A054723 -id:A054723 - cp's OEIS frontend

A244453 Prime factors of 2^A054723(n)-1, ordered by increasing n, then by increasing size of the factors.

23, 89, 47, 178481, 233, 1103, 2089, 223, 616318177, 13367, 164511353, 431, 9719, 2099863, 2351, 4513, 13264529, 6361, 69431, 20394401, 179951, 3203431780337, 193707721, 761838257287, 228479, 48544121, 212885833

Offset: 1

Felix Fröhlich, Jun 28 2014

Subsequence of A060443.

Prime factors of composite Mersenne numbers; A089162 with the Mersenne primes A000668 removed. - Jens Kruse Andersen, Jul 11 2014

			A054723(1) = 11. 2^11-1 = 2047 = 23*89. - _Jens Kruse Andersen_, Jul 11 2014
Triangle begins:
23, 89;
47, 178481;
233, 1103, 2089;
223, 616318177;
13367, 164511353;
431, 9719, 2099863;
2351, 4513, 13264529;
6361, 69431, 20394401;

Jens Kruse Andersen, Table of n, a(n) for n = 1..577
Sergey Nikitin, Euler-Fermat algorithm and some of its applications, 2018.
Sam Wagstaff, The Cunningham Project

Cf. A003260, A016047, A046800, A089162.

Map[FactorInteger, Select[2^Prime@Range@20 - 1, CompositeQ]][[All, All, 1]] // Flatten (* Michael De Vlieger, Nov 20 2018 *)

forprime(n=1, 100, m=2^n-1; if(!isprime(m), f=factor(m); for(i=1, #f~, print1(f[i,1]", ")))) \\ Jens Kruse Andersen, Jul 11 2014

A289982 Lesser member p of twin primes in A054723 (Prime exponents of composite Mersenne numbers).

Original entry on oeis.org

41, 71, 101, 137, 149, 179, 191, 197, 227, 239, 269, 281, 311, 347, 419, 431, 461, 569, 599, 617, 641, 659, 809, 821, 827, 857, 881, 1019, 1031, 1049, 1061, 1091, 1151, 1229, 1289, 1301, 1319, 1427, 1451, 1481, 1487, 1607, 1619, 1667, 1697, 1721, 1787, 1871, 1877

Offset: 1

Muniru A Asiru, Jul 17 2017

nonn

2^p-1 is composite. p is the lesser of twin primes in A001359 and a prime exponent of a Mersenne number in A054723.

			p=41 is a member because 41 is a lesser of twin prime and 2^41 - 1 = 13367*164511353 is composite.
Similarly, p=227 is a member because 227 is a lesser of twin prime and 2^227 - 1 is composite.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Subsequence of A054723.

Cf. A001097, A001359, A006512, A077800.

P1:=Difference(Filtered([1..100000],IsPrime),[2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607, 1279, 2203, 2281, 3217, 4253, 4423, 9689, 9941, 11213, 19937, 21701,23209, 44497, 86243]);;
P2:=List([1..Length(P1)-1],i->[P1[i],P1[i+1]]);;
P3:=List(Positions(List(P2,i->i[2]-i[1]),2),i->P2[i][1]);

Function[s, Flatten@ Map[s[[#, 1]] &, Position[Most@ s, d_ /; Quiet@ Differences@ d == {2}, {1}]]]@ Partition[#, 2, 1] &@ Select[Prime@ Range@ 360, ! PrimeQ[2^# - 1] &] (* Michael De Vlieger, Jul 17 2017 *)
Select[Partition[Module[{nn=20,mp},mp=MersennePrimeExponent[Range[nn]];Complement[Prime[Range[PrimePi[Last[mp]]]],mp]],2,1],#[[2]]-#[[1]]==2 && AllTrue[#,PrimeQ]&][[All,1]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Apr 10 2019 *)

isok(n) = isprime(n) && isprime(n+2) && !isprime(2^n-1) && !isprime(2^(n+2)-1); \\ Michel Marcus, Jul 19 2017

A001262 Strong pseudoprimes to base 2.

