A136801 -id:A136801 - cp's OEIS frontend

A158034 Integers n for which f = (4^n - 2^n + 8n^2 - 2) / (2n * (2n + 1)) is an integer.

3, 11, 23, 83, 131, 179, 191, 239, 243, 251, 359, 419, 431, 443, 491, 659, 683, 719, 743, 891, 911, 1019, 1031, 1103, 1223, 1439, 1451, 1499, 1511, 1539, 1559, 1583, 1811, 1931, 2003, 2039, 2063, 2211, 2339, 2351, 2399, 2459, 2511, 2543, 2699, 2819, 2903

Offset: 1

Reikku Kulon, Mar 11 2009

Superset of A002515; 2n + 1 is prime. A recursive search for members of this sequence results in the infinite series of very large primes A145918. Most members of this sequence are also prime, but five members less than 10000 are composite:
.. . 243 = 3^5
.. . 891 = 3^4 * 11
. . 1539 = 3^4 * 19
. . 2211 = 3 * 11 * 67
. . 2511 = 3^4 * 31
The polygonal number with f sides of length 2n + 1 is (2^n - 1)(2^(n - 1)).
Contribution from Reikku Kulon, May 19 2009: (Start)
The average difference between successive composite terms gradually increases, remaining near their order of magnitude. Roughly 3% of all primes less than 20 billion belong to this sequence or the 2n + 1 sequence. The interval between composite terms 12228632879 and 13169544651 contains 1113606 primes, accounting for 2.75% of the primes in the interval and 1.42% of the primes between 24457265759 and 26339089303.
Prime factors are most often congruent to 3 (mod 4), but some factors are congruent to 1 (mod 4), especially when a term has an even number of not necessarily distinct factors. The most common factor is 3, and often a large power of 3 is a divisor. 5, 7, 13, and 17 are never factors.
The ones digit of composite terms is most often 1, and becomes progressively more likely to be 1. It is never 5. It cannot be 7, because 2n + 1 would then be divisible by 5. The lack of solutions with n divisible by 5 appears crucial to the consistent primality of 2n + 1.
The tens digit is odd if the ones digit is 1 or 9; it is even if the ones digit is 3. This is a consequence of congruence to 3 (mod 4).
The most common least significant two digits of composite terms are 51.
The least significant digits of prime terms do not follow an obvious distribution.
This is the simplest and possibly most productive member of a family of similar sequences defined by f = (s + 8n^2 - 2) / (2n * (2n + 1)), where s is pronic. For these sequences, 2n + 1 is dominated by primes.
=====================================
Large sequences of consecutive primes
=====================================
.   Initial term    Primes   Predecessor     Successor          Gap
.   ---------------------------------------------------------------
.     1529648303    157285    1529648231    1639846391    110198160
.     3832649339    473045    3832647111    4193496803    360849692
.     5897103683    411434    5897102751    6223464171    326361420
.     6543227423    445293    6543226251    6899473631    356247380
.     8126586971    913506    8126586711    8871331491    744744780
.     9533381219    689395    9533380131   10103115231    569735100
.    11576086883    708712   11576086731   12171829419    595742688
.    12228633251   1113606   12228632879   13169544651    940911772
.    21315457451   2328623   21315457251   23375077119   2059619868
(End)

			ngon(f, k) = k * (f * (k - 1) / 2 - k + 2)
. . . 3 = (4^3 - 2^3 + 8 * 9 - 2) / (6 * 7)
. . . . = (2 * 28 + 70) / 42
. . 126 = (2 * 28 + 70)
.. . 28 = (2^3 - 1) * 2^2
. . . . = 126 - 70 - 28
. . . . = 7 * (18 - 10 - 4)
. . . . = 7 * (3 * 6 - 3 * 3 - 5)
. . . . = 7 * (3 * 3 - 7 + 2)
.. 8287 = (4^11 - 2^11 + 8 * 121 - 2) / (22 * 23)
. . . . = (2 * 2096128 + 966) / 506
4193222 = (2 * 2096128 + 966)
2096128 = (2^11 - 1) * 2^10
. . . . = 4193222 - 2096128 - 966
. . . . = 23 * (182314 - 91136 - 42)
. . . . = 23 * (8287 * 22 - 8287 * 11 - 21)
. . . . = 23 * (8287 * 11 - 23 + 2)
Coincidentally, 8287 = 129 * 64 + 31 = 257 * 32 + 63 is prime, and may be the largest value of f that is.
1031 = 257 * 4 + 3 and 2063 = 1031 * 2 + 1 are both members of this sequence, 4127 = 2063 * 2 + 1 is prime, and 8287 = (4127 + 16) * 2 + 1.

