A005099 -id:A005099 - cp's OEIS frontend

A101794 Numbers k such that 4k-1, 8k-1, 16k-1 and 32k-1 are all primes.

45, 90, 675, 885, 3030, 4290, 6870, 13410, 14460, 15855, 17850, 18675, 20625, 21885, 25350, 26820, 26925, 28230, 30525, 30705, 31710, 31785, 33375, 34860, 41685, 41940, 57435, 63420, 63570, 71805, 74025, 78585, 83865, 85230, 93075

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 45 is a term.

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

Cf. A002515, A101795, A101796, A101797, A101798.
Subsequence of A005099, A005122 and A101790.
Subsequence: A101994.

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

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

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

Original entry on oeis.org

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

A123986 Numbers n for which 4n+1 and 4n+3 are primes.

Original entry on oeis.org

1, 4, 7, 10, 25, 34, 37, 49, 67, 70, 115, 130, 142, 154, 160, 202, 205, 214, 220, 262, 265, 307, 319, 322, 325, 370, 424, 430, 469, 487, 499, 520, 532, 535, 559, 577, 595, 637, 664, 682, 697, 700, 742, 814, 832, 847, 865, 889, 895, 955, 979, 982, 1000, 1012, 1039

Offset: 1

Artur Jasinski, Oct 30 2006

nonn

All terms == 1 mod 3. - Zak Seidov, Dec 02 2011

Intersection of A005098 and A095278. - Michel Marcus, Jan 31 2015

Zak Seidov, Table of n, a(n) for n = 1..10000

Cf. A005098, A005099, A095278.

[n: n in [0..1100] |IsPrime(4*n+1) and IsPrime(4*n+3)]; // Vincenzo Librandi, Feb 01 2015

Select[Range[1100], And @@ PrimeQ /@ ({1, 3} + 4#) &] (* Ray Chandler, Nov 05 2006 *)
nn=10000;k=0;x=1;re=Reap[While[kZak Seidov, Dec 02 2011 *)

Extended by Ray Chandler, Nov 05 2006

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

A089986 Numbers n such that 4n + 7 is prime.

Original entry on oeis.org

-1, 0, 1, 3, 4, 6, 9, 10, 13, 15, 16, 18, 19, 24, 25, 30, 31, 33, 36, 39, 40, 43, 46, 48, 51, 54, 55, 58, 61, 64, 66, 69, 75, 76, 81, 85, 88, 90, 93, 94, 103, 106, 108, 109, 114, 115, 118, 120, 121, 123, 124, 129, 135, 139, 141, 145, 148, 150, 153, 156, 159, 160, 163, 169

Offset: 1

Giovanni Teofilatto, Jan 13 2004

sign

M. Cerasoli, F. Eugeni and M. Protasi, Elementi di Matematica Discreta, Bologna 1988.
Emanuele Munarini and Norma Zagaglia Salvi, Matematica Discreta,UTET, CittaStudiEdizioni, Milano 1997.

Cf. A005099 ((( Primes = -1 mod 4 ) + 1)/4), A005098 (4n+1 is prime), A095278 (4n+3 is prime), A111215 (4n+5 is prime).

[ n: n in [-1..170] | IsPrime(4*n+7) ];

A089986:=n->`if`(isprime(4*n+7), n, NULL): seq(A089986(n), n=-1..200); # Wesley Ivan Hurt, Sep 18 2014

Select[Range[-1, 200], PrimeQ[4 # + 7] &] (* Wesley Ivan Hurt, Sep 18 2014 *)

is(n)=isprime(4*n+7) \\ Charles R Greathouse IV, Feb 20 2017

a(n) = A005099(n) - 2 = A095278(n) - 1.

Edited and extended by Klaus Brockhaus, Dec 22 2008

A156287 Numbers k such that 4*k-5 is a prime number.

Original entry on oeis.org

2, 3, 4, 6, 7, 9, 12, 13, 16, 18, 19, 21, 22, 27, 28, 33, 34, 36, 39, 42, 43, 46, 49, 51, 54, 57, 58, 61, 64, 67, 69, 72, 78, 79, 84, 88, 91, 93, 96, 97, 106, 109, 111, 112, 117, 118, 121, 123, 124, 126, 127, 132, 138, 142, 144, 148, 151, 153, 156, 159, 162, 163, 166

Offset: 1

Vincenzo Librandi, Feb 07 2009

Two more than the associated A095278, one more than the associated A005099. - R. J. Mathar, Jan 05 2011

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A156288, A005099, A095278.

[n: n in [1..400] | IsPrime(4*n - 5)]; // Vincenzo Librandi, Feb 16 2013

Select[Range[600], PrimeQ[4 # - 5]&] (* Vincenzo Librandi, Feb 16 2013 *)

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

A190105 a(n) = (3*A002145(n) - 1)/4.

Original entry on oeis.org

2, 5, 8, 14, 17, 23, 32, 35, 44, 50, 53, 59, 62, 77, 80, 95, 98, 104, 113, 122, 125, 134, 143, 149, 158, 167, 170, 179, 188, 197, 203, 212, 230, 233, 248, 260, 269, 275, 284, 287, 314, 323, 329, 332, 347, 350, 359, 365, 368, 374, 377, 392, 410, 422, 428, 440

Offset: 1

J. M. Bergot, May 04 2011

For primes p of the form 4n+3, in the order of A002145, let us seek solutions for prime p|(a^x + b^y) or p|(a^y + b^x) subject to the conditions p = a+b = x+y and 0 < a,b,x,y < p. The larger of the two exponents x and y is inserted into the sequence.

