A101794 -id:A101794 - cp's OEIS frontend

A101994 Numbers k such that 4k-1, 8k-1, 16k-1, 32k-1 and 64*k-1 are all primes.

45, 13410, 15855, 31710, 31785, 63570, 74025, 85230, 151830, 202635, 267300, 280665, 399675, 405405, 455250, 466560, 478170, 480240, 511335, 534600, 539475, 561330, 569520, 589305, 666945, 716460, 743160, 748215, 766785, 799350, 860835

Offset: 1

Douglas Stones (dssto1(AT)student.monash.edu.au), Dec 23 2004

			4*45-1 = 179, 8*45-1 = 359, 16*45-1 = 719, 32*45-1 = 1439 and 64*45-1 = 2879 are primes, so 45 is a term.

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

Cf. A002515, A101995, A101996, A101997, A101998, A101999.

Subsequence of A005099, A005122, A101790 and A101794.

Select[Range[10^6], And @@ PrimeQ[2^Range[2, 6]*# - 1] &] (* Amiram Eldar, May 13 2024 *)

is(k) = isprime(4*k-1) && isprime(8*k-1) && isprime(16*k-1) && isprime(32*k-1) && isprime(64*k-1); \\ Amiram Eldar, May 13 2024

A101790 Numbers k such that 4k-1, 8k-1 and 16*k-1 are all primes.

Original entry on oeis.org

3, 45, 90, 180, 255, 258, 363, 378, 453, 483, 615, 675, 705, 873, 885, 978, 1350, 1533, 1770, 1788, 2673, 2793, 2868, 3030, 3225, 3240, 4203, 4290, 4548, 4830, 4998, 5103, 5253, 5295, 5568, 5775, 5955, 6060, 6138, 6870, 7383, 7713, 8133, 8370, 8580, 9000

Offset: 1

Douglas Stones (dssto1(AT)student.monash.edu.au), Dec 16 2004

			4*3 - 1 = 11, 8*3 - 1 = 23 and 16*3 - 1 = 47 are primes, so 3 is a term.

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

Cf. A002515, A101791, A101792, A101793.
Subsequence of A005099 and A005122.
Subsequences: A101794, A101994.

[n: n in [0..10000] | IsPrime(4*n-1) and IsPrime(8*n-1) and IsPrime(16*n-1)]; // Vincenzo Librandi, Nov 17 2010

Select[Range[10^4], And @@ PrimeQ[2^Range[2, 4]*# - 1] &] (* Amiram Eldar, May 12 2024 *)

is(k) = isprime(4*k-1) && isprime(8*k-1) && isprime(16*k-1); \\ Amiram Eldar, May 12 2024

A101995 Primes of the form 4k-1 such that 8k-1, 16k-1, 32k-1 and 64*k-1 are also primes.

Original entry on oeis.org

179, 53639, 63419, 126839, 127139, 254279, 296099, 340919, 607319, 810539, 1069199, 1122659, 1598699, 1621619, 1820999, 1866239, 1912679, 1920959, 2045339, 2138399, 2157899, 2245319, 2278079, 2357219, 2667779, 2865839

Offset: 1

Douglas Stones (dssto1(AT)student.monash.edu.au), Dec 23 2004

			4*45-1 = 179, 8*45-1 = 359, 16*45-1 = 719, 32*45-1 = 1439 and 64*45-1 = 2879 are primes, so 179 is a term.

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

Cf. A002515, A101794.
Cf. A101994, A101996, A101997, A101998, A101999.
Subsequence of A002145, A101791 and A101795.

4 * Select[Range[10^5], And @@ PrimeQ[2^Range[2, 6]*# - 1] &] - 1 (* Amiram Eldar, May 13 2024 *)

is(k) = if(k % 4 == 3, my(m = k\4 + 1); isprime(4*m-1) && isprime(8*m-1) && isprime(16*m-1) && isprime(32*m-1) && isprime(64*m-1), 0); \\ Amiram Eldar, May 13 2024

a(n) = 4*A101994(n) - 1. - Amiram Eldar, May 13 2024

A101795 Primes of the form 4k-1 such that 8k-1, 16k-1 and 32k-1 are also primes.

Original entry on oeis.org

179, 359, 2699, 3539, 12119, 17159, 27479, 53639, 57839, 63419, 71399, 74699, 82499, 87539, 101399, 107279, 107699, 112919, 122099, 122819, 126839, 127139, 133499, 139439, 166739, 167759, 229739, 253679, 254279, 287219, 296099

Offset: 1

Douglas Stones (dssto1(AT)student.monash.edu.au), Dec 16 2004

			4*45-1 = 179, 8*45-1 = 359, 16*45-1 = 719 and 32*45-1 = 1439 are primes, so 179 is a term.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harvey P. Dale)

Cf. A002515, A101794, A101796, A101797, A101798.
Subsequence of A002145 and A101791.
Subsequence: A101995.

