A054723 -id:A054723 - cp's OEIS frontend

A135979 Indices n such that 2^prime(n)-1 has exactly 2 distinct prime factors.

5, 9, 12, 13, 17, 19, 23, 25, 26, 27, 29, 32, 33, 34, 35, 39, 45, 46, 49, 53, 57, 58, 60, 62, 69, 74, 75, 82, 88, 93, 99, 129, 140, 152, 164, 166, 168, 178, 179

Offset: 1

Artur Jasinski, Dec 09 2007

a(40)>=206. - Amiram Eldar, Sep 29 2018

Cf. A000225, A065341, A054723, A134852, A135975, A135976, A135977, A135978.

k = {}; Do[If[ ! PrimeQ[2^Prime[n] - 1], c = FactorInteger[2^Prime[n] - 1]; d = Length[c]; If[d == 2, AppendTo[k, n]]], {n, 1, 40}]; k
Select[Range[40],PrimeNu[2^Prime[#]-1]==2&] (* Harvey P. Dale, Jul 07 2013 *)

Equals {k: A001221(A001348(k)) = 2}. a(n) = A049084(A135978(n)). - R. J. Mathar, May 03 2008

Edited by R. J. Mathar, May 03 2008
a(17)-a(34) from Donovan Johnson, Jun 14 2009
a(35)-a(39) from Amiram Eldar, Sep 29 2018

A155151 Triangle T(n, k) = 4nk + 2n + 2k + 2, read by rows.

Original entry on oeis.org

10, 16, 26, 22, 36, 50, 28, 46, 64, 82, 34, 56, 78, 100, 122, 40, 66, 92, 118, 144, 170, 46, 76, 106, 136, 166, 196, 226, 52, 86, 120, 154, 188, 222, 256, 290, 58, 96, 134, 172, 210, 248, 286, 324, 362, 64, 106, 148, 190, 232, 274, 316, 358, 400, 442, 70, 116, 162

Offset: 1

Vincenzo Librandi, Jan 21 2009

First column: A016957, second column: A017341, third column: 2*A017029, fourth column: A082286. - Vincenzo Librandi, Nov 21 2012
Conjecture: Let p = prime number. If 2^p belongs to the sequence, then 2^p-1 is not a Mersenne prime. - Vincenzo Librandi, Dec 12 2012
Conjecture is true because if T(n, k) = 2^p with p prime, then 2^p-1 = 4*n*k + 2*n + 2*k + 1 = (2*n+1)*(2*k+1) hence 2^p-1 is not prime. - Michel Marcus, May 31 2015
It appears that T(m,p) = 2^p for Lucasian primes (A002515) greater than 3. For instance: T(44, 11) = 2^11, T(89240, 23) = 2^23. - Michel Marcus, May 28 2015
For n > 1, ascending numbers along the diagonal are also terms of the even principal diagonal of a 2n X 2n spiral (A137928). - Avi Friedlich, May 21 2015

			Triangle begins
  10;
  16,  26;
  22,  36,  50;
  28,  46,  64,  82;
  34,  56,  78, 100, 122;
  40,  66,  92, 118, 144, 170;
  46,  76, 106, 136, 166, 196, 226;
  52,  86, 120, 154, 188, 222, 256, 290;
  58,  96, 134, 172, 210, 248, 286, 324, 362;
  64, 106, 148, 190, 232, 274, 316, 358, 400, 442;

Vincenzo Librandi, Rows n = 1..100, flattened

Cf. A000043, A000668, A016957, A017029, A017341, A054723, A082286, A144650.

[4*n*k + 2*n + 2*k + 2: k in [1..n], n in [1..11]]; // Vincenzo Librandi, Nov 21 2012

seq(seq( 2*(2*n*k+n+k+1), k=1..n), n=1..15) # G. C. Greubel, Mar 21 2021

T[n_,k_]:=4*n*k + 2*n + 2*k + 2; Table[T[n, k], {n, 11}, {k, n}]//Flatten (* Vincenzo Librandi, Nov 21 2012 *)

flatten([[2*(2*n*k+n+k+1) for k in (1..n)] for n in (1..15)]) # G. C. Greubel, Mar 21 2021

T(n, k) = 2*A144650(n, k).

Sum_{k=1..n} T(n,k) = n*(2*n^2 + 5*n + 3) = n*A014105(n+2) =

Edited by Robert Hochberg, Jun 21 2010

A135980 Numbers k such that the Mersenne number 2^prime(k)-1 is composite.

Original entry on oeis.org

5, 9, 10, 12, 13, 14, 15, 16, 17, 19, 20, 21, 22, 23, 25, 26, 27, 29, 30, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78

Offset: 1

Artur Jasinski, Dec 09 2007

nonn

A135979 is a subsequence of this sequence.

Cf. A000225, A065341, A054723, A134852, A135975, A135976, A135977, A135978, A135979.

k = {}; Do[If[ ! PrimeQ[2^Prime[n] - 1], AppendTo[k, n]], {n, 1, 40}]; k
m = PrimePi @ MersennePrimeExponent @ Range[13]; Complement[Range[m[[-1]]], m] (* Amiram Eldar, Mar 12 2020 *)

isok(k) = !isprime(2^prime(k)-1); \\ Michel Marcus, Mar 12 2020

prime(a(n)) = A054723(n).

a(n) = pi(A054723(n)).