If either of (a,b) is a primitive root of p, there is a unique solution, either p|(a^x + b^y) or p|(a^y + b^x). If neither is a primitive root (see A060749), there are multiple solutions and p|(a^x + b^y) or p|(a^y + b^x) will always be one of them for some of the given exponents (x,y) contributing to the sequence.

			For p=43=A002145(7), (x,y)=(11,32) because 43-(43+1)/4=32; hence x=43-32.  With (a,b)=(12,31) the unique solution is 43|(12^11 + 31^32) because 12 is a primitive root of 43. The larger of 11 and 32 is a(7)=32 in the sequence. For 43 multiple solutions occur when neither of the pairs (a,b) is a primitive root of 43: p divides each of 11^4 + 32^39, 11^18 + 32^25, 11^32 + 32^11; note that the exponents (11,32) occur in the last entry.

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A005099 is the list of x in (x,y).

for n from 1 to 200 do p:=4*n-1: if(isprime(p))then printf("%d, ", (3*p-1)/4); fi: od: # Nathaniel Johnston, May 18 2011

A002145 := Select[4 Range[300] - 1, PrimeQ]; Table[(3*A002145[[n]] - 1)/4, {n, 1, 60}] (* G. C. Greubel, Nov 07 2018 *)

A045752 4n-1 is composite.

Original entry on oeis.org

4, 7, 9, 10, 13, 14, 16, 19, 22, 23, 24, 25, 28, 29, 30, 31, 34, 36, 37, 39, 40, 43, 44, 46, 47, 49, 51, 52, 54, 55, 58, 59, 61, 62, 64, 65, 67, 69, 70, 72, 73, 74, 75, 76, 79, 80, 81, 82, 84, 85, 86, 88, 89, 91, 93, 94, 97, 98, 99, 100

Offset: 1

Felice Russo

Apparently the same as "numbers k that can be written as 4xy + x - y for x>0,y>0". - Ron R Spencer, Jul 28 2016
From Wolfdieter Lang, Aug 30 2016: (Start)
Proof: If the 3 (mod 4) number  4*k-1 is composite it can be written as a product of a number a == 3 (mod 4) and powers of numbers 1 (mod 4), that is as a product of
  a = 4*x-1 and b = 4*y+1. Then  4*k-1 = (4*x-1)*(4*y+1) or k = 4*x*y + x - y. And conversely, if k = 4*x*y + x - y then 4*k-1 = (4*x-1)*(4*y+1), that is composite.
The example of Vincenzo Librandi below is equivalent to "numbers m that can be written as 4*H*K + 3*H + K +1 for H>0, K>0" (consider h, k of opposite parity, which is necessary to have even 2*h*k + k + h + 1. W.l.o.g. take h = 2*H and k = 2*K+1). Then 4*m - 1 = (4*K+3)*(4*H+1). This is equivalent to Ron R Spencer's statement with K=x-1, H=y. (End)

			7 belongs to the sequence because 7*4-1=27 is not a prime.
Distribution of the positive terms in the following triangular array:
*;
4,*;
*,9,*;
7,*,16,*;
*,14,*,25,*;
10,*,23,*,36,*; etc.
where * marks the non-integer values of (2*h*k + k + h + 1)/2 with h >= k >= 1. - _Vincenzo Librandi_, Jul 29 2016

Robert Israel, Table of n, a(n) for n = 1..10000

Complement of A005099.

[n: n in [1..120] |not IsPrime(4*n-1)]; // Vincenzo Librandi, Jul 29 2016

remove(t -> isprime(4*t-1), [$1..1000]); # Robert Israel, Jul 29 2016

Select[Range@ 100, CompositeQ[4 # - 1] &] (* Michael De Vlieger, Jul 28 2016 *)

isok(n) = ! isprime(4*n-1); \\ Michel Marcus, Sep 28 2013

A333721 Numbers k such that k + 1, 2k + 1, 3k + 1, 4k + 1, and 6k + 1 are all prime.

Original entry on oeis.org

1530, 4260, 25410, 26040, 78540, 111720, 174990, 211050, 214830, 395430, 403260, 409290, 459690, 487830, 512820, 711120, 779790, 910560, 1023750, 1135950, 1280370, 1312350, 1451520, 1464810, 1487070, 1563510, 1623360, 1698060, 1824330, 1933680, 2006340, 2097480

Offset: 1

Pedro Caceres, May 04 2020

nonn

All terms are multiples of 6.

All terms are multiples of 30. - Robert Israel, Jun 17 2020

			25410 is in the sequence because 25411, 50821, 76231, 101641, 152461 are all prime.

Robert Israel, Table of n, a(n) for n = 1..10000
Pedro Caceres, 30 Ideas about Prime Numbers

Cf. A006093, A005097, A024892, A087370, A005098, A005099, A024898, A024899.

select(t -> andmap(isprime, [t+1,2*t+1,3*t+1,4*t+1,6*t+1]), [seq(i,i=30..3*10^6,30)]); # Robert Israel, Jun 17 2020

isok(m)={for(i=1, 6, if(i<>5&&!isprime(i*m+1), return(0))); 1}
{ forstep(n=0, 3*10^6, 6, if(isok(n), print1(n, ", "))) } \\ Andrew Howroyd, May 04 2020

cp's OEIS Frontend

A101794 Numbers k such that 4*k-1, 8*k-1, 16*k-1 and 32*k-1 are all primes.

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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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A123986 Numbers n for which 4n+1 and 4n+3 are 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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A089986 Numbers n such that 4n + 7 is prime.

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A156287 Numbers k such that 4*k-5 is a prime number.

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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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A190105 a(n) = (3*A002145(n) - 1)/4.

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A101794 Numbers k such that 4k-1, 8k-1, 16k-1 and 32k-1 are all primes.

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.

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