A050414 -id:A050414 - cp's OEIS frontend

A067192 Composite c such that sigma(c)==2 (mod phi(c)).

10, 20, 52, 232, 976, 1332, 65152, 261376, 4191232, 67096576, 274877120512, 4398043365376

Offset: 1

Benoit Cloitre, Feb 19 2002

a(12) > 10^12. 4398043365376, 70368731594752 and 72057593635274752 are also terms. - Donovan Johnson, Feb 29 2012

a(13) > 10^13. If 2^k-3 is prime (A050414), then 2^(k-2)*(2^k-3) is a term. Up to 10^13 the only term not of this form is 1332. - Giovanni Resta, Mar 29 2020

Cf. A050414, A067193.

isok(c) = !isprime(c) && ((sigma(c) % eulerphi(c)) == 2); \\ Michel Marcus, Feb 17 2021

a(8)-a(10) from Donovan Johnson, Dec 14 2009
a(11) from Donovan Johnson, Feb 29 2012
a(12) from Giovanni Resta, Mar 29 2020

A176680 Primes p == 2 (mod 3) of the form 2^n-3.

Original entry on oeis.org

5, 29, 509, 536870909, 13164036458569648337239753460458804039861886925068638906788872189, 3369993333393829974333376885877453834204643052817571560137951281149, 13803492693581127574869511724554050904902217944340773110325048447598589

Offset: 1

Roger L. Bagula, Apr 23 2010

nonn

Also primes of the form 2^(2k+1)-3, since 2^n-3 == 2 (mod 3) iff n is odd. - Robert Israel, Feb 17 2016

Robert Israel, Table of n, a(n) for n = 1..11

Cf. A050414.

[2^n-3: n in [1..300] | IsPrime(2^n-3) and (2^n-3) mod 3 eq 2] - Jaroslav Krizek, Feb 17 2016

map(t -> 2^t-3, select(t -> isprime(2^t-3), [seq(t,t=1..2000, 2)])); # Robert Israel, Feb 17 2016

Flatten[Table[If[IntegerQ[Rationalize[N[ (1 + 2^n)/ 3]]] && PrimeQ[2^n - 3], 2^n - 3, {}], {n, 2, 100000}]]

Definition corrected by Jaroslav Krizek, Feb 17 2016

a(5)-a(7) from Jaroslav Krizek, Feb 17 2016

A217353 Numbers k such that 8^k - 3 is prime.

Original entry on oeis.org

1, 2, 3, 4, 8, 50, 58, 71, 112, 1079, 1318, 2252, 3524, 4800, 5560, 6919, 11484, 12184, 41099, 94711, 375460, 449248

Offset: 1

Vincenzo Librandi, Oct 02 2012

3*A217353 is a subsequence of A050414. - Bruno Berselli, Oct 02 2012

Cf. A050414, A059266, A217354.

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

is(n)=ispseudoprime(8^n-3) \\ Charles R Greathouse IV, May 22 2017

a(15)-a(17), a(19)-a(20) using A050414 by Bruno Berselli, Oct 02 2012

a(18), a(21)-a(22) using A050414 by Michael S. Branicky, Sep 15 2024

A238797 Smallest k such that 2^k - (2n+1) and (2n+1)2^k - 1 are both prime, k <= 2n+1, or 0 if no such k exists.

Original entry on oeis.org

0, 3, 4, 0, 0, 0, 0, 5, 6, 5, 7, 6, 9, 5, 0, 7, 6, 6, 0, 0, 10, 0, 6, 0, 7, 9, 6, 7, 8, 0, 17, 8, 0, 0, 7, 0, 0, 18, 0, 0, 0, 8, 0, 10, 8, 9, 18, 0, 0, 7, 0, 0, 8, 12, 0, 7, 0, 11, 16, 0, 21, 0, 0, 0, 8, 14, 0, 0, 18, 9, 10, 8, 77, 0, 0, 0, 12, 8, 0, 11, 18, 0