More terms from Amiram Eldar, Mar 12 2020

A221902 Primes of the form 2n^2 + 10n + 3.

Original entry on oeis.org

31, 103, 211, 751, 1291, 2371, 2803, 3271, 5503, 6151, 8311, 9103, 9931, 17851, 23971, 25303, 32503, 42331, 49603, 51511, 68071, 82003, 94603, 97231, 105331, 119551, 122503, 137803, 157351, 167611, 171103, 174631, 192811, 204151

Offset: 1

Vincenzo Librandi, Jan 31 2013

Conjecture: After the first term, 2^a(n)-1 is not prime; in other words, these primes (except 31) are included in A054723.

2*a(n) + 19 is a square. - Vincenzo Librandi, Apr 10 2015

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. Primes of the form 2*n^2+2*(2*k+3)*n+(2*k+1): A176549 (k=0), A154577 (k=2), A154592 (k=3), A154601 (k=4), A217494 (k=7), A217495 (k=10), A217496 (k=11), A217497 (k=12), A217498 (k=13), A217499 (k=16), A217500 (k=17), A217501 (k=18), A217620 (k=19), A217621 (k=21).

Cf. A054723 (Prime exponents of nonprime Mersenne numbers).

[a: n in [1..500] | IsPrime(a) where a is 2*n^2 + 10*n + 3];

Select[Table[2 n^2 + 10 n + 3,{n, 500}],PrimeQ]

A330382 Composite numbers k such that k-1 divides 2^k-2.

Original entry on oeis.org

55, 295, 343, 1027, 1135, 1315, 1807, 2059, 2395, 3403, 4375, 5335, 6175, 6499, 7183, 7939, 9235, 10207, 12643, 13123, 14155, 16003, 16255, 19495, 21547, 23815, 27595, 27703, 30619, 35479, 37927, 43219, 45487, 48007, 48763, 50275, 55567, 58483, 64387, 64639, 74899

Offset: 1

Amiram Eldar and Thomas Ordowski, Dec 12 2019

nonn

If k is in this sequence, then 2^k-1 is also a term, so this sequence is infinite.
Also 2^p-1 is in this sequence for such prime p in A069051 that 2^p-1 is composite.
Theorem: if k-1 | 2^k-2, then m-1 | 2^m-2, where m = 2^k-1.
Conjecture: k-1 | 2^k-2 for k = (2^n-1)^3 if and only if n(n-1) | 2^n-2 for n > 2.
It seems that A007013(n)^3 for n > 1 and A007013(n) for n > 4 are in this sequence.
These are the composites k for which M - 1 divides 2^M - 2 where M = 2^k - 1. - Thomas Ordowski, Jul 01 2024

Amiram Eldar, Table of n, a(n) for n = 1..5000

Cf. A007013, A014741, A054723, A069051, A217468.

A217468 is a subsequence.

Select[Range[75000], CompositeQ[#] && Divisible[PowerMod[2, #, # - 1] - 2, # - 1] &]

forcomposite(k=1,75000,if(!((2^k-2)%(k-1)),print1(k,", "))) \\ Hugo Pfoertner, Dec 12 2019

Composites of A014741(n) + 1. - Thomas Ordowski, Jul 01 2024

A045762 Numbers k such that 2^k - 1 is not a prime.

Original entry on oeis.org

0, 1, 4, 6, 8, 9, 10, 11, 12, 14, 15, 16, 18, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79

Offset: 1

Felice Russo

			8 belongs to the sequence because 2^8 - 1 = 255 is not a prime.

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

Cf. A000043 (Mersenne prime exponents), A054723 (primes of this sequence).

[n: n in [0..100]| not IsPrime(2^n-1)]; // Vincenzo Librandi, Jan 28 2011

Select[Range[0,100],!PrimeQ[2^#-1]&]  (* Harvey P. Dale, Mar 22 2011 *)

Complement of A000043.

More terms from Jennifer D. Secor (s1175994(AT)cedarville.edu)

A079324 k such that 2kp+1 is the first factor of a nonprime Mersenne number M(p) = 2^p - 1.

Original entry on oeis.org

1, 1, 4, 3, 163, 5, 25, 60, 1525, 1445580, 1609, 3, 17, 1, 59, 36793758459, 12379533, 3421967, 15, 1, 116905896337578232, 20236572837, 290792847537859675, 60, 2713800, 461, 7033, 2112, 1, 120, 1, 35807, 19, 413328944, 36, 41, 59441263078804, 3284, 3, 1, 45644

Offset: 1

Jon Perry, Feb 12 2003

nonn

a(188) = 216 = k = (f-1)/2p for p=1231, f=531793. Although Mersenne numbers with p = 1213, 1217, 1229, 1231 are not fully factored, we know their smallest factors. One factor is known for p=1237 but it is not certain that it is the smallest. - Gord Palameta, Sep 26 2018

			2^11 - 1 = 23*89, 23 = 2*1*11 + 1, therefore a(1) = 1.

