A243358 -id:A243358 - cp's OEIS frontend

A243370 Decimal expansion of the number A = 1.8252076... which generates the densest possibly infinite sequence of primes a(n) = floor[A^(C^n)] for A < 2. That prime sequence is A243358.

1, 8, 2, 5, 2, 0, 7, 6, 3, 4, 7, 6, 9, 3, 3, 5, 0, 6, 8, 0, 5, 1, 8, 3, 4, 1, 5, 5, 7, 8, 3, 3, 4, 2, 4, 8, 6, 2, 2, 8, 9, 5, 8, 9, 7, 7, 4, 9, 7, 8, 6, 2, 8, 5, 6, 9, 6, 5, 4, 5, 0, 0, 8, 0, 5, 0, 0, 5, 0, 9, 8, 2, 2, 4, 9, 2, 8, 1, 2, 5, 3, 5, 7, 5, 9, 9, 0

Offset: 1

Andrey V. Kulsha, Jun 04 2014

It is very likely, but not yet proved, that the sequence of primes A243358 is actually infinite. But it's clear that if such an infinite sequence exists, then its density parameter C should be larger than C_0 = 1.2209864... (see A117739).

Andrey V. Kulsha, Table of n, a(n) for n = 1..50000

Cf. A117739, A243358.

A = 84^(1/C_0^10), where C_0 (mentioned above) is given in A117739.

A117739 Decimal expansion of the largest C_0 = 1.2209864... such that for C < C_0 and A < 2 the sequence a(n) = floor[A^(C^n)] can't contain only prime terms.

Original entry on oeis.org

1, 2, 2, 0, 9, 8, 6, 4, 0, 7, 1, 3, 9, 5, 5, 0, 2, 4, 4, 2, 7, 3, 7, 0, 1, 4, 5, 1, 8, 8, 3, 5, 5, 8, 1, 4, 1, 6, 4, 6, 2, 4, 7, 5, 4, 0, 6, 0, 2, 9, 3, 8, 4, 4, 4, 7, 9, 1, 9, 7, 2, 9, 2, 5, 3, 7, 5, 1, 0, 3, 8, 7, 9, 7, 4, 6, 0, 0, 9, 1, 9, 1, 0, 3, 4, 2

Offset: 1

Martin Raab, May 04 2006

It is not proved that for C > C_0 the mentioned infinite sequence of primes actually exists. However, heuristics show that A243358 could be infinite (the decimal expansion of corresponding A value is A243370).

Andrey V. Kulsha, Table of n, a(n) for n = 1..50000
Chris K. Caldwell, A proof of a generalization of Mills' Theorem

Cf. A243358 (primes), A243370 (value of A), A051021 (Mills' constant)

C_0 can be estimated as (logP/log84)^(1/k), where P is k+10th term of A243358.

Terms after a(18) from Andrey V. Kulsha, Jun 03 2014

A382261 a(n) = floor(x^(phi^n)), where phi = (1+sqrt(5))/2 and x is the constant A382260.

Original entry on oeis.org

2, 3, 7, 23, 163, 3803, 620549, 2359981439, 1464484123012601, 3456155348019933976288373, 5061484633840283809323162088349619180781, 17493277186167814180104995425523045477935447066389138909089293633

Offset: 1

Thomas Scheuerle, Mar 19 2025

nonn

Conjecture: All terms are prime numbers. For details see A382260.

Cf. A001622, A059784, A051254, A243358, A382260. A090253.

Cf. A090253 ( similar growth ).

nextprime(a(n-2)*a(n-1)) <= a(n) < nextprime((a(n-2)+1)*a(n-1)).

cp's OEIS Frontend

A243370 Decimal expansion of the number A = 1.8252076... which generates the densest possibly infinite sequence of primes a(n) = floor[A^(C^n)] for A < 2. That prime sequence is A243358.

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A117739 Decimal expansion of the largest C_0 = 1.2209864... such that for C < C_0 and A < 2 the sequence a(n) = floor[A^(C^n)] can't contain only prime terms.

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A382261 a(n) = floor(x^(phi^n)), where phi = (1+sqrt(5))/2 and x is the constant A382260.

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