A088250 -id:A088250 - cp's OEIS frontend

A088251 A088250(n) + 1.

2, 2, 3, 331, 10831, 25411, 512821, 512821, 12960606121, 434491727671, 1893245380951, 71023095613471, 878232256181281

Offset: 1

Amarnath Murthy, Sep 26 2003

nonn

The n-th row of the following triangle contains smallest set of n primes which form n successive terms of an arithmetic progression from the 2nd to (n+1)th term with the first term 1. 2 2 3 3 5 7 331 661 991 1321 ... Sequence contains the first column.
Conjecture: (1) Sequence is infinite. (2) For every n there are infinitely many arithmetic progressions with n successive primes.
Minimal primes p beginning a chain of n primes in an arithmetic progression of common difference  p-1. - Robin Garcia, Jun 22 2013
Least prime p such that pi = i*p-i+1 is prime for i = 2 to i = n. - Robin Garcia, Jun 22 2013
a(n) is 1 mod 10 for n > 3 because if p is 3 mod 10, then all (2+5*t)*p -(1+5*t) for t=0,1,2,... are 5 mod 10; if p is 7 mod 10, all (4+5*t)*p -(3+5*t) are 5 mod 10 for t=0,1,2...;  if p is 9 mod 10, all (3+5*t)*p - (2+5*t) are 5 mod 10 for t=0,1,2... - Robin Garcia, Jun 22 2013

			The n-th row of the following triangle contains smallest set of n primes which form n successive terms of an arithmetic progression from the 2nd to (n+1)-st term with the first term 1.
2
2 3
3 5 7
331 661 991 1321
...
Sequence contains the first column.

Cf. A002110, A088250, A088252.

More terms from Don Reble and Farideh Firoozbakht, Feb 17 2004

A071576 a(n) = least k such that 2ik + 1 is prime for all 1 <= i <= n.

Original entry on oeis.org

1, 1, 1, 165, 5415, 12705, 256410, 256410, 6480303060, 217245863835, 946622690475, 35511547806735, 439116128090640, 5714676453270219435

Offset: 1

Benoit Cloitre, May 31 2002

Cf. A088250, A124516, A124522, A124522, A063983.

Cf. A005097, A123998, A124408-A124411.

k = 1; Do[ While[p = Table[2*i*k + 1, {i, 1, n}]; Union[ PrimeQ[p]] != {True}, k++ ]; Print[k], {n, 1, 15}] (* Robert G. Wilson v *)

for(n=1,6,s=1; while(sum(i=1,n,isprime(2*s*i+1))

Extended by Robert G. Wilson v, Jun 06 2002
a(9) from Ryan Propper, Jun 20 2005
a(10)-a(13) from Don Reble, Nov 05 2006
a(14) from Giovanni Resta, Apr 01 2017

A088651 a(n) = smallest number k such that rk-1 is prime for all r = 1 to n.

Original entry on oeis.org

3, 3, 4, 6, 6, 154770, 2894220, 2894220, 407874180, 214580145780, 9448481062020, 247236503934420, 2545206711847800, 18178612369988250180, 53792264108455702830

Offset: 1

Amarnath Murthy, Oct 29 2003

nonn

Is the sequence finite?

			a(5)=6 because 1*6-1, 2*6-1, 3*6-1, 4*6-1, 5*6-1 are all prime.
a(11)=9448481062020 because all eleven numbers 1*9448481062020-1, 2*9448481062020-1,3*9448481062020-1,...,10*9448481062020-1 & 11*9448481062020-1 are prime and 9448481062020 is the smallest number with such property.

Carlos Rivera, Problems & Puzzles.

Cf. A088652, A125838, A125839, A088250, A101779.

Corrected and extended by Ray Chandler, Nov 01 2003
More terms from Don Reble, Nov 23 2003
Entry revised by N. J. A. Sloane, Jan 05 2007
a(14)-a(15) from Jens Kruse Andersen, May 02 2008

A088252 n-th row of the following triangle contains smallest set of n primes which form n successive terms of an arithmetic progression from the 2nd to (n+1)th term with the first term 1. Sequence contains the leading diagonal.

Original entry on oeis.org

2, 3, 7, 1321, 54151, 152461, 3589741, 4102561, 116645455081, 4344917276701, 20825699190451, 852277147361641, 11417019330356641

Offset: 1

Amarnath Murthy, Sep 26 2003

nonn

Conjectures: (1) Sequence is infinite. (2) For every n there are infinitely many arithmetic progressions with n successive primes.