Offset: 0

Ilya Lopatin and Juri-Stepan Gerasimov, Mar 05 2014

nonn

Numbers n such that 2^k - (2*n+1) and (2*n+1)*2^k - 1 are both prime:
For k = 0: 2, 3, 5, 7, 13, 17, ...   Intersection of A000043 and A000043
for k = 1: 3, 4, 6, 94, ...          Intersection of A050414 and A002235
for k = 2: 4, 8, 10, 12, 18, 32, ... Intersection of A059608 and A001770
for k = 3:                           Intersection of A059609 and A001771
for k = 4: 21, ...                   Intersection of A059610 and A002236
for k = 5:                           Intersection of A096817 and A001772
for k = 6:                           Intersection of A096818 and A001773
for k = 7: 5, 10, 14, ...            Intersection of A059612 and A002237
for k = 8: 6, 16, 20, 36, ...        Intersection of A059611 and A001774
for k = 9: 5, 21, ...                Intersection of A096819 and A001775
for k = 10: 7, 13, ...               Intersection of A096820 and A002238
for k = 11: 6, 8, 12, ...
for k = 12: 9, ...
for k = 13: 5, 8, 10, ...

			a(1) = 3 because 2^3 - (2*1+1) = 5 and (2*1+1)*2^3 - 1 = 23 are both prime, 3 = 2*1+1,
a(2) = 4 because 2^4 - (2*2+1) = 11 and (2*2+1)*2^4 - 1 = 79 are both prime, 4 < 2*2+1 = 5.

Giovanni Resta, Table of n, a(n) for n = 0..1000

Cf. A238748, A238904 (smallest k such that 2^k + (2n+1) and (2n+1)*2^k + 1 are both prime, k <= n, or -1 if no such k exists).

a[n_] := Catch@ Block[{k = 1}, While[k <= 2*n+1, If[2^k - (2*n + 1) > 0 && PrimeQ[2^k - (2*n+1)] && PrimeQ[(2*n + 1)*2^k-1], Throw@k]; k++]; 0]; a/@ Range[0, 80] (* Giovanni Resta, Mar 15 2014 *)

a(0), a(19), a(20) corrected by Giovanni Resta, Mar 13 2014

A093810 Smallest prime factor of 2^n-3.

Original entry on oeis.org

1, 5, 13, 29, 61, 5, 11, 509, 1021, 5, 4093, 19, 16381, 5, 13, 53, 11, 5, 1048573, 773, 4194301, 5, 16777213, 479, 37, 5, 11, 536870909, 23, 5, 9241, 29, 5113, 5, 242819, 47189, 11, 5, 13, 23, 47, 5, 5927, 2087, 227, 5, 11, 19, 59, 5, 13, 2203, 36217, 5, 181

Offset: 2

Yasutoshi Kohmoto, May 11 2004

nonn

Amiram Eldar, Table of n, a(n) for n = 2..626 (terms 2..200 from Vincenzo Librandi)

Cf. A020639, A036563, A050414, A050415, A093817.

PrimeFactors[n_] := Flatten[ Table[ # [[1]], {1}] & /@ FactorInteger[n]]; Table[ PrimeFactors[2^n - 3][[1]], {n, 2, 60}] (* Robert G. Wilson v, May 24 2004 *)
FactorInteger[#][[1,1]]&/@(2^Range[2,60]-3) (* Harvey P. Dale, Aug 21 2016 *)

a(n) = A020639(A036563(n)). - Amiram Eldar, Sep 12 2022

More terms from Robert G. Wilson v, May 24 2004

A093817 Largest prime factor of 2^n-3.

Original entry on oeis.org

1, 5, 13, 29, 61, 5, 23, 509, 1021, 409, 4093, 431, 16381, 6553, 71, 2473, 23831, 97, 1048573, 2713, 4194301, 1677721, 16777213, 70051, 5197, 31033, 1877171, 536870909, 46684427, 22605091, 464773, 296204641, 3360037, 6871947673, 283007

Offset: 2

Yasutoshi Kohmoto, May 11 2004

nonn

Amiram Eldar, Table of n, a(n) for n = 2..626 (terms 2..200 from Vincenzo Librandi)

Cf. A006530, A036563, A050414, A050415, A093810.

PrimeFactors[n_] := Flatten[ Table[ # [[1]], {1}] & /@ FactorInteger[n]]; Table[ PrimeFactors[2^n - 3][[ -1]], {n, 2, 46}] (* Robert G. Wilson v, May 24 2004 *)
Table[FactorInteger[2^n-3][[-1,1]],{n,2,40}] (* Harvey P. Dale, Feb 01 2015 *)

a(n) = A006530(A036563(n)). - Amiram Eldar, Sep 12 2022

More terms from Robert G. Wilson v, May 24 2004

A283266 Prime numbers p such that 2^p - 3 is prime.