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
Mario Raso, Integer Sequences in Cryptography: A New Generalized Family and its Application, Ph. D. Thesis, Sapienza University of Rome (Italy 2025). See p. 112.

Cf. A002515 (Lucasian primes)
Cf. A145918 (exponential Sophie Germain primes)
Cf. A139601 (polygonal numbers)
Cf. A046318, A139876 (related to composite members 243, 891, 1539, and 2511)
Cf. A060210, A002034, A109833, A136801 (their factors)
Cf. A039506 (3, 8287)
Cf. A006516 (2^n - 1)(2^(n - 1))
Cf. A000051 (Fermat numbers), A019434 (Fermat primes)
Cf. A142291 (prime sequence 257, 1031, 2063, 4127)
Cf. A235540 (nonprimes), A002943.

a158034 n = a158034_list !! (n-1)
a158034_list = [x | x <- [1..],
                    (4^x - 2^x + 8*x^2 - 2) `mod` (2*x*(2*x + 1)) == 0]
-- Reinhard Zumkeller, Jan 12 2014

A136798 First term in a sequence of at least 3 consecutive composite integers.

Original entry on oeis.org

8, 14, 20, 24, 32, 38, 44, 48, 54, 62, 68, 74, 80, 84, 90, 98, 104, 110, 114, 128, 132, 140, 152, 158, 164, 168, 174, 182, 194, 200, 212, 224, 230, 234, 242, 252, 258, 264, 272, 278, 284, 294, 308, 314, 318, 332, 338, 350, 354, 360, 368, 374, 380, 384, 390, 398

Offset: 1

Enoch Haga, Jan 21 2008

The meaning of "first" is that the run of composites is started with this term, that is, it is the one after a prime.
The number of terms in any run of composites is odd, because the difference between the relevant consecutive primes is even.
Composite numbers m such that m+1 is also composite, but m-1 is not. - Reinhard Zumkeller, Aug 04 2015

			a(1)=8 because 8 is the first term in a sequential run of 3 composites, 8,9,10

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Carlos Rivera, Puzzle 430, Grimm's Conjecture, Prime puzzles and problems connection.

Cf. A136799, A136800, A136801.
Cf. A049591, A010051.
a(n) = 2 * A104280(n).

import Data.List (elemIndices)
a136798 n = a136798_list !! (n-1)
a136798_list = tail $ map (+ 1) $ elemIndices 1 $
   zipWith (*) (0 : a010051_list) $ map (1 -) $ tail a010051_list
-- Reinhard Zumkeller, Aug 04 2015

Prime/@Flatten[Position[Differences[Prime[Range[80]]],?(#>2&)]]+1 (* _Harvey P. Dale, Jun 19 2013 *)

a(n) = A049591(n)+1. - R. J. Mathar, Jan 23 2008

A010051(a(n)-1) * (1 - A010051(a(n)) - A010051(a(n)+1)) = 1. - Reinhard Zumkeller, Aug 04 2015

Edited by R. J. Mathar, May 27 2009

A136799 Last term in a sequence of at least 3 consecutive composite integers.

Original entry on oeis.org

10, 16, 22, 28, 36, 40, 46, 52, 58, 66, 70, 78, 82, 88, 96, 100, 106, 112, 126, 130, 136, 148, 156, 162, 166, 172, 178, 190, 196, 210, 222, 226, 232, 238, 250, 256, 262, 268, 276, 280, 292, 306, 310, 316, 330, 336, 346, 352, 358, 366, 372, 378, 382, 388, 396

Offset: 1

Enoch Haga, Jan 21 2008

An equivalent definition is "Last term in a sequence of at least 2 consecutive composite integers". - Jon E. Schoenfield, Dec 04 2017
The BASIC program below is useful in testing Grimm's Conjecture, subject of Carlos Rivera's Puzzle 430
Use the program with lines 30 and 70 enabled in the first run and then disabled with lines 31 and 71 enabled in the second run.
Composite numbers m such that m-1 is composite, and m+1 is not. - Martin Michael Musatov, Oct 24 2017

			a(1)=10 because 10 is the last term in a run of three composites beginning with 8 and ending with 10 (8,9,10).

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Carlos Rivera, Puzzle 430, Grimm's Conjecture, Prime puzzles and problems connection.

Cf. A136798, A136800, A136801.