A088252(n)=n*A088250(n)+1=n*A088251(n)-n+1. - Farideh Firoozbakht, Feb 21 2004

			2
2 3
3 5 7
331 661 991 1321
...

Cf. A002110, A088250, A088251.

More terms from Farideh Firoozbakht, Feb 21 2004

A125838 a(n) is the smallest number m such that k*m - 1 for k=2,3,...,n is prime.

Original entry on oeis.org

2, 2, 2, 6, 120, 120, 2894220, 397073040, 1236161850, 764907546690, 8955490023480, 138393712627170, 8047290924923250

Offset: 2

Carlos Rivera, Jan 01 2007

			a(7)=120 because 2*120-1, 3*120-1, 4*120-1, 5*120-1, 6*120-1 & 7*120-1 are prime and 120 is the smallest such number.

Carlos Rivera, Problems & Puzzles.

Cf. A125839, A088651, A088250.

a[n_] := Block[{k=2}, While[! AllTrue[k Range[2, n] - 1, PrimeQ], k += 2]; k]; a /@
Range[2, 8] (* Giovanni Resta, Mar 29 2017 *)

a(8) from Luke Pebody (luke.pebody(AT)gmail.com)
a(9)-a(10) from Vladimir Trushkov (vladimir(AT)trushkov.botik.ru)
a(11) from Farideh Firoozbakht
a(5) corrected by and a(12)-a(14) from Giovanni Resta, Mar 29 2017

A125839 a(n) is the smallest number m such that k*m - 1 is prime for all k=3,4,...,n.

Original entry on oeis.org

1, 1, 6, 18, 120, 1260, 1485540, 28667100, 28667100, 842889105240, 2281585556250, 163881570370980, 45187548280664790

Offset: 3

Luke Pebody (luke.pebody(AT)gmail.com), Jan 02 2007

For n > 6, 10 divides a(n).

			a(11)=28667100 because 3*28667100-1, 4*28667100-1, 5*28667100-1, 6*28667100-1, 7*28667100-1, 8*28667100-1, 9*28667100-1, 10*28667100-1 & 11*28667100-1 are prime and 28667100 is the smallest number with this property.

Carlos Rivera, Problems & puzzles.

Cf. A125838, A088651, A088250.

a[n_] := Block[{k = If[n<5, 1, 6], s}, s = k;  While[! AllTrue[k Range[3, n] - 1, PrimeQ], k += s]; k]; a /@ Range[3, 9] (* Giovanni Resta, Mar 29 2017 *)

a(12)-a(13) from Farideh Firoozbakht

a(14)-a(15) from Giovanni Resta, Mar 30 2017

A237189 Numbers k such that k+1, 2k+1, 3k+1, 4k+1 are all prime.

Original entry on oeis.org

330, 1530, 3060, 4260, 4950, 6840, 10830, 15390, 18120, 23010, 25410, 26040, 31770, 33300, 40110, 41490, 45060, 49830, 53880, 59340, 65850, 70140, 73770, 78540, 88740, 95460, 96930, 109470, 111720, 112620, 117720, 131310, 133200, 134730, 135300, 150150, 165900

Offset: 1

Alex Ratushnyak, Feb 04 2014

nonn

A subsequence of A064238.

All terms are divisible by 30, and b(n)=a(n)/30 begins: 11, 51, 102, 142, 165, 228, 361, 513, 604, 767, 847, 868, 1059, 1110, 1337, 1383, 1502, 1661, 1796, 1978, 2195, ...