Original entry on oeis.org

3, 5, 29, 233, 42689, 69337

Offset: 1

Dmitry Ezhov, Mar 04 2017

Let W = 2^p - 3 and s = (W+1)/(2*p), then 3^s == -2 (mod W) for terms 1..6.

a(7) > 2086750 using A050414. - Michael S. Branicky, Jan 27 2025

Prime terms in A050414.

forprime(p=2, 10^5, W= 2^p-3; if(ispseudoprime(W), print1(p, ", ")))

A302990 a(n) = index of first odd prime number in the (n-th)-order Fibonacci sequence Fn, or 0 if no such index exists.

Original entry on oeis.org

0, 0, 4, 6, 9, 10, 40, 14, 17, 19, 361, 23, 90, 26, 373, 47, 288, 34, 75, 38, 251, 43, 67, 47, 74, 310, 511, 151534, 57, 20608, 1146, 62, 197, 94246, 9974, 287, 271172, 758

Offset: 0

Jacques Tramu, Apr 17 2018

Fn is defined by: Fn(0) = Fn(1) = ... = Fn(n-2) = 0, Fn(n-1) = 1, and Fn(k+1) = Fn(k) + Fn(k-1) + ... + Fn(k-n+1).
In general, Fn(k) is odd iff k == -1 or -2 (mod n+1), therefore a(n) = k*(n+1) - (1 or 2) for all n. Since Fn(n-1) = F(n) = 1, we must have a(n) >= 2n. Since Fn(k) = 2^(k-n) for n <= k < 2n, Fn(2n) = 2^n-1, so a(n) = 2n exactly for the Mersenne prime exponents A000043, while a(n) = 2n+1 when n is not in A000043 but n+1 is in A050414. - M. F. Hasler, Apr 18 2018
Further terms of the sequence: a(38) > 62000, a(39) > 72000, a(40) = 285, a(41) > 178000, a(42) = 558, a(44) = 19529, a(46) = 33369, a(47) = 239, a(48) = 6368, a(53) = 2860, a(54) = 2418, a(58) = 176, a(59) = 18418, a(60) = 1463, a(61) = 122, a(62) = 8755, a(63) = 5118,  a(64) = 25089, a(65) = 988, a(66) = 333, a(67) = 406, a(70) = 1632, a(74) = 374, a(76) = 13704, a(77) = 4991, a(86) = 347, a(89) = 178, a(92) = 1114, a(93) = 187, a(98) = 395, a(100) > 80000; a(n) > 10^4 for all other n up to 100. - Jacques Tramu and M. F. Hasler, Apr 18 2018

			a(2) = 4 because F2 (Fibonacci) = 0, 1, 1, 2, 3, 5, 8, ... and F2(4) = 3 is prime.
a(3) = 6 because F3 (tribonacci) = 0, 0, 1, 1, 2, 4, 7, 13, ... and F3(6) = 7 is prime.
a(4) = 9 because F4 (tetranacci) = 0, 0, 0, 1, 1, 2, 4, 8, 15, 29, 56, ...  and F4(9) = 29 is prime.
From _M. F. Hasler_, Apr 18 2018: (Start)
We see that Fn(k) = 2^(k-n) for n <= k < 2n and thus Fn(2n) = 2^n-1, so a(n) = 2n exactly for the Mersenne prime exponents A000043.
a(n) = 2n + 1 when 2^(n+1) - 3 is prime (n+1 in A050414) but 2^n-1 is not, i.e., n = 4, 8, 9, 11, 21, 23, 28, 93, 115, 121, 149, 173, 212, 220, 232, 265, 335, 451, 544, 688, 693, 849, 1735, ...
For other primes we have: a(29) = 687*30 - 2, a(37) = 20*38 - 2, a(41) > 10^4, a(43) > 10^4, a(47) = 5*48 - 1, a(53) = 53*54 - 2, a(59) = 307*60 - 2, a(67) = 6*67 - 1. (End)

Cf. A000045 (F2), A000073 (F3), A000078 (F4), A001591 (F5), A001592 (F6), A122189(F7), A079262 (F8), A104144 (F9), A122265 (F10).
(According to the definition, F0 = A000004 and F1 = A000012.)
Cf. A001605 (indices of prime numbers in F2).
Cf. A000043, A050414.

