Andrey V. Kulsha - 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.

A243358 The densest possibly infinite sequence of primes of the form a(n) = floor[A^(C^n)] for A < 2. The density parameter C here approaches its minimal possible value C_0 = 1.2209864... (A117739), while the corresponding value of A is 1.8252076... (A243370).

Original entry on oeis.org

2, 2, 2, 3, 5, 7, 11, 19, 37, 83, 223, 739, 3181, 18911, 166657, 2375617, 60916697, 3199316947, 403223394631, 147983594957101, 200280265936061027, 1333721075205083093951, 62146579709944366260614273, 31146685223026045243771057244741

Offset: 1

Andrey V. Kulsha, Jun 03 2014

nonn

Double-checked by David J. Broadhurst. Terms from a(61) to a(67) from David J. Broadhurst. Terms after a(52) are strong probable primes.

It is very likely, but not yet proved, that the sequence is infinite. However, it is clear that for density parameter C < C_0 = 1.2209864... (see A117739) such a sequence must contain nonprime terms.

Andrey V. Kulsha, Table of n, a(n) for n = 1..40
Andrey V. Kulsha and David J. Broadhurst, Table of n, a(n) for n = 1..67
Chris K. Caldwell, A proof of a generalization of Mills' Theorem

Cf. A060699, A117739, A243370.

Once the terms up to the prime 223 are known, the following algorithm works:
1. assign P:=(the largest prime currently in the sequence)
2. assign k:=(the distance between 83 and P in the sequence)
3. assign C:=(logP/log84)^(1/k)
4. assign P:=P^C
5. if floor[P] is prime, add it to the sequence and go to 4
6. add nextprime[P] to the sequence and go to 1
That algorithm gives heuristically as many terms as needed because the increment of C at step 3 becomes so tiny that the values of 84^(C^n) for n < k don't jump over integers anymore (although there's no proof).
So we have a(n) = floor[(84-0)^(C_0^(n-10))], where C_0 = 1.2209864... (see A117739), and "84-0" notation means that when C approaches C_0 from above, the necessary value of A brings A^(C^10) to 84 from below.

A209839 Integers n such that for all i > n the largest prime factor of product(i+k, {k,0,15}) exceeds the largest prime factor of product(n+k, {k,0,15}).

Original entry on oeis.org

1, 3, 7, 13, 15, 21, 25, 27, 31, 37, 43, 45, 51, 55, 57, 63, 67, 73, 81, 85, 87, 91, 93, 97, 111, 115, 121, 123, 133, 135, 141, 147, 151, 157, 163, 165, 175, 177, 181, 183, 195, 207, 211, 213, 217, 223, 225, 235, 241, 247, 253, 1330, 1383, 4179, 4180, 4181, 5243, 12985, 48506, 87161, 211750

Offset: 1

Andrey V. Kulsha, Mar 14 2012

nonn

Heuristics show that these terms are valid, but a strict proof is yet to be done.

Andrey V. Kulsha, Table of n, a(n) for n = 1..80

Cf. A209838.

A209838 Integers n such that for all i > n the largest prime factor of product(i+k, {k,0,14}) exceeds the largest prime factor of product(n+k, {k,0,14}).

Original entry on oeis.org

2, 4, 8, 14, 16, 22, 26, 28, 32, 38, 44, 46, 52, 56, 58, 64, 68, 74, 82, 86, 88, 92, 94, 98, 112, 116, 122, 124, 134, 136, 142, 148, 152, 158, 164, 166, 176, 178, 182, 184, 196, 208, 1330, 1331, 48503, 48507, 87162, 211751, 279366, 1845847, 3177028, 4309596, 4363994, 9034134, 61575286

Offset: 1

Andrey V. Kulsha, Mar 14 2012

nonn

Heuristics show that these terms are valid, but a strict proof is yet to be done.

Andrey V. Kulsha, Table of n, a(n) for n = 1..70

Cf. A209837, A209839.

A209837 Integers n such that for all i > n the largest prime factor of product(i+k, {k,0,13}) exceeds the largest prime factor of product(n+k, {k,0,13}).

Original entry on oeis.org

3, 5, 9, 15, 17, 23, 27, 29, 33, 39, 45, 47, 53, 57, 59, 65, 69, 75, 83, 87, 89, 93, 95, 99, 113, 117, 123, 125, 135, 137, 143, 149, 153, 159, 165, 167, 177, 179, 183, 185, 197, 209, 1331, 1332, 48504, 48508, 87163, 416318, 780357, 1845848, 3177027, 3177029, 4309597, 12369422

Offset: 1

Andrey V. Kulsha, Mar 14 2012

nonn

Heuristics show that these terms are valid, but a strict proof is yet to be done.

Andrey V. Kulsha, Table of n, a(n) for n = 1..72

Cf. A200570, A209838.

A200570 Integers n such that for all i > n the largest prime factor of product(i+k, {k,0,12}) exceeds the largest prime factor of product(n+k, {k,0,12}).