Original entry on oeis.org

2047, 3277, 4033, 4681, 8321, 15841, 29341, 42799, 49141, 52633, 65281, 74665, 80581, 85489, 88357, 90751, 104653, 130561, 196093, 220729, 233017, 252601, 253241, 256999, 271951, 280601, 314821, 357761, 390937, 458989, 476971, 486737

Offset: 1

N. J. A. Sloane

The number 2^k-1 is in the sequence iff k is in A054723 or in A001567. - Thomas Ordowski, Sep 02 2016

The number (2^k+1)/3 is in the sequence iff k is in A127956. - Davide Rotondo, Aug 13 2021

			From _Michael B. Porter_, Sep 04 2016: (Start)
For k = 577, k-1 = 576 = 9*2^6. Since 2^(9*2^3) = 2^72 == -1 (mod 577), 577 passes the primality test, but since it is actually prime, it is not in the sequence.
For k = 3277, k-1 = 3276 = 819*2^2, and 2^(819*2) == -1 (mod 3277), so k passes the primality test, and k = 3277 = 29*113 is composite, so 3277 is in the sequence. (End)

R. K. Guy, Unsolved Problems Theory Numbers, A12.
P. Ribenboim, The Book of Prime Number Records. Springer-Verlag, NY, 2nd ed., 1989, p. 95.

T. D. Noe, Table of n, a(n) for n = 1..10000 (using data from A001567)
Joerg Arndt, Matters Computational (The Fxtbook), section 39.10, pp. 786-792.
Chris Caldwell, Strong probable prime.
Eric Weisstein's World of Mathematics, Strong Pseudoprime.
OEIS Wiki, Strong Pseudoprime.
Wikipedia, Strong pseudoprime.
Index entries for sequences related to pseudoprimes

Cf. A001567 (pseudoprimes to base 2), A020229 (strong pseudoprimes to base 3), A020231 (base 5), A020233 (base 7).

Cf. A072276 (SPP to base 2 and 3), A215568 (SPP to base 2 and 5), A056915 (SPP to base 2,3 and 5), A074773 (SPP to base 2,3,5 and 7).

A007814 := proc(n) padic[ordp](n,2) ; end proc:
isStrongPsp := proc(n,b) local d,s,r; if type(n,'even') or n<=1 then return false; elif isprime(n) then return false; else s := A007814(n-1) ; d := (n-1)/2^s ; if modp(b &^ d,n) = 1 then return true; else for r from 0 to s-1 do if modp(b &^ d,n) = n-1 then return true; end if; d := 2*d ; end do: return false; end if; end if; end proc:
isA001262 := proc(n) isStrongPsp(n,2) ; end proc:
for n from 1 by 2 do if isA001262(n) then print(n); end if; end do:
# R. J. Mathar, Apr 05 2011

sppQ[n_?EvenQ, ] := False; sppQ[n?PrimeQ, ] := False; sppQ[n, b_] := (s = IntegerExponent[n-1, 2]; d = (n-1)/2^s; If[PowerMod[b, d, n] == 1, Return[True], Do[If[PowerMod[b, d, n] == n-1, Return[True]]; d = 2*d, {s}]]); lst = {}; k = 3; While[k < 500000, If[sppQ[k, 2], Print[k]; AppendTo[lst, k]]; k += 2]; lst (* Jean-François Alcover, Oct 20 2011, after R. J. Mathar *)

isStrongPsp(n,b)={
        my(s,d,r,bm) ;
        if( (n% 2) ==0 || n <=1, return(0) ;) ;
        if(isprime(n), return(0) ;) ;
        s = valuation(n-1,2) ;
        d = (n-1)/2^s ;
        bm = Mod(b,n)^d ;
        if ( bm == Mod(1,n), return(1) ;) ;
        for(r=0,s-1,
                bm = Mod(b,n)^d ;
                if ( bm == Mod(-1,n),
                        return(1) ;
                ) ;
                d *= 2;
        ) ;
        return(0);
}
isA001262(n)={
        isStrongPsp(n,2)
}
{
for(n=1,10000000000,
    if(isA001262(n),
        print(n)
    ) ;
) ;
} \\ R. J. Mathar, Mar 07 2012

is_A001262(n,a=2)={ (bittest(n,0) && !isprime(n) && n>8) || return; my(s=valuation(n-1,2)); if(1==a=Mod(a,n)^(n>>s),return(1)); while(a!=-1 && s--, a=a^2); a==-1} \\ M. F. Hasler, Aug 16 2012

More terms from David W. Wilson, Aug 15 1996

A065341 Mersenne composites: 2^prime(m) - 1 is not a prime.

