A065341 -id:A065341 - cp's OEIS frontend

A089158 Second prime factor, if it exists, of Mersenne numbers.

89, 178481, 1103, 616318177, 164511353, 9719, 4513, 69431, 3203431780337, 761838257287, 48544121, 2298041, 202029703, 57912614113275649087721, 13842607235828485645766393, 341117531003194129, 3976656429941438590393

Offset: 1

Cino Hilliard, Dec 06 2003

nonn

			The 5th Mersenne number 2^11 - 1 = 23*89 and 89 is the second prime divisor.
The 9th Mersenne number 2^23 - 1 = 47*178481 and 178481 is the second prime divisor.
Notice 23, 89 congruent to 1 mod 11 and 47, 178481 congruent to 1 mod 23.

Amiram Eldar, Table of n, a(n) for n = 1..183
Chris Caldwell, Mersenne Primes: History, Theorems and Lists.

Cf. A001348, A016047, A065341, A089159, A089992, A135978.

mersenne(b,n,d) = { c=0; forprime(x=2,n, c++; y = b^x-1; f=factor(y); v=component(f,1); ln = length(v); if(ln>=d,print1(v[d]",")); ) }

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

A145099 a(n) = 1 if the largest proper divisor of Mersenne composite A145097(n) is prime and a(n) = 0 in opposite case.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0

Offset: 1

Artur Jasinski, Oct 01 2008

nonn

Amiram Eldar, Table of n, a(n) for n = 1..191

Cf. A065341, A135975, A135980, A145097, A136031.

a = {}; Do[m = 2^Prime[n] - 1; k = Divisors[m][[ -2]]; If[PrimeQ[m], null, If[PrimeQ[k], AppendTo[a, 1], AppendTo[a, 0]]], {n, 1, 50}]; a

Name clarified and more terms added by Amiram Eldar, Mar 12 2020

A344515 Primes p such that 2^p-1 has exactly 3 distinct prime factors.

Original entry on oeis.org

29, 43, 47, 53, 71, 73, 79, 179, 193, 211, 257, 277, 283, 311, 331, 349, 353, 389, 409, 443, 467, 499, 563, 577, 599, 613, 631, 643, 647, 683, 709, 751, 769, 829, 919, 941, 1039, 1103, 1117, 1123, 1171, 1193

Offset: 1

Amiram Eldar, May 21 2021

The corresponding Mersenne numbers are in A135977.
a(43) >= 1237.
The following primes are also terms of this sequence: 1301, 1303, 1327, 1459, 1531, 1559, 1907, 2311, 2383, 2887, 3041, 3547, 3833, 4127, 4507, 4871, 6883, 7673, 8233.

			29 is a term since 2^29-1 = 536870911 = 233 * 1103 * 2089 has exactly 3 distinct prime factors.

Subsequence of A054723.

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

Select[Range[200], PrimeQ[#] && PrimeNu[2^# - 1] == 3 &]

2^a(n) - 1 = A135977(n).

A367229 Fermat pseudoprimes to base 2 that are products of two Mersenne numbers (not necessarily distinct) that are larger than 1.

Original entry on oeis.org

1905, 15841, 129921, 8322945, 66977281, 4395899025409, 4398012825601, 140735340806145, 36892925197465616385, 2342736497361113055105, 4951750712408555360305545217, 39304596247310823728047193985, 2535301191011725837253847547905, 1298074214624262174166747352924161

Offset: 1

Amiram Eldar, Nov 11 2023

nonn

Without the restriction to Mersenne numbers that are larger than 1 all the composite Mersenne numbers (A065341) will be terms.
Szymiczek (1964) proved that if p is a prime == 7 (mod 8) (A007522) and t = 2^phi((p-1)/2), then M(p)*M(t) is a Fermat pseudoprime to base 2, where phi is the Euler totient function (A000010) and M(n) = 2^n-1 = A000225(n) is the n-th Mersenne number. The smallest pseudoprime that is generated by this rule, for p = 7 and t = 2^phi((7-1)/2) = 4, is M(7) * M(4) = 1905. The next two, corresponding to p = 23 and 31, have 316 and 87 digits, respectively.
Rotkiewicz and Makowski (1966) proved that if p is a prime or a Fermat pseudoprime to base 2 such that o(p), the multiplicative order of 2 modulo p, is odd (A014663 for primes, A367230 for pseudoprimes), then for each positive k <= p/o(o(p)), if t = 2^(k*o(o(p))) then M(p)*M(t) is a Fermat pseudoprime to base 2. For example, for p = 7, p/o(o(7)) = 7/2, so for k = 1, 2 and 3 the resulting pseudoprimes are 1905, 8322945 and 2342736497361113055105, respectively.

			a(1) = 1905 = (2^4-1) * (2^7-1).
a(2) = 15841 = (2^5-1) * (2^9-1).