A302990(n,L=oo,a=vector(n+1,i,if(i1 && for(i=-2+2*n+=1,L, ispseudoprime(a[i%n+1]=2*a[(i-1)%n+1]-a[i%n+1]) && return(i))} \\ Testing primality only for i%n>n-3 is not faster, even for large n. - M. F. Hasler, Apr 17 2018; improved Apr 18 2018

a(n) == -1 or -2 (mod n+1). a(n) >= 2n, with equality iff n is in A000043. a(n) <= 2n+1 for n+1 in A050414. - M. F. Hasler, Apr 18 2018

a(29) from Jacques Tramu, Apr 19 2018
a(33) from Daniel Suteu, Apr 20 2018
a(36) from Jacques Tramu, Apr 25 2018

A167917 Mersenne primes that belong to Cunningham chains = {3, 7} U {Mp | 2Mp - 1 is prime. (Mp a Mersenne prime)}.

Original entry on oeis.org

3, 7, 31, 8191, 524287

Offset: 1

Washington Bomfim, Nov 15 2009

If p is prime, p >= 5, and Mp belongs to a chain, Mp is always the first term of a chain of the second kind. This is true since (Mp+1)/2 = (2^p - 1 +1)/2 = 2^(p-1), which is composite for p >= 3. (Mp-1)/2 = (2^p - 1 -1)/2 = 2^(p-1)-1 = a. For p >= 5, a is composite since a>3, and a mod 3 = 0. Finally 2Mp + 1 = 2(2^p - 1)+1 = 2^(p+1)-1 = a. If p>=3, a is composite because a > 3, and a mod 3 = 0. We can conclude that beginning with 31, a Mersenne prime can only starts a Cunningham chain of the second kind. If Mp >= 31 starts a chain, the second term of this chain is 2Mp -1=2(2^p - 1)-1 = 2^(p+1) - 3.
That is a number of the form 2^N - 3, even N, so also of the form a^2 - 3, a = 2^(N/2). In this case any factor f of the second term of a chain satisfies f mod 24=1, or f mod 24=11, or f mod 24=13, or f mod 24=23. (1) The next term of this sequence is an unknown Mersenne prime. Probably many primes of this kind will be determined until this term be found. In the work with the known Mersenne primes, M42643801 gives T=2^(42643801+1) -3. The smallest factor of T is f = 38334482051, which is greater than 2^35.
Considering the probabilities given in the second reference, one can conclude that before T was identified as composed (by the exam of all the primes less than f satisfying (1)), the probability of prime T reached a value of 1 in 609,197. This probability is small, but not negligible. Note that the largest known Cunningham chain of length 2 has starting prime 607095* 2^176311 - 1. This is a "very small chain" compared with a chain beginning with a new Mersenne prime.

			a(1) = 3 since 2*3 - 1 = 5. a(2) = 7 because 2*7 - 1 = 13.

Niccolò Castronuovo, On the number of fixed points of the map gamma, arXiv:2102.02739 [math.NT], 2021. Mentions this sequence.
Mathpages, Some Properties of the Lucas Sequence
GIMPS, Mathematics and Research Strategy
Wikipedia, Cunningham chain

Cf. A000043, A000668, A050415, A050414.

A172041 Primes of the form 2^p-3 with p also prime.

Original entry on oeis.org

5, 29, 536870909, 13803492693581127574869511724554050904902217944340773110325048447598589

Offset: 1

Vincenzo Librandi, Jan 24 2010

nonn

Next two terms are 2^42689-3 and 2^69337-3.
Subsequence of A050415.
Corresponding p are the primes in A050414.
The next term has 12851 digits. - Harvey P. Dale, Aug 27 2023

Select[Table[2^p-3,{p,Prime[Range[100]]}],PrimeQ] (* Harvey P. Dale, Aug 27 2023 *)

Definition clarified by R. J. Mathar, Jan 28 2010

cp's OEIS Frontend

A067192 Composite c such that sigma(c)==2 (mod phi(c)).

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A176680 Primes p == 2 (mod 3) of the form 2^n-3.

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Magma

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A217353 Numbers k such that 8^k - 3 is prime.

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A238797 Smallest k such that 2^k - (2*n+1) and (2*n+1)*2^k - 1 are both prime, k <= 2*n+1, or 0 if no such k exists.

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A093810 Smallest prime factor of 2^n-3.

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A093817 Largest prime factor of 2^n-3.

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A283266 Prime numbers p such that 2^p - 3 is prime.

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PARI

A302990 a(n) = index of first odd prime number in the (n-th)-order Fibonacci sequence Fn, or 0 if no such index exists.

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PARI

A238797 Smallest k such that 2^k - (2n+1) and (2n+1)2^k - 1 are both prime, k <= 2n+1, or 0 if no such k exists.