Original entry on oeis.org

2047, 8388607, 536870911, 137438953471, 2199023255551, 8796093022207, 140737488355327, 9007199254740991, 576460752303423487, 147573952589676412927, 2361183241434822606847, 9444732965739290427391

Offset: 1

Labos Elemer, Oct 30 2001

nonn

For the number of prime factors in a(n) see A135975. For indices of primes n in composite 2^prime(n)-1 see A135980. For smallest prime divisors of Mersenne composites see A136030. For largest prime divisors of Mersenne composites see A136031. For largest divisors see A145097. - Artur Jasinski, Oct 01 2008

All the terms are Fermat pseudoprimes to base 2 (A001567). For a proof see, e.g., Jaroma and Reddy (2007). - Amiram Eldar, Jul 24 2021

			2^11 - 1 = 2047 = 23*89.

Muniru A Asiru, Table of n, a(n) for n = 1..110
John H. Jaroma and Kamaliya N. Reddy, Classical and alternative approaches to the Mersenne and Fermat numbers, The American Mathematical Monthly, Vol. 114, No. 8 (2007), pp. 677-687.

Cf. A054723, A000043, A000668, A001348, A001567.

Cf. A135975, A135980, A145097, A136031. - Artur Jasinski, Oct 01 2008

A065341 := proc(n) local i;
i := 2^(ithprime(n))-1:
if (not isprime(i)) then
   RETURN (i)
fi: end: seq(A065341(n), n=1..21); # Jani Melik, Feb 09 2011

Select[Table[2^Prime[n]-1,{n,30}],!PrimeQ[#]&] (* Harvey P. Dale, May 06 2018 *)

a(n) = 2^A054723(n) - 1.

A176549 Primes of the form 2n^2+6n+1.

Original entry on oeis.org

37, 109, 541, 757, 1009, 1297, 1621, 2377, 6841, 7561, 8317, 9109, 11701, 12637, 15661, 16741, 19009, 23977, 25309, 28081, 34057, 38917, 40609, 42337, 44101, 47737, 51517, 55441, 57457, 59509, 65881, 70309, 72577, 82009, 84457, 99901

Offset: 1

Vincenzo Librandi, Apr 20 2010

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

2*a(n) + 7 is a square. - Vincenzo Librandi, Apr 09 2015

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

Primes in A059993.
Subsequence of A093838.
Cf. Primes of the form 2*n^2+2*(2*k+3)*n+(2*k+1): this sequence (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).

[a: n in [0..300] | IsPrime(a) where a is 2*n^2+6*n+1]; // Vincenzo Librandi, Jul 26 2012

Select[Table[2 n^2 + 6 n + 1, {n, 2000}], PrimeQ] (* Vincenzo Librandi, Jul 26 2012 *)

Removed an obviously incorrect part of the definition - R. J. Mathar, Apr 21 2010

A217494 Primes of the form 2n^2 + 34n + 15.

Original entry on oeis.org

631, 883, 1171, 2251, 2683, 8191, 9811, 12511, 20071, 25183, 30871, 33931, 38791, 40483, 57331, 61471, 70183, 81883, 94483, 105211, 125371, 150571, 157231, 167491, 188983, 292483, 315883, 340183, 360271, 423991, 440731, 469351, 481051, 510931

Offset: 1

Vincenzo Librandi, Oct 08 2012

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

2*a(n)+259 is a square. - Vincenzo Librandi, Mar 04 2013

Vincenzo Librandi, Table of n, a(n) for n = 1..3000

Cf. A054723, A176549.

Subsequence of A002145.

[a: n in [1..500] | IsPrime(a) where a is 2*n^2+34*n+15];

Select[Table[2 n^2 + 34 n + 15, {n, 500}], PrimeQ]

A135975 Number of prime factors (without multiplicity) in Mersenne composites A065341.

Original entry on oeis.org

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

Offset: 1

Artur Jasinski, Dec 09 2007

nonn

Currently the smallest prime exponent p for which 2^p-1 is incompletely factored is p = 1213. - Gord Palameta, Aug 06 2018

Gord Palameta, Table of n, a(n) for n = 1..183
GIMPS, Status of M1213
S. S. Wagstaff, Jr., Main Tables from the Cunningham Project: cofactor of M1213 is C297.

Cf. A000225, A001221, A065341, A054723, A134852.

k = {}; Do[If[ ! PrimeQ[2^Prime[n] - 1], c = FactorInteger[2^Prime[n] - 1]; d = Length[c]; AppendTo[k, d]], {n, 1, 40}]; k
(PrimeNu /@ Select[2^Prime[Range[40]] - 1, ! PrimeQ[#] &]) (* Jean-François Alcover, Aug 13 2014 *)

forprime(p=1, 1e3, if(!ispseudoprime(2^p-1), print1(omega(2^p-1), ", "))) \\ Felix Fröhlich, Aug 12 2014

a(n) = A001221(A065341(n)). - Michel Marcus, Aug 07 2018

a(29)-a(46) from Felix Fröhlich, Aug 12 2014

a(47)-a(100) from Gord Palameta, Aug 07 2018

A217496 Primes of the form 2n^2 + 50n + 23.

