A057331 -id:A057331 - 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

A059766 Initial (unsafe) primes of Cunningham chains of first type with length exactly 6.

Original entry on oeis.org

89, 63419, 127139, 405269, 810809, 1069199, 1178609, 1333889, 1598699, 1806089, 1958249, 2606069, 2848949, 3241289, 3339989, 3784199, 3962039, 4088879, 4444829, 4664249, 4894889, 4897709, 5132999, 5215499, 5238179, 6026309, 6059519, 6088529, 6490769, 6676259

Offset: 1

Labos Elemer, Feb 21 2001

nonn

Special terms of A059453. Not identical to A023330 of which 1122659, 2164229, 2329469, ..., etc. are omitted since they have exact length 7 or larger.

Unsafe primes starting complete chains of length 6.

			89 is a term because (89-1)/2 = 44 and 64*89+63 = 5759 = 13*443 are composites, while 89, 179, 359, 719, 1439, and 2879 are primes.
1122659 is not a term because it initiates a chain of length 7.
4658939 is not a term because (4658939-1)/2 = 2329469 is prime. - _Sean A. Irvine_, Oct 09 2022

Amiram Eldar, Table of n, a(n) for n = 1..1000
Chris Caldwell's Prime Glossary, Cunningham chains.

Cf. A023272, A023302, A023330, A005384, A005385, A059452, A059453, A059454, A059455, A007700, A059759, A059760, A059761, A059762, A059763, A059764, A038397, A104349, A091314, A069362, A016093, A014937, A057326.

Cf. A005602, A053176, A059500, A057331, A059688, A059690.

Entry revised by N. J. A. Sloane Apr 01 2006

a(12) onward corrected and extended by Sean A. Irvine, Oct 09 2022

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

A339581 Indices of records in A063377.

Original entry on oeis.org

1, 2, 89, 1122659, 19099919, 85864769, 26089808579, 554688278429, 4090932431513069, 95405042230542329

Offset: 1

N. J. A. Sloane, Dec 24 2020

The records themselves begin 0,5,6,7,8,9,10,12,13,14.
a(11) <= 90616211958465842219 = A005602(15). Between a(10) and this upper bound could be another record which might not be listed in A005602.
a(n) == 9 mod 10 for n > 2 (see A063377). - Michael S. Branicky, Dec 24 2020

Carl Pomerance, Problem 81:21 (= 321), in R. K. Guy link.

R. K. Guy, editor, Western Number Theory Problems, 1985-12-21 & 23, Typescript, Jul 13 1986, Dept. of Math. and Stat., Univ. Calgary, 11 pages. Annotated scan of pages 1, 3, 7, 9, with permission.

Cf. A005602, A057331, A063377, A063378, A339579, A339580.

a(n) = A057331(n + 2) for n >= 2. - David A. Corneth, Dec 25 2020

a(6) corrected and a(7) found by David A. Corneth, Dec 24 2020.

a(8)-a(10) were taken from A057331 and the bound on a(11) was taken from A005602. - David A. Corneth and Amiram Eldar, Dec 25 2020

A339580 Indices of records in A339579.

Original entry on oeis.org

1, 3, 90, 1122660, 19099920, 85864770, 26089808580, 554688278430, 4090932431513070, 95405042230542330

Offset: 1

N. J. A. Sloane, Dec 24 2020

The records themselves begin 4,5,6,7,8,9,10,12,13,14.

a(11) <= 90616211958465842220.

			90 is in the sequence as A339579(90) = 6 (90*2^k - 1 is prime for k = 0..5 and composite for k = 6) and A339579(m) < 6 for m < 90. - _David A. Corneth_, Dec 24 2020

Carl Pomerance, Problem 81:21 (= 321), in R. K. Guy problem list.

R. K. Guy, editor, Western Number Theory Problems, 1985-12-21 & 23, Typescript, Jul 13 1986, Dept. of Math. and Stat., Univ. Calgary, 11 pages. Annotated scan of pages 1, 3, 7, 9, with permission.

Cf. A063377, A124514, A125113, A339579, A339581.

a(n) = A339581(n) + 1 for n >= 2. - David A. Corneth, Dec 24 2020

a(8)-a(10) were taken from A057331 and the bound on a(11) was taken from A005602. - David A. Corneth and Amiram Eldar, Dec 25 2020

A059690 Number of distinct Cunningham chains of first kind whose initial prime (cf. A059453) <= 2^n.

Original entry on oeis.org

1, 2, 2, 2, 3, 5, 7, 13, 20, 31, 52, 83, 142, 242, 412, 742, 1308, 2294, 4040, 7327, 13253, 24255, 44306, 81700, 150401, 277335, 513705, 954847, 1780466, 3325109, 6224282, 11676337, 21947583, 41327438

Offset: 1

Labos Elemer, Feb 06 2001

			a(11)-a(10) = 21 means that between 1024 and 2048 exactly 21 primes introduce Cunningham chains: {1031, 1049, 1103, 1223, 1229, 1289, 1409, 1451, 1481, 1499, 1511, 1559, 1583, 1601, 1733, 1811, 1889, 1901, 1931, 1973, 2003}.
Their lengths are 2, 3 or 4. Thus the complete chains spread over more than one binary size-zone: {1409, 2819, 5639, 11279}. The primes 1439 and 2879 also form a chain but 1439 is not at the beginning of that chain, 89 is.

C. K. Caldwell, Cunningham Chains
W. Roonguthai, Yves Gallot's Proth.exe and Cunningham Chains

Cf. A023272, A023302, A023330, A005602, A007700, A053176, A059452-A059456, A059500, A057331, A059688, A007053, A036378, A029837, A007053.

c = 0; k = 1; Do[ While[k <= 2^n, If[ PrimeQ[k] && !PrimeQ[(k - 1)/2] && PrimeQ[2k + 1], c++ ]; k++ ]; Print[c], {n, 1, 29}]

from itertools import count, islice
from sympy import isprime, primerange
def c(p): return not isprime((p-1)//2) and isprime(2*p+1)
def agen():
    s = 1
    for n in count(2):
        yield s; s += sum(1 for p in primerange(2**(n-1)+1, 2**n) if c(p))
print(list(islice(agen(), 20))) # Michael S. Branicky, Oct 09 2022

Edited and extended by Robert G. Wilson v, Nov 23 2002
Title and a(30)-a(31) corrected, and a(32) from Sean A. Irvine, Oct 02 2022
a(33)-a(34) from Michael S. Branicky, Oct 09 2022

cp's OEIS Frontend

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

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A059766 Initial (unsafe) primes of Cunningham chains of first type with length exactly 6.

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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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A339581 Indices of records in A063377.

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