A057330 -id:A057330 - cp's OEIS frontend

A084960 Initial prime of a prime chain of length n under the iteration x->5x+4.

2, 3, 5, 83, 263, 5333, 5333, 6714497, 42360737, 3757699889, 3757699889, 1431898413161, 5654774136689, 12756824771254199, 184574272412533499

Offset: 1

W. Edwin Clark, Jun 14 2003

This is a special case of prime chains generated by f(x) = c*x + d.

			a(3) = 5 since 5, f(5) = 29 and f(29) = 149 are primes when f(x) = 5x+4.

D. H. Lehmer, On certain chains of primes, Proc. London Math. Soc. (3) 14a 1965 183-186.

Cf. A057330, A057331, A083388, A084954, A084955, A084956, A084957, A084958, A084959, A084961.

t[p_] := Block[{c=1, q = 5*p+4}, While[ PrimeQ@q, q = 5*q + 4; c++]; c]; a[n_] := Block[{p = 2}, While[t[p] < n, p = NextPrime@ p]; p]; Array[a, 8] (* Giovanni Resta, Mar 21 2017 *)

a(9) from Stefan Steinerberger, May 18 2007
a(10)-a(11) from Donovan Johnson, Sep 27 2008
a(12)-a(13) from Giovanni Resta, Mar 21 2017
a(14)-a(15) from Bert Dobbelaere, May 30 2025

A084958 Initial prime of a prime chain of length n under the iteration x->5x+2.

Original entry on oeis.org

2, 3, 13, 19, 373, 135859, 135859, 18235423, 26588257, 93112729, 376038903103, 7087694466289, 120223669028389

Offset: 1

W. Edwin Clark, Jun 14 2003

This is a special case of prime chains generated by f(x) = cx + d.

a(11) > 8695354111. - Donovan Johnson, Sep 27 2008

			a(3)=13 since 13, f(13)=67 and f(67)=337 are primes when f(x) = 5x+2.

D. H. Lehmer, On certain chains of primes, Proc. London Math. Soc. (3) 14a 1965 183-186.

Cf. A057330, A057331, A083388, A084954, A084955, A084956, A084957, A084959, A084960, A084961.

c[p_] := Block[{k = 1, q = 5*p+2}, While[ PrimeQ[q], q = 5*q+2; k++]; k]; a[n_] := Block[{p = 2}, While[c[p] < n, p = NextPrime@ p]; p]; Array[a, 7] (* Giovanni Resta, Mar 21 2017 *)

a(10) from Donovan Johnson, Sep 27 2008
a(11)-a(12) from John Cerkan, Jan 20 2017
a(13) from Giovanni Resta, Mar 21 2017

A084956 Initial prime of the first prime chain of length n under the iteration x -> 3x+4.

Original entry on oeis.org

2, 3, 3, 23, 3203, 34613, 165443, 1274803, 26314573, 26314573, 590256673403, 15113026057043, 334156170011893, 3998669569752373

Offset: 1

W. Edwin Clark, Jun 14 2003

This is a special case of prime chains generated by f(x) = cx + d.

a(11) > 8695354111. - Donovan Johnson, Sep 27 2008

			a(3) = 3 since 3, f(3) = 13 and f(13) = 43 are primes when f(x) = 3*x + 4.

D. H. Lehmer, On certain chains of primes, Proc. London Math. Soc. (3) 14a 1965 183-186.

Cf. A057330, A057331, A083388, A084954, A084955, A084957, A084958, A084959, A084960, A084961.

c[p_] := Block[{k=1, q=3*p + 4}, While[PrimeQ[q], q=3*q+4; k++]; k]; a[n_] := Block[{p = 2}, While[c[p] < n, p = NextPrime[p]]; p]; Array[a, 7] (* Giovanni Resta, Mar 22 2017 *)

a(9)-a(10) from Donovan Johnson, Sep 27 2008
a(11)-a(12) from John Cerkan, Jan 13 2017
a(13)-a(14) from Giovanni Resta, Mar 22 2017

A084957 Initial prime of the first prime chain of length n under the iteration x -> 4x + 3.

Original entry on oeis.org

2, 2, 2, 2, 1447, 9769, 17231, 17231, 32611, 18527009, 161205841, 3123824801, 26813406071, 4398156030379, 4398156030379

Offset: 1

W. Edwin Clark, Jun 14 2003

This is a special case of prime chains generated by f(x) = c*x + d.

			a(3) = 2 since 2, f(2) = 11, and f(11) = 47 are primes when f(x) = 4*x + 3.