Select[Table[4n-1,{n,75000}],AllTrue[(#+1)*{1,2,4,8}-1,PrimeQ]&] (* Harvey P. Dale, Apr 23 2019 *)

is(k) = if(k % 4 == 3, my(m = k\4 + 1); isprime(4*m-1) && isprime(8*m-1) && isprime(16*m-1) && isprime(32*m-1), 0); \\ Amiram Eldar, May 13 2024

a(n) = 4*A101794(n) - 1. - Amiram Eldar, May 13 2024

A101796 Primes of the form 8k-1 such that 4k-1, 16k-1 and 32k-1 are also primes.

Original entry on oeis.org

359, 719, 5399, 7079, 24239, 34319, 54959, 107279, 115679, 126839, 142799, 149399, 164999, 175079, 202799, 214559, 215399, 225839, 244199, 245639, 253679, 254279, 266999, 278879, 333479, 335519, 459479, 507359, 508559

Offset: 1

Douglas Stones (dssto1(AT)student.monash.edu.au), Dec 16 2004

			4*45-1 = 179, 8*45-1 = 359, 16*45-1 = 719 and 32*45-1 = 1439 are primes, so 359 is a term.

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

Cf. A002515, A101794, A101795, A101797, A101798.
Subsequence of A007522 and A101792.
Subsequence: A101996.

8 * Select[Range[10^5], And @@ PrimeQ[2^Range[2, 5]*# - 1] &] - 1 (* Amiram Eldar, May 13 2024 *)

is(k) = if(k % 8 == 7, my(m = k\8 + 1); isprime(4*m-1) && isprime(8*m-1) && isprime(16*m-1) && isprime(32*m-1), 0); \\ Amiram Eldar, May 13 2024

a(n) = 8*A101794(n) - 1 = 2*A101795(n) + 1. - Amiram Eldar, May 13 2024

A101797 Primes of the form 16k-1 such that 4k-1, 8k-1 and 32k-1 are also primes.

Original entry on oeis.org

719, 1439, 10799, 14159, 48479, 68639, 109919, 214559, 231359, 253679, 285599, 298799, 329999, 350159, 405599, 429119, 430799, 451679, 488399, 491279, 507359, 508559, 533999, 557759, 666959, 671039, 918959, 1014719, 1017119, 1148879

Offset: 1

Douglas Stones (dssto1(AT)student.monash.edu.au), Dec 16 2004

			4*45-1 = 179, 8*45-1 = 359, 16*45-1 = 719 and 32*45-1 = 1439 are primes, so 719 is a term.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harvey P. Dale)

Cf. A002515, A101794, A101795, A101796, A101798.
Subsequence of A127576 and A101793.
Subsequence: A101997.

16#-1&/@Select[Range[80000],AllTrue[#*2^Range[2,5]-1,PrimeQ]&] (* Harvey P. Dale, Apr 25 2015 *)

is(k) = if(k % 16 == 15, my(m = k\16 + 1); isprime(4*m-1) && isprime(8*m-1) && isprime(16*m-1) && isprime(32*m-1), 0); \\ Amiram Eldar, May 13 2024

a(n) = 16*A101794(n) - 1 = 4*A101795(n) + 3 = 2*A101796(n) + 1. - Amiram Eldar, May 13 2024

A101798 Primes of the form 32k-1 such that 4k-1, 8k-1 and 16k-1 are also primes.

Original entry on oeis.org

1439, 2879, 21599, 28319, 96959, 137279, 219839, 429119, 462719, 507359, 571199, 597599, 659999, 700319, 811199, 858239, 861599, 903359, 976799, 982559, 1014719, 1017119, 1067999, 1115519, 1333919, 1342079, 1837919, 2029439, 2034239

Offset: 1

Douglas Stones (dssto1(AT)student.monash.edu.au), Dec 16 2004

			4*45-1 = 179, 8*45-1 = 359, 16*45-1 = 719 and 32*45-1 = 1439 are primes, so 1439 is a term.

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

Cf. A002515, A101794, A101795, A101796, A101797.
Subsequence of A127578.
Subsequence: A101998.

32 * Select[Range[10^5], And @@ PrimeQ[2^Range[2, 5]*# - 1] &] - 1 (* Amiram Eldar, May 13 2024 *)
Select[Prime[Range[200000]],Mod[#,32]==31&&AllTrue[{4,8,16} (#+1)/32-1,PrimeQ]&] (* Harvey P. Dale, Feb 20 2025 *)

is(k) = if(k % 32 == 31, my(m = k\32 + 1); isprime(4*m-1) && isprime(8*m-1) && isprime(16*m-1) && isprime(32*m-1), 0); \\ Amiram Eldar, May 13 2024

a(n) = 32*A101794(n) - 1 = 8*A101795(n) + 7 = 4*A101796(n) + 3 = 2*A101797(n) + 1. - Amiram Eldar, May 13 2024

A101320 Numbers k such that 4k-1, 8k-1, 16k-1, 32k-1, 64k-1 and 128k-1 are all primes.

