A101998 -id:A101998 - 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

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

A101996 Primes of the form 8k-1 such that 4k-1, 16k-1, 32k-1 and 64*k-1 are also primes.

Original entry on oeis.org

359, 107279, 126839, 253679, 254279, 508559, 592199, 681839, 1214639, 1621079, 2138399, 2245319, 3197399, 3243239, 3641999, 3732479, 3825359, 3841919, 4090679, 4276799, 4315799, 4490639, 4556159, 4714439, 5335559, 5731679

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 359 is a term.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..750 from Harvey P. Dale)

Cf. A101789, A101994, A101995, A101997, A101998, A101999.

Subsequence of A007522, A101792 and A101796.

8#-1&/@Select[Range[720000],AllTrue[{4,8,16,32,64}#-1,PrimeQ]&] (* Harvey P. Dale, Jan 17 2023 *)
Select[Table[2^Range[2,6] n-1,{n,750000}],AllTrue[#,PrimeQ]&][[;;,2]] (* Harvey P. Dale, Jun 03 2023 *)

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) && isprime(64*m-1), 0); \\ Amiram Eldar, May 13 2024

a(n) = 8*A101994(n) - 1 = 2*A101995(n) + 1. - Amiram Eldar, May 13 2024

Corrected by T. D. Noe, Nov 15 2006

A101997 Primes of the form 16k-1 such that 4k-1, 8k-1, 32k-1 and 64*k-1 are also primes.

Original entry on oeis.org

719, 214559, 253679, 507359, 508559, 1017119, 1184399, 1363679, 2429279, 3242159, 4276799, 4490639, 6394799, 6486479, 7283999, 7464959, 7650719, 7683839, 8181359, 8553599, 8631599, 8981279, 9112319, 9428879, 10671119

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 719 is a term.

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

Cf. A101994, A101995, A101996, A101998, A101999.

Subsequence of A127576, A101793 and A101797.

Select[With[{c=2^Range[2,6]},Table[c n-1,{n,700000}]],AllTrue[#,PrimeQ]&][[All,3]] (* Harvey P. Dale, Nov 29 2018 *)

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) && isprime(64*m-1), 0); \\ Amiram Eldar, May 13 2024

a(n) = 16*A101994(n) - 1 = 4*A101995(n) + 3 = 2*A101996(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

A101999 Primes of the form 64k-1 such that 4k-1, 8k-1, 16k-1 and 32*k-1 are also primes.

Original entry on oeis.org

2879, 858239, 1014719, 2029439, 2034239, 4068479, 4737599, 5454719, 9717119, 12968639, 17107199, 17962559, 25579199, 25945919, 29135999, 29859839, 30602879, 30735359, 32725439, 34214399, 34526399, 35925119, 36449279

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 2879 is a term.

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

Cf. A101994, A101995, A101996, A101997, A101998.

Subsequence of A127579.

64#-1&/@Select[Range[570000],AllTrue[#*2^Range[2,6]-1,PrimeQ]&] (* Harvey P. Dale, Aug 07 2021 *)

is(k) = if(k % 64 == 63, my(m = k\64 + 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) = 64*A101994(n) - 1 = 16*A101995(n) + 15 = 8*A101996(n) + 7 = 4*A101997(n) + 3 = 2*A101998(n) + 1. - Amiram Eldar, May 13 2024

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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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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A101996 Primes of the form 8*k-1 such that 4*k-1, 16*k-1, 32*k-1 and 64*k-1 are also primes.

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A101997 Primes of the form 16*k-1 such that 4*k-1, 8*k-1, 32*k-1 and 64*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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A101999 Primes of the form 64*k-1 such that 4*k-1, 8*k-1, 16*k-1 and 32*k-1 are also 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.

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

A101996 Primes of the form 8k-1 such that 4k-1, 16k-1, 32k-1 and 64*k-1 are also primes.

A101997 Primes of the form 16k-1 such that 4k-1, 8k-1, 32k-1 and 64*k-1 are also primes.

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

A101999 Primes of the form 64k-1 such that 4k-1, 8k-1, 16k-1 and 32*k-1 are also primes.