Amiram Eldar, Table of n, a(n) for n = 1..242
Andrzej Rotkiewicz and Andrzej Makowski, On Pseudoprime Numbers of the Form M_p M_t, Elemente der Mathematik, Vol. 21 (1966), pp. 133-134.
Kazimierz Szymiczek, On prime numbers p,q and r such that pq, pr, and qr are pseudoprimes, Colloquium Mathematicum, Vol. 13 (1964), pp. 259-263; alternative link.

Cf. A000010, A000225, A001567, A007522, A014663, A065341, A367230.

With[{max = 110}, m = 2^Range[2, max] - 1; Sort@ Select[Times @@@ Subsets[m, {2}], # < m[[-1]] && PowerMod[2, # - 1, #] == 1 &]]

A186645 Numbers k such that 2^(k-1) == 1 + b*k (mod k^2), where b divides k-1.

Original entry on oeis.org

3, 7, 11, 13, 19, 29, 31, 37, 71, 127, 379, 491, 2047, 2633, 2659, 3373, 8191, 13249, 26893, 70687, 74597, 87211, 131071, 184511, 524287, 642581, 1897121, 2676301, 2703739, 8388607, 15456151, 52368101, 102785339, 126233057, 193481677, 536870911, 856645921, 1552107133, 2001907169, 2147483647, 2935442621, 3668158729, 6004262437

Offset: 1

Alzhekeyev Ascar M, Feb 25 2011

nonn

All composites in this sequence are 2-pseudoprimes, A001567.
The sequence contains all Mersenne numbers, A001348, k=2^p-1 for primes p (for which b=(k-1)/p). Correspondingly, the composites in this sequence contain all terms of A065341.
The sequence also contains composites of the form 2^A001567(j) - 1, which do not belong to A065341. The existence of composites in the sequence that are not of the form 2^x-1 is unclear.
The sequence contains A125854 as a subsequence.

Cf. A001348, A001567, A125854.

isA186645 := proc(n)
        if Power(2,n-1) mod n = 1 then
                x := Power(2,n-1) mod (n^2) ;
                b := (x-1)/n ;
                if b>0 then if modp(n-1,b) = 0 then true; else false; end if;
                else false;
                end if;
        else
                false;
        end if;
end proc:
for n from 1 do if isA186645(n) then printf("%d,\n",n); end if; end do: # R. J. Mathar, Mar 09 2011

Edited and more terms added by Max Alekseyev, Mar 14 2011

A225721 Starting with x = n, the number of iterations of x := 2x - 1 until x is prime, or -1 if no prime exists.

Original entry on oeis.org

-1, 0, 0, 1, 0, 1, 0, 2, 1, 1, 0, 1, 0, 2, 1, 1, 0, 3, 0, 6, 1, 1, 0, 1, 2, 2, 1, 2, 0, 1, 0, 8, 3, 1, 2, 1, 0, 2, 5, 1, 0, 1, 0, 2, 1, 2, 0, 583, 1, 2, 1, 1, 0, 1, 1, 4, 1, 2, 0, 5, 0, 4, 7, 1, 2, 1, 0, 2, 1, 1, 0, 3, 0, 2, 1, 1, 4, 3, 0, 2, 3, 1, 0, 1, 2, 4

Offset: 1

Kevin L. Schwartz and Christian N. K. Anderson, May 13 2013

sign

This appears to be a shifted variant of A040076. - R. J. Mathar, May 28 2013

If n is prime, then a(n) = 0. If the sequence never reaches a prime number (for n = 1) or the prime number has more than 1000 digits, -1 is used instead. There are 22 such numbers for n < 10000.

			For a(20), the trajectory is 20->39->77->153->305->609->1217, a prime number. That required 6 steps, so a(20)=6.

Christian N. K. Anderson, Table of n, a(n) for n = 1..10000

Cf. A050921 (primes obtained).
Cf. A040081, A038699, A050412, A052333, A046069 (related to the Riesel problem).
Cf. A000668, A000043, A065341 (Mersenne primes), A000079 (powers of 2).
Cf. A007770 (happy numbers), A031177 (unhappy numbers).
Cf. A037274 (home primes), A037271 (steps), A037272, A037272.

y=as.bigz(rep(0,500)); ys=rep(0,500);
for(i in 1:500) { n=as.bigz(i); k=0;
    while(isprime(n)==0 & ndig(n)<1000 & k<5000) { k=k+1; n=2*n-1 }
    if(ndig(n)>=1000 | k>=5000) { ys[i]=-1; y[i]=-1;
    } else {ys[i]=k; y[i]=n; }
}

A319908 Lesser of twin primes pair p, such that the Mersenne numbers 2^p - 1 and 2^(p+2) - 1 have the same number of prime factors.