Original entry on oeis.org

15855, 31785, 267300, 280665, 399675, 561330, 946050, 990510, 1022220, 1082115, 1164735, 1283250, 1303875, 1309545, 1514880, 1669065, 1924410, 2850225, 3078675, 3092760, 3492270, 3536385, 3611205, 3920670, 4148970, 4454775

Offset: 1

Douglas Stones (dssto1(AT)student.monash.edu.au), Dec 23 2004

			4*15855-1, 8*15855-1, 16*15855-1, 32*15855-1, 64*15855-1 and 128*15855-1 are primes, so 15855 is a term.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Iain Fox)

Cf. A002515.

Subsequence of A005099, A005122, A101790, A101794 and A101994.

Select[Range[10^6], And @@ PrimeQ[2^Range[2, 7]*# - 1] &] (* Amiram Eldar, May 23 2024 *)

lista(nn) = for(n=1, nn, if(ispseudoprime(4*n-1) && ispseudoprime(8*n-1) && ispseudoprime(16*n-1) && ispseudoprime(32*n-1) && ispseudoprime(64*n-1) && ispseudoprime(128*n-1), print1(n, ", "))) \\ Iain Fox, Nov 23 2017

A101236 Smallest i such that i2^(2)-1, ..., i2^(n+2)-1 are primes.

Original entry on oeis.org

1, 1, 3, 45, 45, 15855, 280665, 4774980, 4393585185, 6522452145, 166260770280, 4321816939440, 15939674132892510, 22654052989616460555, 22654052989616460555, 202608454566431632290

Offset: 0

Douglas Stones (dssto1(AT)student.monash.edu.au), Dec 16 2004; revised Dec 31 2004

(2^2)*3-1=11, (2^3)*3-1=23 and (2^4)*3-1=47 are primes so 3 is the third entry.
For every x in A001122, the x-th term of this sequence and every succeeding term is divisible by x. For example 3 divides the 3rd and every succeeding term, 5 divides the 5th and every succeeding term.
The sequences of primes generated by these numbers are a type of Cunningham chain of the first kind (CC1). Since the longest known CC1 chain is of length 16, the next terms are currently unknown. - Douglas Stones (dssto1(AT)student.monash.edu.au), Mar 16 2005

Paul Jobling, NewPGen
G. Loeh, Long chains of nearly doubled primes, Mathematics of Computation, Vol. 53, No. 188, Oct 1989, pp. 751-759.

Cf. A002515, A101790, A101794.

More terms from Douglas Stones (dssto1(AT)student.monash.edu.au), Mar 16 2005

cp's OEIS Frontend

A101994 Numbers k such that 4*k-1, 8*k-1, 16*k-1, 32*k-1 and 64*k-1 are all primes.

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A101790 Numbers k such that 4*k-1, 8*k-1 and 16*k-1 are all primes.

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A101995 Primes of the form 4*k-1 such that 8*k-1, 16*k-1, 32*k-1 and 64*k-1 are also primes.

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A101795 Primes of the form 4*k-1 such that 8*k-1, 16*k-1 and 32*k-1 are also primes.

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A101796 Primes of the form 8*k-1 such that 4*k-1, 16*k-1 and 32*k-1 are also primes.

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A101797 Primes of the form 16*k-1 such that 4*k-1, 8*k-1 and 32*k-1 are also primes.

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A101798 Primes of the form 32*k-1 such that 4*k-1, 8*k-1 and 16*k-1 are also primes.

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A101320 Numbers k such that 4*k-1, 8*k-1, 16*k-1, 32*k-1, 64*k-1 and 128*k-1 are all primes.

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A101994 Numbers k such that 4k-1, 8k-1, 16k-1, 32k-1 and 64*k-1 are all primes.

A101790 Numbers k such that 4k-1, 8k-1 and 16*k-1 are all primes.

A101995 Primes of the form 4k-1 such that 8k-1, 16k-1, 32k-1 and 64*k-1 are also primes.

A101795 Primes of the form 4k-1 such that 8k-1, 16k-1 and 32k-1 are also primes.

A101796 Primes of the form 8k-1 such that 4k-1, 16k-1 and 32k-1 are also primes.

A101797 Primes of the form 16k-1 such that 4k-1, 8k-1 and 32k-1 are also primes.

A101798 Primes of the form 32k-1 such that 4k-1, 8k-1 and 16k-1 are also primes.

A101320 Numbers k such that 4k-1, 8k-1, 16k-1, 32k-1, 64k-1 and 128k-1 are all primes.

A101236 Smallest i such that i2^(2)-1, ..., i2^(n+2)-1 are primes.