A050522 -id:A050522 - cp's OEIS frontend

A001770 Numbers k such that 5*2^k - 1 is prime.

2, 4, 8, 10, 12, 14, 18, 32, 48, 54, 72, 148, 184, 248, 270, 274, 420, 1340, 1438, 1522, 1638, 1754, 1884, 2014, 2170, 2548, 2622, 2652, 2704, 13510, 21738, 25624, 41934, 51478, 52540, 53230, 172300, 245728, 350028, 1194164

Offset: 1

N. J. A. Sloane

A084213(a(n)+1) is in A136539, for all n. - Farideh Firoozbakht and M. F. Hasler, Nov 03 2012

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).

Y. Gallot, Proth.exe: Windows Program for Finding Large Primes
Wilfrid Keller, List of primes k.2^n - 1 for k < 300
H. C. Williams and C. R. Zarnke, A report on prime numbers of the forms M = (6a+1)*2^(2m-1)-1 and (6a-1)*2^(2m)-1, Math. Comp., 22 (1968), 420-422.
Index entries for sequences of n such that k*2^n-1 (or k*2^n+1) is prime

Cf. A002254 (5*2^n+1 is prime), A050522 (primes of the form 5*2^n - 1).

A001770:=n->`if`(isprime(5*2^n-1),n,NULL): seq(A001770(n), n=1..1000); # Wesley Ivan Hurt, Oct 15 2014

Select[Range[1000], PrimeQ[5*2^# - 1] &] (* Vaclav Kotesovec, Apr 28 2014 *)

is(n)=isprime(5*2^n-1) \\ Charles R Greathouse IV, Feb 07 2017

More terms from Hugo Pfoertner, Jun 23 2004

a(40) from the Wilfrid Keller link by Robert Price, Dec 22 2018

A234504 Number of ways to write n = k + m with k > 0 and m > 0 such that 2^(phi(k) + phi(m)/4) - 5 is prime, where phi(.) is Euler's totient function.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 2, 1, 1, 3, 2, 3, 2, 3, 2, 3, 4, 5, 5, 4, 5, 6, 7, 4, 5, 6, 7, 6, 5, 7, 8, 5, 7, 9, 8, 8, 6, 8, 7, 10, 7, 10, 10, 9, 9, 8, 9, 10, 5, 10, 10, 9, 10, 10, 9, 10, 9, 7, 12, 14, 10, 9, 5, 11, 7, 13, 8, 13, 6, 9, 11, 11, 14, 15, 9, 13

Offset: 1

Zhi-Wei Sun, Dec 26 2013

nonn

Conjecture: a(n) > 0 for all n > 10.

We have verified this for n up to 50000. The conjecture implies that there are infinitely many primes of the form 2^n - 5.

			a(15) = 2 since 2^(phi(2) + phi(13)/4) - 5 = 2^4 - 5 = 11 and 2^(phi(3) + phi(12)/4) - 5 = 2^3 - 5 = 3 are both prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

Cf. A000010, A000040, A000079, A050522, A156560, A234309, A234451, A234470, A234475, A234503, A236358.

f[n_,k_]:=2^(EulerPhi[k]+EulerPhi[n-k]/4)-5
a[n_]:=Sum[If[PrimeQ[f[n,k]],1,0],{k,1,n-1}]
Table[a[n],{n,1,100}]

A332216 Fixed points of A332221: Numbers k such that A156552(sigma(k)) is equal to k.

Original entry on oeis.org

2, 3, 5, 7, 11, 19, 23, 31, 47, 55, 79, 87, 127, 191, 383, 1279, 5119, 6143, 8191, 20479, 81919, 131071, 524287, 786431, 1310719, 2147483647

Offset: 1

Antti Karttunen, Feb 10 2020

Equally, numbers k such that sigma(k) is equal to A005940(1+k).
The primes in this sequence are obtained by subtracting 1 from those terms of A029747 that are one more than a prime.
Questions: Are there other composite terms than 55 and 87? Are there other even terms than 2? (All such even terms should also occur in A332218).

Cf. A000203, A005940, A029747, A156552, A332217, A332218, A332221.

Subsequences: A000668, A007505, A050522.

