A344385 -id:A344385 - cp's OEIS frontend

A344384 Prime numbers p such that p-1 or p+1 is a number of least prime signature (A025487).

2, 3, 5, 7, 11, 13, 17, 23, 29, 31, 37, 47, 59, 61, 71, 73, 97, 127, 179, 181, 191, 193, 211, 239, 241, 257, 359, 383, 419, 421, 431, 433, 479, 577, 719, 769, 839, 863, 1151, 1153, 1259, 1297, 1439, 1801, 2161, 2309, 2311, 2521, 2591, 2593, 2879, 3359, 3361

Offset: 1

Hal M. Switkay, May 16 2021

nonn

The corresponding numbers of least prime signature are A344385.
19 is the first prime not in this sequence.
This sequence unites many familiar sequences of primes, including Fermat primes (A019434), Mersenne primes (A000668), primorial primes (A018239 and A057705), factorial primes (A088054), A007505, and A039687.
Questions: 1) Is this sequence infinite? 2) Is log(a(n)) = O(log(n)^2)?

			17 is a term because 17 - 1 = 16 is a number of least prime signature.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..158 from Hal M. Switkay)

Cf. A025487, A344385.

Cf. A000668, A007505, A018239, A019434, A039687, A057705, A088054.

{2}~Join~Select[Prime@ Range[2, 900], AnyTrue[# + {-1, 1}, Times @@ MapIndexed[Prime[First[#2]]^#1 &, Sort[FactorInteger[#][[All, -1]], Greater] ] == # &] &] (* Michael De Vlieger, May 16 2021 *)

A354604 Midpoints of record gaps between primes: a(n) = (A000101(n) + A002386(n))/2 for n > 1.

Original entry on oeis.org

4, 9, 26, 93, 120, 532, 897, 1140, 1344, 9569, 15705, 19635, 31433, 155964, 360701, 370317, 492170, 1349592, 1357267, 2010807, 4652430, 17051797, 20831428, 47326803, 122164858, 189695776, 191912907, 387096258, 436273150, 1294268635, 1453168287, 2300942709, 3842610941, 4302407536, 10726904850, 20678048489, 22367085156, 25056082315, 42652618575

Offset: 2

Donghwi Park, Jul 08 2022

nonn

In the displayed portion of the sequence, the only numbers of least prime signature (A025487) are 4 and 120. This is noteworthy because numbers of least prime signature frequently are adjacent to primes (see A344385). It appears to be far more rare for a number of least prime signature to be at the center of a maximal prime gap. With 4 being a term in A344385, 120 seems to have a unique status. - Hal M. Switkay, Mar 13 2025

Hugo Pfoertner, Table of n, a(n) for n = 2..83
Hugo Pfoertner, Gap sizes vs location, using Plot 2.

Subsequence of A024675.

Cf. A000040, A000101, A002386, A344385.

A375197 Terms k in A025487 such that k-1 and k+1 are twin primes.

Original entry on oeis.org

4, 6, 12, 30, 60, 72, 180, 192, 240, 420, 432, 1152, 2310, 2592, 3360, 6300, 7560, 9240, 15360, 21600, 23040, 26880, 55440, 100800, 110880, 138240, 180180, 241920, 264600, 345600, 415800, 453600, 737280, 786432, 995328, 1088640, 1921920, 1940400, 2116800, 3456000

Offset: 1

Amiram Eldar, Aug 04 2024

nonn

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

Intersection of A014574 and A025487.
Subsequence of A344385.
A068507 is a subsequence.
Cf. A001097, A001359, A006512, A344384, A375198.

Select[Cases[Import["https://oeis.org/A025487/b025487.txt", "Table"], {, }][[;; , 2]], And @@ PrimeQ[# + {-1, 1}] &]

a(n) = A025487(A375198(n)).

cp's OEIS Frontend

A344384 Prime numbers p such that p-1 or p+1 is a number of least prime signature (A025487).

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A354604 Midpoints of record gaps between primes: a(n) = (A000101(n) + A002386(n))/2 for n > 1.

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A375197 Terms k in A025487 such that k-1 and k+1 are twin primes.

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