Original entry on oeis.org

4, 6, 10, 16, 18, 24, 28, 30, 34, 40, 46, 48, 114, 118, 124, 126, 136, 138, 294, 318, 1332, 1333, 48505, 48509, 87164, 135990, 416319, 436744, 780358, 8763442, 25039772, 35176558, 44444364, 61575288, 100918165, 846178875, 1214569400, 2644138052, 9381631594

Offset: 1

Andrey V. Kulsha, Nov 19 2011

nonn

Heuristics show that these terms are valid, but a strict proof is yet to be done.

Andrey V. Kulsha, Table of n, a(n) for n = 1..43

Cf. A200569, A209837.

A200569 Integers n such that for all i > n the largest prime factor of product(i+k, {k,0,11}) exceeds the largest prime factor of product(n+k, {k,0,11}).

Original entry on oeis.org

1, 5, 7, 11, 17, 19, 25, 29, 31, 35, 41, 47, 49, 115, 119, 125, 127, 137, 139, 295, 319, 1333, 1334, 3470, 8470, 48506, 85759, 87165, 416320, 721404, 2731942, 3500623, 6804174, 8763443, 18567909, 25039772, 25039773, 37980379, 162799068, 304261639, 475527535

Offset: 1

Andrey V. Kulsha, Nov 19 2011

nonn

Heuristics show that these terms are valid, but a strict proof is yet to be done.

Andrey V. Kulsha, Table of n, a(n) for n = 1..49

Cf. A200568, A200570.

A200568 Integers n such that for all i > n the largest prime factor of product(i+k, {k,0,10}) exceeds the largest prime factor of product(n+k, {k,0,10}).

Original entry on oeis.org

2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42, 48, 50, 116, 200, 212, 296, 320, 527, 3471, 8471, 48507, 85760, 104661, 283968, 1487448, 2982728, 3500623, 6748298, 6804175, 11135837, 11421278, 18567910, 25039773, 31512846, 203403620, 1170278793, 2809596041

Offset: 1

Andrey V. Kulsha, Nov 19 2011

nonn

Heuristics show that these terms are valid, but a strict proof is yet to be done.

Andrey V. Kulsha, Table of n, a(n) for n = 1..44

Cf. A200567, A200569.

A200567 Integers n such that for all i > n the largest prime factor of product(i+k, {k,0,9}) exceeds the largest prime factor of product(n+k, {k,0,9}).

Original entry on oeis.org

1, 3, 7, 9, 13, 19, 21, 27, 31, 33, 37, 43, 49, 51, 117, 201, 527, 867, 3472, 8472, 48508, 146064, 373824, 591215, 4325170, 6629044, 6823283, 30188529, 105066156, 203403620, 203403621, 323547634, 439515072, 478478082, 585101566, 1170278794, 2809596042

Offset: 1

Andrey V. Kulsha, Nov 19 2011

nonn

Heuristics show that these terms are valid, but a strict proof is yet to be done.

Andrey V. Kulsha, Table of n, a(n) for n = 1..45

Cf. A200566, A200568.

A200566 Integers n such that for all i > n the largest prime factor of product(i+k, {k,0,8}) exceeds the largest prime factor of product(n+k, {k,0,8}).

Original entry on oeis.org

2, 4, 8, 10, 14, 20, 22, 28, 32, 34, 38, 44, 50, 52, 182, 242, 284, 528, 868, 5246, 8473, 8784, 48503, 48509, 146065, 438122, 591216, 1312724, 4325171, 6629045, 6823284, 56075700, 154392832, 248932942, 251237694, 1913253200, 3123337916, 3180792660

Offset: 1

Andrey V. Kulsha, Nov 19 2011

nonn

Heuristics show that these terms are valid, but a strict proof is yet to be done.

Andrey V. Kulsha, Table of n, a(n) for n = 1..48

Cf. A199407, A200567.

cp's OEIS Frontend

User: Andrey V. Kulsha

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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A243358 The densest possibly infinite sequence of primes of the form a(n) = floor[A^(C^n)] for A < 2. The density parameter C here approaches its minimal possible value C_0 = 1.2209864... (A117739), while the corresponding value of A is 1.8252076... (A243370).

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A209839 Integers n such that for all i > n the largest prime factor of product(i+k, {k,0,15}) exceeds the largest prime factor of product(n+k, {k,0,15}).

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A209838 Integers n such that for all i > n the largest prime factor of product(i+k, {k,0,14}) exceeds the largest prime factor of product(n+k, {k,0,14}).

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A209837 Integers n such that for all i > n the largest prime factor of product(i+k, {k,0,13}) exceeds the largest prime factor of product(n+k, {k,0,13}).

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A200570 Integers n such that for all i > n the largest prime factor of product(i+k, {k,0,12}) exceeds the largest prime factor of product(n+k, {k,0,12}).

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A200569 Integers n such that for all i > n the largest prime factor of product(i+k, {k,0,11}) exceeds the largest prime factor of product(n+k, {k,0,11}).

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A200568 Integers n such that for all i > n the largest prime factor of product(i+k, {k,0,10}) exceeds the largest prime factor of product(n+k, {k,0,10}).

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A200567 Integers n such that for all i > n the largest prime factor of product(i+k, {k,0,9}) exceeds the largest prime factor of product(n+k, {k,0,9}).

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A200566 Integers n such that for all i > n the largest prime factor of product(i+k, {k,0,8}) exceeds the largest prime factor of product(n+k, {k,0,8}).

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