Jon E. Schoenfield, Table of n, a(n) for n = 1..10000 (first 1000 terms from Giovanni Resta)

Cf. A088250, A064238, A105653, A124409.

import sympy
from sympy import isprime
for n in range(0,100000,2):
    if isprime(n+1) and isprime(2*n+1) and isprime(3*n+1) and isprime(4*n+1):
        print(str(n), end=',')

a(n) = 2*(A105653(n) + 1) = 2*A124409(n). - Hugo Pfoertner, May 03 2021

A164325 a(n) is the smallest number m such that (2k-1)m+1 is prime for all 0

Original entry on oeis.org

1, 2, 2, 6, 1170, 64590, 25153800, 25153800, 4747505070, 207187349040, 6703860240000, 26997529639080, 1760354281625940, 1760354281625940, 10718654377787155800

Offset: 1

Farideh Firoozbakht, Sep 15 2009

Cf. A088250, A071576, A164326, A088251, A088651, A125838, A125839, A088252.

a(5) corrected by Zak Seidov, Sep 16 2009
a(10) and a(11) from Zak Seidov, Sep 17 2009
a(12)=26997529639080 from Zak Seidov, Sep 25 2009
a(13)-a(15) from Giovanni Resta, Apr 01 2017

A237190 Numbers k such that k+1, 2k+1, 3k+1, 4k+1, 5k+1 are five primes.

Original entry on oeis.org

10830, 25410, 26040, 88740, 165900, 196560, 211050, 224400, 230280, 247710, 268500, 268920, 375480, 377490, 420330, 451410, 494340, 512820, 592620, 604170, 735750, 751290, 765780, 799170, 808080, 952680, 975660, 1053690, 1064190, 1132860, 1156170, 1532370, 1559580

Offset: 1

Alex Ratushnyak, Feb 04 2014

nonn

A subsequence of A237189.

All terms are divisible by 30, and b(n) = a(n)/30 begins: 361, 847, 868, 2958, 5530, 6552, 7035, 7480, 7676, 8257, 8950, 8964, 12516, 12583, 14011, ...

Giovanni Resta, Table of n, a(n) for n = 1..1000

Cf. A088250, A064238, A124410, A237189.

Select[30*Range[52000],AllTrue[#*Range[5]+1,PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Dec 31 2017 *)

from sympy import isprime
for n in range(2000000):
    if isprime(n+1) and isprime(2*n+1) and isprime(3*n+1) and isprime(4*n+1) and isprime(5*n+1):
        print(n, end=', ')

A372238 Least number m such that 9km+1 is prime for k=1..n.

Original entry on oeis.org

2, 2, 4, 170, 9860, 23450, 56980, 56980, 6723767050, 48276858630, 77460393371130, 97581361797920, 97581361797920, 1269928100726715430

Offset: 1

Jean-Marc Rebert, Apr 23 2024

			a(1) = 2, because 9*1*2 + 1 = 19 is prime and no lesser number has this property.

David A. Corneth, PARI program

Cf. A372188, A088250.

p[m_, n_] := AllTrue[Range[n], PrimeQ[9*#*m + 1] &];
a[n_] := a[n] = Module[{m = 1}, While[! p[m, n], m++]; m]
Table[a[n], {n, 1, 9}] (* Robert P. P. McKone, May 02 2024 *)

is(n,m)=my(u=vector(n,k,9*k*m+1));for(i=1,n,if(!isprime(u[i]),return(0)));1
a(n)=my(pas=1);if(n<15,if(n>2,pas=factorback(primes(primepi(n)));pas/=3;my(m=pas));forstep(m=pas,+oo,pas,if(is(n,m),return(m))))

PARI
```
See PARI link
```

If A088250(n) is divisible by 9, then a(n) = A088250(n) / 9. - Jason Yuen, Apr 25 2024

a(11)-a(13) from David A. Corneth, Apr 24 2024

a(14) from Jason Yuen, Apr 25 2024

cp's OEIS Frontend

A088251 A088250(n) + 1.

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A071576 a(n) = least k such that 2ik + 1 is prime for all 1 <= i <= n.

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A088651 a(n) = smallest number k such that rk-1 is prime for all r = 1 to n.

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A088252 n-th row of the following triangle contains smallest set of n primes which form n successive terms of an arithmetic progression from the 2nd to (n+1)th term with the first term 1. Sequence contains the leading diagonal.

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A125838 a(n) is the smallest number m such that k*m - 1 for k=2,3,...,n is prime.

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A125839 a(n) is the smallest number m such that k*m - 1 is prime for all k=3,4,...,n.

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A237189 Numbers k such that k+1, 2k+1, 3k+1, 4k+1 are all prime.

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A164325 a(n) is the smallest number m such that (2k-1)m+1 is prime for all 0

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A237190 Numbers k such that k+1, 2k+1, 3k+1, 4k+1, 5k+1 are five primes.

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A372238 Least number m such that 9km+1 is prime for k=1..n.