[p-1: p in PrimesInInterval(4, 420) | not IsPrime(p - 2)]; // Vincenzo Librandi, Apr 11 2019

Select[Prime@ Range@ 78, CompositeQ[# - 2] &] - 1 (* Michael De Vlieger, Oct 23 2015, after PARI *)

forprime(p=5, 1000, if(isprime(p-2)==0, print1(p-1, ", "))) \\ Altug Alkan, Oct 23 2015

10 'puzzle 430 (gap finder) 20 N=1 30 A=1:S=sqrt(N):print N; 31 'A=1:S=N\2:print N; 40 B=N\A 50 if B*A=N and B=prmdiv(B) then print B; 60 A=A+1 70 if A<=sqrt(N) then 40 71 'if A<=N\2 then 40 80 C=C+1:print C 90 N=N+1: if N=prmdiv(N) then C=0:print:stop:goto 90:else 30

a(n) = A025584(n+2) - 1. - R. J. Mathar, Jan 24 2008

a(n) ~ n log n. - Charles R Greathouse IV, Oct 27 2015

Edited by R. J. Mathar, May 27 2009

a(53) corrected by Bill McEachen, Oct 27 2015

A136800 Number of composites in prime gaps of size 3 or larger, in order of appearance.

Original entry on oeis.org

3, 3, 3, 5, 5, 3, 3, 5, 5, 5, 3, 5, 3, 5, 7, 3, 3, 3, 13, 3, 5, 9, 5, 5, 3, 5, 5, 9, 3, 11, 11, 3, 3, 5, 9, 5, 5, 5, 5, 3, 9, 13, 3, 3, 13, 5, 9, 3, 5, 7, 5, 5, 3, 5, 7, 3, 7, 9, 9, 5, 3, 5, 7, 3, 3, 11, 7, 3, 7, 3, 5, 11

Offset: 1

Enoch Haga, Jan 22 2008

The sequence counts the terms in the runs of composites associated with A136798-A136799.
A129856 is obtained by removing the composites (9, 15 etc.) from this sequence.
This is sequence A046933, with the zero and all the 1's deleted. - R. J. Mathar, Jan 24 2008

			a(1)=3 because in the run 8, 9, 10 there are three terms.

Cf. A136798, A136799, A136801.

Select[#[[2]]-#[[1]]-1&/@Partition[Prime[Range[100]],2,1],#>2&] (* Harvey P. Dale, Apr 08 2015 *)

a(n)=A136799(n)-A136798(n)+1.

Edited by R. J. Mathar, May 27 2009

A136802 The composite with the largest prime factor in the n-th prime gap larger than 2.

Original entry on oeis.org

10, 14, 22, 26, 34, 38, 46, 51, 58, 62, 69, 74, 82, 86, 94, 99, 106, 111, 122, 129, 134, 146, 155, 158, 166, 172, 178, 183, 194, 206, 218, 226, 232, 237, 249, 254, 262, 267, 274, 278, 291, 302, 309, 314, 326, 334, 346, 351, 358, 362, 371, 376, 382, 386, 394

Offset: 1

Enoch Haga, Jan 24 2008

Pick the number in the interval [A136798(n),A136799(n)] with the largest prime factor.

The sequence is obtained from A114331 by removing terms in prime gaps of size 2.

			a(1)=10 because at N=10 the largest prime factor is 5.

Cf. A136798, A136799, A136800, A136801.

A006530 := proc(n) max( op(numtheory[factorset](n))) ; end:
A136798 := proc(n) local a; if n = 1 then 8; else a := nextprime( procname(n-1))+1 ; while nextprime(a)-a <=2 do a := nextprime(a)+1 ; od; RETURN(a) ; fi; end:
A136802 := proc(n) local c,lpf,a; c := A136798(n) ; lpf := A006530(c) ; a := c; while not isprime(c+1) do c := c+1 ; if A006530(c) > lpf then a := c ; lpf := A006530(c) ; fi; od: a ; end:
seq(A136802(n),n=1..80) ; # R. J. Mathar, May 27 2009

A006530(a(n)) = A136801(n).

Edited by R. J. Mathar, May 27 2009

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A158034 Integers n for which f = (4^n - 2^n + 8n^2 - 2) / (2n * (2n + 1)) is an integer.

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A136798 First term in a sequence of at least 3 consecutive composite integers.

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A136799 Last term in a sequence of at least 3 consecutive composite integers.

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A136800 Number of composites in prime gaps of size 3 or larger, in order of appearance.

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A136802 The composite with the largest prime factor in the n-th prime gap larger than 2.

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