10.3.1.

Cf. A000203, A033879 (2n - sigma(n)).

For negative values of n see A234286.

Edited by Max Alekseyev, Aug 29 2025

A133830 Least positive number k < n such that the binary trinomial 1 + 2^n + 2^k is prime, or 0 if there is no such k.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 0, 3, 3, 2, 1, 4, 5, 1, 1, 11, 1, 6, 5, 4, 7, 3, 9, 0, 17, 15, 1, 15, 1, 6, 0, 4, 9, 14, 13, 3, 11, 25, 0, 6, 7, 0, 17, 7, 15, 2, 0, 30, 23, 6, 21, 2, 33, 1, 0, 3, 0, 14, 5, 6, 21, 19, 0, 30, 3, 1, 5, 34, 19, 26, 17, 19, 17, 5, 33, 15, 23, 27, 33, 4, 3, 26, 1, 39, 35, 19, 9, 18

Offset: 2

T. D. Noe, Sep 26 2007

nonn

Sequence A081504 gives the n such that a(n) = 0. For those n, A133831(n) gives the least k > n for which the binary trinomial is prime.

T. D. Noe, Table of n, a(n) for n=2..2000

Cf. A057732, A059242, A057196, A057200, A081091 (various forms of prime binary trinomials).

Cf. A095056, A133831, A133832 (k > n equivalent).

Mathematica
```
Table[s=1+2^n; k=1; While[k
				
```

Edited by Peter Munn, Sep 30 2024

A133831 Least positive number k != n such that the binary trinomial 1 + 2^n + 2^k is prime, or 0 if there is no such k.

Original entry on oeis.org

2, 1, 1, 1, 2, 1, 1, 9, 3, 3, 2, 1, 4, 5, 1, 1, 11, 1, 6, 5, 4, 7, 3, 9, 27, 17, 15, 1, 15, 1, 6, 458465, 4, 9, 14, 13, 3, 11, 25, 57, 6, 7, 46, 17, 7, 15, 2, 1009, 30, 23, 6, 21, 2, 33, 1, 1265, 3, 69, 14, 5, 6, 21, 19, 2241, 30, 3, 1, 5, 34, 19, 26, 17, 19, 17, 5, 33, 15, 23, 27

Offset: 1

T. D. Noe, Sep 26 2007

nonn

Does such k exist (so that a(n) is nonzero) for all n? These binary trinomials can also be written as f*2^n+1, where f=2^m+1 for some m, which is reminiscent of the Sierpinski problem (see A076336). Hence if there are no Sierpinski numbers of the form 2^m+1, then a(n) is nonzero for all n.

The PFGW program was used to find a(32), which produces a 138012-digit probable prime. If a(256) is nonzero, it is greater than 10^6.

T. D. Noe, Table of n, a(n) for n = 1..255
Henri Lifchitz and Renaud Lifchitz (Editors), Search for 2^n+2^m+1, PRP Top Records.

Cf. A057732, A059242, A057196, A057200, A081091 (various forms of prime binary trinomials).
Closely related problems: A040076 (see also A076336), A067760, A133830 (k < n), A133832 (k > n).
Cf. A095056.

mx=4000; Table[s=1+2^n; k=1; While[k==n || (k

Edited by Peter Munn, Sep 29 2024

A133832 Least number k > n such that the binary trinomial 1 + 2^n + 2^k is prime, or 0 if there is no such k.

Original entry on oeis.org

2, 3, 5, 13, 6, 7, 9, 9, 18, 19, 14, 13, 15, 17, 17, 81, 20, 19, 30, 33, 26, 27, 38, 81, 27, 35, 31, 33, 35, 31, 42, 458465, 36, 45, 47, 37, 67, 53, 41, 57, 42, 45, 46, 69, 54, 57, 53, 1009, 100, 119, 55, 73, 83, 67, 57, 1265, 74, 69, 66, 113, 75, 101, 66, 2241, 68, 67, 70

Offset: 1

T. D. Noe, Sep 26 2007

nonn

Conjecture: a(n) is nonzero for all n. These binary trinomials can also be written as f*2^n+1, where f=2^m+1 for some m, which is reminiscent of the Sierpinski problem (see A076336). The conjecture is equivalent to no Sierpinski numbers of the form 2^m+1 existing.

The PFGW program was used to find a(32), which produces a 138012-digit probable prime.

T. D. Noe, Table of n, a(n) for n=1..255
Henri Lifchitz and Renaud Lifchitz (Editors), Search for 2^n+2^m+1, PRP Top Records.

Cf. A057732, A059242, A057196, A057200, A081091 (various forms of prime binary trinomials).

Cf. A095056, A133830 (k < n equivalent), A133831.

mx=4000; Table[s=1+2^n; k=n+1; While[k

Edited by Peter Munn, Sep 29 2024

cp's OEIS Frontend

A089437 Numbers k such that 4^k + 3 is prime.

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A135535 Primes of the form 4^k - 3.

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A217354 Numbers n such that 8^n + 3 is prime.

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A257273 a(n) = 2^(n-1)*(2^n+3).

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Magma

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A057736 Primes p such that 2^p + 3 is prime.

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Python

A134916 Numbers n such that both 3^n+2 and 2^n+3 are primes.

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Mathematica

PARI

A234285 Positive odd numbers n such that sigma(m) - 2m is never equal to n, where sigma(.) is the sum of divisors function A000203. Conjectural.

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Keywords

Comments

Links

Crossrefs

Extensions

A133830 Least positive number k < n such that the binary trinomial 1 + 2^n + 2^k is prime, or 0 if there is no such k.

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Author

Keywords

Comments

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