A338944 -id:A338944 - cp's OEIS frontend

A338945 Lengths of Cunningham chains of the first kind that are sorted by first prime in the chain.

5, 2, 1, 1, 1, 2, 1, 1, 3, 1, 2, 1, 1, 1, 1, 1, 6, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 2, 2, 1, 2, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 1, 2, 1

Offset: 1

Michael De Vlieger, Nov 17 2020

nonn

Row lengths of A075712.

			We begin with the smallest prime 2. Since 2(5) + 1 = 11 is prime, further, 2(11) + 1 = 23, 2(23) + 1 = 47 are prime but 2(47) + 1 = 95 is not, we have the complete chain {2, 5, 11, 23, 47} of length 5, thus a(1) = 5.
We resume with 3, since 3 has not appeared in any chain generated thus far. Since 2(3) + 1 = 7, but 2(7) + 1 = 15, we have the complete chain {3, 7}, therefore a(2) = 2.
Starting from 13, we find 2(13) + 1 = 27, thus we have a singleton chain and have a(3) = 1, etc.

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Chris K. Caldwell, Cunningham Chain (PrimePages, Prime Glossary).
Wikipedia, Cunningham chain.

Cf. A075712, A338944, A338946.

Block[{a = {2}, b = {}, j = 0, k, p}, Do[k = Length@ b + 1; If[PrimeQ@ a[[-1]], AppendTo[a, 2 a[[-1]] + 1]; j++, While[! FreeQ[a, Set[p, Prime[k]]], k++]; AppendTo[b, j]; Set[j, 0]; Set[a, Append[a[[1 ;; -2]], p]]], {10^3}]; b]

A338946 Lengths of Cunningham chains of the second kind that are sorted by first prime in the chain.

Original entry on oeis.org

3, 2, 1, 1, 3, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 3, 2, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 3, 1, 1, 1, 1, 1, 1

Offset: 1

Michael De Vlieger, Nov 17 2020

nonn

Row lengths of A338944.

			We start with p = 2. Since 2(2) - 1 = 3 is prime, and further 2(3) - 1 = 5 is prime, but 2(5) - 1 is composite, we have chain length 3, so a(1) = 3.
p = 7 is the smallest prime that hasn't appeared in a chain thus far; since 2(7) - 1 = 13 is prime but 2(13) - 1 = 25 is composite, we have a chain of length 2, so a(2) = 2.
p = 11 is the smallest prime that hasn't appeared in a chain; 2(11) - 1 = 21 is composite, so we have a singleton chain, thus a(3) = 1, etc.

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Chris K. Caldwell, Cunningham Chain (PrimePages, Prime Glossary).
Wikipedia, Cunningham chain.

Cf. A075712, A338944, A338945.

Block[{a = {2}, b = {}, j = 0, k, p}, Do[k = Length@ b + 1; If[PrimeQ@ a[[-1]], AppendTo[a, 2 a[[-1]] - 1]; j++, While[! FreeQ[a, Set[p, Prime[k]]], k++]; AppendTo[b, j]; Set[j, 0]; Set[a, Append[a[[1 ;; -2]], p]]], {10^3}]; b]

cp's OEIS Frontend

A338945 Lengths of Cunningham chains of the first kind that are sorted by first prime in the chain.

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