A057331 -id:A057331 - cp's OEIS frontend

A111145 Length of the Cunningham chain initiated by the n-th Sophie Germain prime.

5, 2, 4, 3, 2, 2, 3, 2, 2, 6, 2, 2, 2, 5, 2, 2, 2, 2, 2, 2, 4, 2, 2, 2, 2, 4, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 3, 3, 2, 2, 2, 4, 2, 4, 2, 3, 3, 2, 3, 2, 2, 2, 2, 3, 3, 3, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 4, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 4, 2, 2, 2, 2, 2, 3, 2, 2, 3, 2, 2, 2, 2

Offset: 1

Christopher M. Tomaszewski (cmt1288(AT)comcast.net), Oct 18 2005

If a(n) is a high-water mark of this sequence, then A057331(a(n)) is the first term of the first Cunningham sequence of length a(n). For example, a(10)=6 is a high-water mark of this sequence and A057331(a(10))=89 is the first term of the first Cunningham sequence of length 6.

			a(10)=6 because 89, the 10th Sophie Germain prime, is the first term of the Cunningham chain 89, 179, 359, 719, 1439, 2879, which consists of 6 terms.

T. D. Noe, Table of n, a(n) for n = 1..10000

Cf. A005384, A057331.

lst=Select[Prime[Range[1000]], PrimeQ[2#+1]&]; Table[p=lst[[i]]; k=1; While[p=2p+1; PrimeQ[p], k++ ]; k, {i,Length[lst]}] (* T. D. Noe, Jun 06 2006 *)
ccl[n_]:=Length[NestWhileList[2#+1&,n,PrimeQ[2#+1]&]]; ccl/@Select[ Prime[ Range[1000]],PrimeQ[2#+1]&] (* Harvey P. Dale, Sep 29 2018 *)

More terms from T. D. Noe, Jun 06 2006

A321058 Indices of records in A067849.

Original entry on oeis.org

1, 2, 44, 561329, 9549959, 42932384, 13044904289, 277344139214, 2045466215756534, 47702521115271164

Offset: 1

Torlach Rush, Oct 26 2018

The corresponding record values are 2,4,6,7,8,9,10,12,13,14.
From David A. Corneth, Nov 10 2018: (Start)
Terms a(n) are of the form 3*k+2 for n > 1.
If 2^k - 1 is composite then a(n) is not divisible by any prime factor of 2^k-1 for n > k. So for example, gcd(a(n), 105) = 1 for n > 5. (End)
From Glen Whitney, Sep 14 2022: (Start)
Similarly to Corneth's observations, modulo any prime p, any residue for a(n) of the form 2^k - 1 mod p is forbidden for n greater than or equal to the number of such residues; for example a(n) may not be congruent to 0, 1, or 3 mod 7 for n >= 3.
For n > 2, if a(n) appears in this sequence, 2a(n) + 1 must appear in A057331. (End)

			2 is a term because A067849(2) = 4 > A067849(1) = 2.
44 is a term because A067849(44) = 6 > A067869(2) = 4.

Cf. A067849, A005602, A057331, A063378.

b(n) = {my(nb = 0, newn); while (isprime(newn=2*n+1), nb++; n = newn); nb;} \\ A067849
lista(nn) = {my(mmax = -1, mm); for (n=1, nn, if ((mm=b(n)) > mmax, mmax = mm; print1(n, ", ")););} \\ Michel Marcus, Nov 10 2018

a(7) from Amiram Eldar, Nov 10 2018

a(8)-a(10) from A057331 by Glen Whitney, Sep 14 2022

cp's OEIS Frontend

A111145 Length of the Cunningham chain initiated by the n-th Sophie Germain prime.

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