Original entry on oeis.org

23, 131, 191, 911, 1223, 1451, 1571, 1823, 3323, 3671, 3851, 5651, 6323, 6791, 7523, 8291, 9371, 10223, 12671, 15731, 16091, 16823, 25931, 28751, 29723, 39191, 43223, 50591, 53831, 55823, 60611, 62723, 64151, 64871, 68531, 73823, 77723, 80111, 87491, 90023

Offset: 1

Vincenzo Librandi, Oct 09 2012

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

2*a(n)+579 is a square. - Vincenzo Librandi, Mar 04 2013

Vincenzo Librandi, Table of n, a(n) for n = 1..3000

Cf. A054723, A176549, A217494.

Subsequence of A002145.

[a: n in [0..200] | IsPrime(a) where a is 2*n^2 + 50*n + 23];

Select[Table[2 n^2 + 50 n + 23, {n, 0, 500}], PrimeQ]

A217497 Primes of the form 2n^2 + 54n + 25.

Original entry on oeis.org

421, 673, 2473, 4561, 5821, 9601, 12301, 14281, 19861, 30661, 32173, 33721, 61261, 67741, 84121, 94273, 107773, 110581, 122173, 134341, 170773, 203821, 207673, 223441, 227473, 265381, 274201, 287701, 344941, 365173, 391273, 396601, 418273, 423781, 469141

Offset: 1

Vincenzo Librandi, Oct 09 2012

Conjecture: 2^a(n)-1 is not prime; in other words, these primes are included in A054723.
2*a(n) + 679 is a square. - Vincenzo Librandi, Apr 10 2015
Equivalently, primes of the form 36*n^2 + 36*n + 9. - Charles R Greathouse IV, Jul 24 2024

Vincenzo Librandi, Table of n, a(n) for n = 1..3000

Subsequence of A002144.

Cf. A054723, A176549, A217494.

[a: n in [1..500] | IsPrime(a) where a is 2*n^2+54*n+25];

Select[Table[2 n^2 + 54 n + 25, {n, 500}], PrimeQ]

list(lim)=my(v=List()); for(n=2,(sqrtint(lim\1*2+679)-27)\6, my(p=18*n^2 + 162*n + 25); if(isprime(p), listput(v,p))); Vec(v) \\ Charles R Greathouse IV, Jul 24 2024

a(n) >> n^2 log n. - Charles R Greathouse IV, Jul 24 2024

A217498 Primes of the form 2n^2 + 58n + 27.

Original entry on oeis.org

151, 367, 619, 907, 1231, 1987, 2887, 3391, 3931, 4507, 5119, 6451, 7927, 8719, 9547, 11311, 13219, 15271, 17467, 21031, 22291, 24919, 27691, 29131, 32119, 35251, 36871, 41947, 43711, 55051, 59119, 63331, 76831, 81619, 84067, 89071, 94219, 96847, 104947

Offset: 1

Vincenzo Librandi, Oct 09 2012

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

2*a(n) + 787 is a square. - Vincenzo Librandi, Apr 10 2015

Vincenzo Librandi, Table of n, a(n) for n = 1..3000

Cf. A054723, A176549, A217494.

Subsequence of A002145.

[a: n in [1..500] | IsPrime(a) where a is 2*n^2+58*n+27];

Select[Table[2 n^2 + 58 n + 27, {n, 500}], PrimeQ]

cp's OEIS Frontend

A244453 Prime factors of 2^A054723(n)-1, ordered by increasing n, then by increasing size of the factors.

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A289982 Lesser member p of twin primes in A054723 (Prime exponents of composite Mersenne numbers).

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A001262 Strong pseudoprimes to base 2.

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A065341 Mersenne composites: 2^prime(m) - 1 is not a prime.

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A176549 Primes of the form 2*n^2+6*n+1.

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Magma

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A217494 Primes of the form 2*n^2 + 34*n + 15.

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Magma

Mathematica

A135975 Number of prime factors (without multiplicity) in Mersenne composites A065341.

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A176549 Primes of the form 2n^2+6n+1.

A217494 Primes of the form 2n^2 + 34n + 15.

A217496 Primes of the form 2n^2 + 50n + 23.

A217497 Primes of the form 2n^2 + 54n + 25.

A217498 Primes of the form 2n^2 + 58n + 27.