D. H. Lehmer, On certain chains of primes, Proc. London Math. Soc. (3) 14a 1965 183-186.

Cf. A057330, A057331, A083388, A084954, A084955, A084956, A084958, A084959, A084960, A084961.

c[p_] := Block[{k=1, q=4*p+3}, While[ PrimeQ[q], q=4*q+3; k++]; k]; a[n_] := Block[ {p=2}, While[c[p] < n, p = NextPrime@ p]; p]; Array[a, 9] (* Giovanni Resta, Mar 21 2017 *)

has(p,n)=for(i=2,n, if(!isprime(p=4*p+3), return(0))); 1
a(n)=forprime(p=2,, if(has(p,n), return(p))) \\ Charles R Greathouse IV, Jan 20 2017

a(11)-a(12) from Donovan Johnson, Sep 27 2008
a(13) from John Cerkan, Jan 20 2017
a(14)-a(15) from Giovanni Resta, Mar 21 2017

A084959 Initial prime of a prime chain of length n under the iteration x->5x+6.

Original entry on oeis.org

2, 5, 7, 7, 79, 79, 345431, 21171649, 34640153, 4174239239, 268130051191, 268130051191, 253134809926049, 253134809926049, 253134809926049

Offset: 1

W. Edwin Clark, Jun 14 2003

This is a special case of prime chains generated by f(x) = cx + d.

a(11) > 8695354111. [Donovan Johnson, Sep 27 2008]

			a(3) = 13 since 7, f(7) = 41, and f(41) = 211 are primes when f(x) = 5*x + 6.

D. H. Lehmer, On certain chains of primes, Proc. London Math. Soc. (3) 14a 1965 183-186.

Cf. A057330, A057331, A083388, A084954, A084955, A084956, A084957, A084958, A084960, A084961.

c[p_] := Block[{k=1, q = 5*p+6}, While[PrimeQ[q], q = 5*q+6; k++]; k]; a[n_] := Block[{p = 2}, While[c[p] < n, p = NextPrime[p]]; p]; Array[a, 7] (* Giovanni Resta, Mar 22 2017 *)

a(7) corrected and a(8)-a(10) from Donovan Johnson, Sep 27 2008
a(11)-a(12) from John Cerkan, Jan 11 2017
a(13)-a(15) from Giovanni Resta, Mar 22 2017

A084961 Initial prime of the first prime chain of length n under the iteration x->6x+5.

Original entry on oeis.org

2, 2, 2, 2, 11, 13, 115571, 23586221, 53165771, 3398453717, 615502598677, 32504183957101, 164289842304587

Offset: 1

W. Edwin Clark, Jun 14 2003

This is a special case of prime chains generated by f(x) = cx + d.

a(11) > 10175130529. [Donovan Johnson, Sep 27 2008]

			a(3) = 2 since 2, f(2) = 17, and f(17) = 107 are primes when f(x) = 6*x + 5.

D. H. Lehmer, On certain chains of primes, Proc. London Math. Soc. (3) 14a 1965 183-186.

Cf. A057330, A057331, A083388, A084954, A084955, A084956, A084957, A084958, A084959, A084960.

c[p_] := Block[{k=1, q=6*p+5}, While[ PrimeQ[q], q = 6*q+5; k++]; k]; a[n_] := Block[ {p=2}, While[c[p] < n, p = NextPrime[p]]; p]; Array[a, 7] (* Giovanni Resta, Mar 22 2017 *)

a(8)-a(10) from Donovan Johnson, Sep 27 2008
a(11)-a(12) from John Cerkan, Jan 11 2017
a(13) from Giovanni Resta, Mar 22 2017

A064812 Smallest prime p such that the infinite sequence {p, p'=2p-1, p''=2p'-1, ...} begins with a string of exactly n primes.

Original entry on oeis.org

5, 3, 2, 2131, 1531, 33301, 16651, 15514861, 857095381, 205528443121, 1389122693971, 216857744866621, 758083947856951, 107588900851484911, 69257563144280941

Offset: 1

David Terr, Oct 21 2002

nonn

Chains of length n of nearly doubled primes.

Smallest prime beginning a complete Cunningham chain of length n of the second kind. (For the first kind see A005602.) - Jonathan Sondow, Oct 30 2015

			a(3) = 2 because 2 is the smallest prime such that the sequence {2, 3, 5, 9, ...} begins with exactly 3 primes, where each term in the sequence is twice the preceding term minus 1.

G. Löh, Long chains of nearly doubled primes, Math. Comp., 53 (1989), 751-759.
Wikipedia, Cunningham chain

A better version of A005603.
Cf. (A005382 and A005383), A057326, A057327, A057328, A057329, A057330, A005602.
Cf. A002808, A006450, A049076-A049081, A050436, A076237-A076239, A050436-A050439, A065860, A181697, A263879.

A231814 Squarefree numbers (from A005117) with prime divisors in a 2p-1 progression.