Gord Palameta, Table of n, a(n) for n = 1..188

Cf. A054723.

forprime (n=3,101,v=2^n-1; if (!isprime(v),print1((factor(v)[,1][1]-1)\(2*n)",")))

More terms from Michel Marcus, Mar 17 2014

A214873 Primes p such that 2*p + 1 is also prime and p + 1 is a highly composite number (definition 1).

Original entry on oeis.org

3, 5, 11, 23, 179, 239, 359, 719, 5039, 55439, 665279, 6486479, 32432399, 698377679, 735134399, 1102701599, 20951330399, 3212537327999, 149602080797769599, 299204161595539199, 2718551763981393634806325317503999

Offset: 1

Arkadiusz Wesolowski, Jul 30 2012

nonn

An equivalent definition of this sequence: odd Sophie Germain primes that differ from a highly composite number by 1.
Intersection of A005384 (Sophie Germain primes) and A072828.
With the exception of 5, a subsequence of A002515 (Lucasian primes).
Except for first two terms, this is a subsequence of A054723.
Except for n = 2, 2*a(n) + 1 is a prime factor of A000225(a(n)) (i.e., 2*23 + 1 divides 2^23 - 1).
Conjecture: for n >= 5, GCD(A000032(a(n)), A000225(a(n))) = 2*a(n) + 1.

			23 is a term because both 23 and 47 are primes and also 24 is a highly composite number.

Amiram Eldar, Table of n, a(n) for n = 1..25
Wikipedia, Sophie Germain prime

Cf. A054723.

lst = {}; a = 0; Do[b = DivisorSigma[0, n + 1]; If[b > a, a = b; If[PrimeQ[n] && PrimeQ[2*n + 1], AppendTo[lst, n]]], {n, 1, 10^6, 2}]; lst

A221903 Primes of the form 2n^2 + 42n + 19.

Original entry on oeis.org

163, 811, 1423, 1783, 2179, 3079, 3583, 9739, 11503, 13411, 14419, 17659, 22483, 25111, 26479, 27883, 42139, 49411, 55243, 57259, 70111, 72379, 77023, 79399, 86743, 97039, 116443, 119359, 125299, 140779, 181603, 188911, 207811

Offset: 1

Vincenzo Librandi, Feb 01 2013

Conjecture: 2^a(n)-1 is not prime; in other words, these primes are included in A054723.

2*a(n) + 403 is a square. - Vincenzo Librandi, Apr 10 2015

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. Primes of the form 2*n^2+2*(2*k+3)*n+(2*k+1): A176549 (k=0), A154577 (k=2), A154592 (k=3), A154601 (k=4), A217494 (k=7), this sequence (k=9), A217495 (k=10), A217496 (k=11), A217497 (k=12), A217498 (k=13), A217499 (k=16), A217500 (k=17), A217501 (k=18), A217620 (k=19), A217621 (k=21).

Cf. A054723.

[a: n in [1..500] | IsPrime(a) where a is 2*n^2 + 42*n + 19];

Select[Table[2 n^2 + 42 n + 19, {n, 500}], PrimeQ]

A243889 Primes of the form 2n^2+30n+13.

Original entry on oeis.org

661, 1201, 2281, 2713, 3181, 4801, 5413, 8221, 9013, 12541, 13513, 17761, 18913, 20101, 32413, 33961, 38821, 51421, 72481, 91921, 94513, 108013, 110821, 134581, 137713, 153913, 167521, 211801, 223681, 265621, 274441, 335281, 345181, 365413, 370561, 440761, 560641

Offset: 1

Vincenzo Librandi, Jun 16 2014

Subsequence of A142104.
Conjecture: except 2281, 2^a(n)-1 is not prime; in other words, these primes are included in A054723.
2*a(n) + 199 is a square. - Vincenzo Librandi, Apr 10 2015

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A142104.

Cf. similar sequences listed in A243888.

[a: n in [1..800] | IsPrime(a) where a is 2*n^2+30*n+13];

Select[Table[2 n^2 + 30 n + 13, {n, 1000}], PrimeQ]

cp's OEIS Frontend

A135979 Indices n such that 2^prime(n)-1 has exactly 2 distinct prime factors.

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A155151 Triangle T(n, k) = 4*n*k + 2*n + 2*k + 2, read by rows.

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A135980 Numbers k such that the Mersenne number 2^prime(k)-1 is composite.

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PARI

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A221902 Primes of the form 2*n^2 + 10*n + 3.

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Magma

Mathematica

A330382 Composite numbers k such that k-1 divides 2^k-2.

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Mathematica

PARI

Formula

A045762 Numbers k such that 2^k - 1 is not a prime.

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Magma

Mathematica

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A079324 k such that 2kp+1 is the first factor of a nonprime Mersenne number M(p) = 2^p - 1.

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PARI

A155151 Triangle T(n, k) = 4nk + 2n + 2k + 2, read by rows.

A221902 Primes of the form 2n^2 + 10n + 3.

A221903 Primes of the form 2n^2 + 42n + 19.

A243889 Primes of the form 2n^2+30n+13.