Original entry on oeis.org

3, 5, 17, 71, 101, 137, 197, 269, 617, 857, 1019, 1049, 1061

Offset: 1

Amiram Eldar, Oct 01 2018

The corresponding number of prime factors is 1, 1, 1, 3, 2, 2, 2, 2, 4, 4, 5, 4, 2, ...

Assuming that Mersenne numbers (2^p-1 with p prime) are always squarefree, the distinction between number of prime factors with multiplicity (A001222) and number of different prime factors (A001221) is inessential.

			3 is in the sequence since 3 and 5 are twin primes pair, and 2^3-1=7 and 2^5-1=31 are both primes, thus having the same number of prime factors.
71 is in the sequence since 71 and 73 are twin primes pair and 2^71-1 and 2^73-1 both have 3 prime factors.

Cf. A000225, A001359, A065341, A135975.

Do[If[PrimeQ[n]&&PrimeQ[n+2]&&PrimeOmega[2^n-1]==PrimeOmega[2^(n+2)-1],Print[n]],{n,1,200}]

isok(p) = isprime(p) && isprime(p+2) && (omega(2^p-1) == omega(2^(p+2)-1)); \\ Michel Marcus, Oct 02 2018

A346567 Fermat pseudoprimes to base 2 that are palindromic in base 2.

Original entry on oeis.org

341, 561, 645, 1105, 2047, 4369, 4681, 5461, 8481, 16705, 33153, 266305, 278545, 526593, 1052929, 1082401, 1398101, 2113665, 2162721, 2290641, 4259905, 6242685, 7674967, 8388607, 16843009, 17895697, 22369621, 34603041, 67371265, 268505089, 280885153, 285212689

Offset: 1

Amiram Eldar, Jul 23 2021

There are 133 terms below 2^64. Only 4 of them are also Carmichael numbers (561, 1105, 278545 and 67371265).

			341 is a term since 341 = 101010101_2 is palindromic in base 2, it is composite (= 11 * 31) and 2^340 == 1 (mod 341).

Amiram Eldar, Table of n, a(n) for n = 1..133

Intersection of A001567 and A006995.
A065341 and A281576 are subsequences.
Cf. A016041, A068445.

pspQ[n_] := CompositeQ[n] && PowerMod[2, n-1, n] == 1; Select[Range[10^6], PalindromeQ[IntegerDigits[#, 2]] && pspQ[#] &]

A236373 Pseudoprimes to base 2 of the form 6p+1 such that 2^(p-1) == 1 (mod p).

Original entry on oeis.org

2047, 8388607, 140737488355327, 576460752303423487

Offset: 1

Alzhekeyev Ascar M, Jan 24 2014

The first four terms are A065341(1), A065341(2), A065341(7), A065341(9) and have the form 2^m-1. Are there terms not of this form?
Composite 2^n-1 belong to this sequence when n is in A005385 (e.g., 2^83-1, 2^167-1, etc.)
No other terms below 2^64. - Max Alekseyev, May 28 2014

			2047=6*341+1; 2^2046 == 1 (mod 2047); 2^340 == 1 (mod 341).

Subsequence of A001567.

a(4) from Max Alekseyev, May 28 2014

cp's OEIS Frontend

A089158 Second prime factor, if it exists, of Mersenne numbers.

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

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A145099 a(n) = 1 if the largest proper divisor of Mersenne composite A145097(n) is prime and a(n) = 0 in opposite case.

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A344515 Primes p such that 2^p-1 has exactly 3 distinct prime factors.

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A367229 Fermat pseudoprimes to base 2 that are products of two Mersenne numbers (not necessarily distinct) that are larger than 1.

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A186645 Numbers k such that 2^(k-1) == 1 + b*k (mod k^2), where b divides k-1.

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Maple

Extensions

A225721 Starting with x = n, the number of iterations of x := 2x - 1 until x is prime, or -1 if no prime exists.

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R

A319908 Lesser of twin primes pair p, such that the Mersenne numbers 2^p - 1 and 2^(p+2) - 1 have the same number of prime factors.

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Author

Keywords

Comments

Examples

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Programs

Mathematica

PARI

A346567 Fermat pseudoprimes to base 2 that are palindromic in base 2.