A085724 -id:A085724 - cp's OEIS frontend

A250203 Numbers n such that the Phi_n(2) is the product of exactly two primes and is divisible by 2n+1.

11, 20, 23, 35, 39, 48, 83, 96, 131, 231, 303, 375, 384, 519, 771, 848, 1400, 1983, 2280, 2640, 2715, 3359, 6144, 7736, 7911, 11079, 13224, 16664, 24263, 36168, 130439, 406583

Offset: 1

Eric Chen, Mar 13 2015

Here Phi_n is the n-th cyclotomic polynomial.
Is this sequence infinite?
Phi_n(2)/(2n+1) is only a probable prime for n > 16664.
a(33) > 2000000.
Subsequence of A005097 (2 * a(n) + 1 are all primes)
Subsequence of A081858.
2 * a(n) + 1 are in A115591.
Primes in this sequence are listed in A239638.
A085021(a(n)) = 2.
All a(n) are congruent to 0 or 3 (mod 4). (A014601)
All a(n) are congruent to 0 or 2 (mod 3). (A007494)
Except the term 20, all even numbers in this sequence are divisible by 8.

			Phi_11(2) = 23 * 89 and 23 = 2 * 11 + 1, so 11 is in this sequence.
Phi_35(2) = 71 * 122921 and 71 = 2 * 35 + 1, so 35 is in this sequence.
Phi_48(2) = 97 * 673 and 97 = 2 * 48 + 1, so 48 is in this sequence.

Eric Chen, Gord Palameta, Factorization of Phi_n(2) for n up to 1280
Will Edgington, Factorization of completely factored Phi_n(2) [from Internet Archive Wayback Machine]
Henri Lifchitz and Renaud Lifchitz, PRP records. Search for (2^a-1)/b
Samuel Wagstaff, The Cunningham project

Cf. A239638, A085724, A072226, A005384, A005097, A081858, A014601, A014664, A001917, A115591.

Select[Range[10000], PrimeQ[2*# + 1] && PowerMod[2, #, 2*# + 1] == 1 &&
PrimeQ[Cyclotomic[#, 2]/(2*#+1)] &]

isok(n) = if (((x=polcyclo(n, 2)) % (2*n+1) == 0) && (omega(x) == 2), print1(n, ", ")); \\ Michel Marcus, Mar 13 2015

A283455 Numbers m such that 2^m - 1 has at most 2 distinct prime factors.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 9, 11, 13, 17, 19, 23, 31, 37, 41, 49, 59, 61, 67, 83, 89, 97, 101, 103, 107, 109, 127, 131, 137, 139, 149, 167, 197, 199, 227, 241, 269, 271, 281, 293, 347, 373, 379, 421, 457, 487, 521, 523, 607, 727, 809, 881, 971, 983, 997, 1061

Offset: 1

Vladimir Shevelev, Mar 08 2017

nonn

The sequence differs from A283364 beginning with a(15). All a(n) > 6 are primes or squares of primes.
As in A283364 one can prove that all a(n) > 6 are odd. It is clear that a(n) is either prime or semiprime. Let us show that in the latter case it is the square of a prime. Indeed, let a(n) = p*q, p < q. Then 2^a(n)-1 is divisible by 2^p-1 < 2^q-1. Thus both of them are Mersenne primes.
Let us show that 2^(p*q)-1 differs from (2^p-1)^u*(2^q-1)^v, u,v >= 1. Indeed the equality is possible only in the case p*u + q*v = p*q. Then p|v and q|u. Let u = q*a, v = p*b. Then a + b = 1, which is impossible for u,v >= 1. Hence, 2^(p*q)-1 has a third prime divisor and p*q is not a member.
Are there terms other than 4, 9 and 49 that are squares of primes? Note that, for prime p, 2^(p^2)-1 differs from (2^p-1)^p, so if p^2 is a term, then for a Mersenne prime 2^p-1 and some t >= 1, the number (2^(p^2)-1)/(2^p-1)^t should be a prime or a power of a prime.
Numbers n such that A046800(n) < 3. - Michel Marcus, Mar 08 2017

The Cunningham Project, The Main Tables, Table 2- Factorizations of 2^n-1, n odd, n<1300.

Cf. A046800, A283364.

Union of {1}, A000043, A085724.

Select[Join[Range@ 6, Range[7, 201, 2]], PrimeNu[2^# - 1] <= 2 &] (* Michael De Vlieger, Mar 08 2017 *)

is(n)=omega(2^n-1)<3 \\ Charles R Greathouse IV, Mar 08 2017

More terms from Peter J. C. Moses, Mar 08 2017
a(48)-a(50) from Charles R Greathouse IV, Mar 08 2017
a(51)-a(57) from Amiram Eldar, Feb 13 2020

A294729 Numbers n such that 2^n - 1 is the product of three primes.

Original entry on oeis.org

6, 8, 10, 14, 15, 25, 26, 27, 29, 34, 38, 43, 47, 53, 62, 65, 71, 73, 79, 85, 93, 122, 133, 179, 193, 211, 254, 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

Offset: 1

Arkadiusz Wesolowski, