Nicholas Leonard - cp's OEIS frontend

A364936 a(n) = minimum number of variables with n possible states in a system such that its solution requires the processing of a transcomputational number of bits.

309, 195, 155, 134, 120, 111, 103, 98, 93, 90, 87, 84, 82, 80, 78, 76, 75, 73, 72, 71, 70, 69, 68, 67, 66, 65, 65, 64, 63, 63, 62, 62, 61, 61, 60, 60, 59, 59, 59, 58, 58, 57, 57, 57, 56, 56, 56, 56, 55, 55

Offset: 2

Nicholas Leonard, Aug 13 2023

The number 10^93, known as Bremermann's limit, is the estimated maximum number of bits able to be processed by a hypothetical Earth-sized computer in a period of time equal to the rough estimate of the Earth's age. All numbers greater than Bremermann's limit are labeled as "transcomputational."

			For k = 2 (i.e., a set of n Boolean variables), 309 is the corresponding term of this sequence as it is the smallest integer which satisfies 10^93 < 2^n.

H. J. Bremermann, "Optimization through evolution and recombination" in Self-Organizing Systems, Spartan Books, 1962, pages 93-106.
G. J. Klir, Facets of Systems Science, Springer, 1991, pages 121-128.

H. J. Bremermann, Optimization through evolution and recombination
Wikipedia, Transcomputational problem

Table[Ceiling[93 Log[10] / Log[n]], {n, 2, 51}]

a(n) = ceiling(93*log(10)/log(n)).

A363752 Primes prime(k) such that prime(k) mod k is prime.

Original entry on oeis.org

5, 7, 17, 19, 23, 41, 47, 53, 61, 71, 79, 89, 101, 107, 113, 127, 131, 137, 139, 151, 163, 167, 173, 181, 191, 193, 197, 211, 223, 227, 229, 233, 239, 241, 257, 269, 277, 281, 313, 317, 347, 359, 367, 373, 383, 397, 421, 433, 443, 457, 463, 479, 503, 521, 541

Offset: 1

Nicholas Leonard, Jun 19 2023

nonn

			The 9th prime is 23 and 23 mod 9 = 5, which is prime, so 23 is a term.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000040, A004648, A363751.

Table[If[PrimeQ[Mod[Prime[k], k]], Prime[k], Nothing], {k, 1, 100}]

from sympy import prime, isprime
a363752=[]
for k in range(1, 101):
    if isprime(prime(k)%k):
        a363752.append(prime(k))

a(n) = A000040(A363751(n)).

A363751 Numbers k such that prime(k) mod k is prime.

Original entry on oeis.org

3, 4, 7, 8, 9, 13, 15, 16, 18, 20, 22, 24, 26, 28, 30, 31, 32, 33, 34, 36, 38, 39, 40, 42, 43, 44, 45, 47, 48, 49, 50, 51, 52, 53, 55, 57, 59, 60, 65, 66, 69, 72, 73, 74, 76, 78, 82, 84, 86, 88, 90, 92, 96, 98, 100, 102, 106, 108, 112, 116, 120, 126, 128, 130

Offset: 1

Nicholas Leonard, Jun 19 2023

nonn

			9 is a term of this sequence as prime(9) mod 9 = 5, which is prime.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000720, A004648, A363752.

Table[If[PrimeQ[Mod[Prime[k], k]], k, Nothing], {k, 1, 100}]

isok(k) = isprime(prime(k) % k); \\ Michel Marcus, Jun 19 2023

from sympy import prime, isprime
a363751=[]
for k in range(1,101):
    if isprime(prime(k)%k):
        a363751.append(k)

a(n) = A000720(A363752(n)).

A357517 Primes that are the average of two consecutive primorial numbers A002110 plus one.

Original entry on oeis.org

5, 19, 270271, 5105101, 103515091681, 3810649312471, 155835500831011, 313986271960080721, 282899575838889614011647241, 113405858671385228324474555982803921209616373612841704311161, 2900763693484834576932132901212043025388720793126978148639249341

Offset: 1

Nicholas Leonard, Oct 01 2022

nonn

a(n) ends with digit 1, for n > 2.

			19 is a term since primorials A002110(2) = 6 and A002110(3) = 30 which give (6 + 30)/2 + 1 = 19 which is prime.

Cf. A002110, A276939.

primorial[n_] := Times@@Table[Prime[k], {k, 1, n}];  Table[If[PrimeQ[(primorial[n] + primorial[n+1])/2 + 1], (primorial[n] + primorial[n+1])/2 + 1, Nothing], {n, 1, 40}]

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User: Nicholas Leonard

A364936 a(n) = minimum number of variables with n possible states in a system such that its solution requires the processing of a transcomputational number of bits.

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A363752 Primes prime(k) such that prime(k) mod k is prime.

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A363751 Numbers k such that prime(k) mod k is prime.

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A357517 Primes that are the average of two consecutive primorial numbers A002110 plus one.

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Mathematica