Select[Range[10^5], DivisorSigma[1, #] == Block[{p = Partition[Split[Join[IntegerDigits[#, 2], {2}]], 2], q}, Times @@ Flatten[Table[q = Take[p, -i]; Prime[Count[Flatten[q], 0] + 1]^q[[1, 1]], {i, Length[p]}]]] &] (* Michael De Vlieger, Feb 12 2020, after Robert G. Wilson v at A005940 *)

A230168 Primes of the form 45*2^n - 1.

Original entry on oeis.org

89, 179, 359, 719, 1439, 2879, 11519, 23039, 737279, 1474559, 2949119, 188743679, 12079595519, 24159191039, 3092376453119, 6184752906239, 810647932926689279, 25940733853654056959, 1740853180245066011576893439, 445658414142736898963684720639

Offset: 1

K. D. Bajpai, Oct 11 2013

nonn

Conjecture: each term in the sequence ends with digit 9.
The expression k*2^n - 1 with k = 45 yields more primes than any other value of k = 1 to  100 and n = 1000.
The term a(44) has 939 digits; a(45) has 1026 digits; a(50) has 2706 digits. - Bajpai
Each term is congruent to 89 mod 90 and therefore each term in the sequence ends in 9. This is a very simple consequence of the definition. - Alonso del Arte, Oct 11 2013

			a(4) = 719: 45*2^4 - 1 = 719, which is prime.
a(9) = 737279: 45*2^14 - 1 = 737279, which is prime.

K. D. Bajpai, Table of n, a(n) for n = 1..44

Cf. A050522.

KD:= proc() local a; a:=45*2^n-1; if isprime(a) then return (a) : fi; end: seq(KD(),n=1..1000);

Select[2^Range[100]45 - 1, PrimeQ] (* Alonso del Arte, Oct 11 2013 *)

A238751 Lesser prime of third Mersenne prime pair {2^m - 5, 5*2^m - 1}.

Original entry on oeis.org

11, 251, 1019, 4091, 65531, 4294967291

Offset: 1

Ilya Lopatin and Juri-Stepan Gerasimov, Mar 04 2014

By comparing A059608 and A001770, the next term, if it exists, is larger than 2^350028. - Giovanni Resta, Mar 06 2014
Lesser prime of Mersenne prime pair of order k and of the form {2^m - (2k - 1), (2k - 1)*2^m - 1}:
for order k = 1: 3, 7, 31, 127, 8191, 131071, ... (Mersenne primes A000668)
for order k = 2: 5, 13, 61, ...
for order k = 3: 11, 251, 1019, 4091, 655531, 4294967291, ... (this sequence)
for order k = 4:
for order k = 5: 2097143, ...
for order k = 6: 3, ...
for order k = 7:
for order k = 8: 17, 1009, 16369, ...
for order k = 9: 47, 65519, 1048559, 68719476719, ...
for order k = 10: 13, 2097133, ...
for order k = 11: 107, 8171, ...
for order k = 12: 41, 233, 4073, ...
for order k = 13: 487, ...
for order k = 14: 5, 229, 997, ...
for order k = 15: 97, ...

			11 is in this sequence because Mersenne prime pair {2^4-(2*3-1) = 11, (2*3-1)*2^4-1 = 79} where 3 is order and 11 is lesser prime (for m = 4).

Cf. A237422, A238694, A238749, A059608, A001770, A156560, A050522.

2^Select[Range[1000], PrimeQ[2^# - 5] && PrimeQ[5*2^# - 1] &] - 5 (* Giovanni Resta, Mar 06 2014 *)

Numbers 2^m - 5 for m belonging to the intersection of A001770 and A059608. - Max Alekseyev, Feb 20 2024

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A001770 Numbers k such that 5*2^k - 1 is prime.

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A234504 Number of ways to write n = k + m with k > 0 and m > 0 such that 2^(phi(k) + phi(m)/4) - 5 is prime, where phi(.) is Euler's totient function.

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A332216 Fixed points of A332221: Numbers k such that A156552(sigma(k)) is equal to k.

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A230168 Primes of the form 45*2^n - 1.

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A238751 Lesser prime of third Mersenne prime pair {2^m - 5, 5*2^m - 1}.

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