Original entry on oeis.org

6, 15, 30, 91, 703, 1891, 2701, 12403, 18721, 38503, 49141, 51319, 79003, 88831, 104653, 146611, 188191, 218791, 226801, 269011, 286903, 385003, 497503, 597871, 665281, 721801, 736291, 765703, 873181, 954271, 1056331, 1314631, 1373653, 1537381, 1755001, 1869211

Offset: 1

Jaroslav Krizek, Nov 13 2013

nonn

Squarefree numbers with k >= 2 prime factors of the form p_1 * p_2 * ... * p_k, where p_1 < p_2 < ... < p_k = primes with p_k = 2 * p_(k-1) - 1.
Each of these numbers is divisible by the arithmetic mean of its proper divisors.
Supersequence of A129521 (numbers of the form p*q, p and q prime with q=2*p-1; see A005382 and A005383).

			51319 = 19*37*73 where 37 = 2*19 - 1, 73 = 2*37 - 1.

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

Cf. A000040, A005117, A005382, A005383, A129521, A231815, A231816.

Cf. A057330 (first prime for such numbers that has n factors).

N:= 10^7: # for terms <= N
p:= 1: S:= NULL: count:= 0:
do
  p:= nextprime(p);
  if p*(2*p-1) > N then break fi;
  q:= p; x:= p;
  do
    q:= 2*q-1;
    if not isprime(q) then break fi;
    x:= x*q;
    if x > N then break fi;
    S:= S,x; count:= count+1;
  od;
od:
sort([S]); # Robert Israel, Mar 24 2023

geomQ[lst_] := Module[{x = lst - 1}, x = x/x[[1]]; Log[2, x] + 1 == Range[Length[x]]]; Select[Range[2, 1000000], ! PrimeQ[#] && SquareFreeQ[#] && geomQ[Transpose[FactorInteger[#]][[1]]] &] (* T. D. Noe, Nov 14 2013 *)

A231816 a(n) = the smallest squarefree number (A005117) with n prime factors in a 2p-1 progression.

Original entry on oeis.org

2, 6, 30, 351137972965951, 8596208716179446431, 698211042943963834650959743951, 744014385572130806167897354113929551, 901203402294977554329263775346819632824908852456695769189267773301

Offset: 1

Jaroslav Krizek, Nov 13 2013

nonn

Smallest squarefree numbers with n >= 2 prime divisors of the form p_1 * p_2 * … * p_n, where p_1 < p_2 < … < p_k = primes with p_k = 2 * p_(k-1) - 1.

Subsequence of A231814.

			8596208716179446431 = 1531*3061*6121*12241*24481, where 3061 = 2*1531 - 1, 6121 = 2*3061 - 1, 12241 = 2*6121 - 1, 24481 = 2*12241 - 1.

Jaroslav Krizek, Table of n, a(n) for n = 1..15

Cf. A005117, A000040, A231814, A231815.

Cf. A057330 (first prime for such numbers that has n factors).

A128509 Initial prime, greater than 7, of a prime chain of length n under the iteration x -> 2x-7.

Original entry on oeis.org

11, 13, 13, 367, 4987, 9697, 78007, 13356127, 13356127, 6753069187, 218617084627, 35679817649197, 92975416840027

Offset: 1

Zak Seidov, May 07 2007

From the definition we see that the sequence is monotonic (but not strictly) increasing. The sequence is a generalized Cunningham chain. - Stefan Steinerberger, May 12 2007

			a(2) = 13 because 13, 13*2-7=19 and 19*2-7=31 are primes.

Cf. A057330, A084954.

k = 5; For[n = 2, n <= 10, n++, While[Union[PrimeQ[NestList[2# - 7 &, Prime[k], n]], PrimeQ[NestList[2# - 7 &, Prime[k], n]]] != {True}, k++ ]; Print[Prime[k]]] (* Stefan Steinerberger, May 12 2007 *)

Edited by Stefan Steinerberger, May 12 2007

Edited by Don Reble, Nov 07 2007

cp's OEIS Frontend

A084960 Initial prime of a prime chain of length n under the iteration x->5x+4.

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A084958 Initial prime of a prime chain of length n under the iteration x->5x+2.

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A084956 Initial prime of the first prime chain of length n under the iteration x -> 3x+4.

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A084957 Initial prime of the first prime chain of length n under the iteration x -> 4x + 3.

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A084959 Initial prime of a prime chain of length n under the iteration x->5x+6.

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A084961 Initial prime of the first prime chain of length n under the iteration x->6x+5.

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A064812 Smallest prime p such that the infinite sequence {p, p'=2p-1, p''=2p'-1, ...} begins with a string of exactly n primes.

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A231814 Squarefree numbers (from A005117) with prime divisors in a 2p-1 progression.

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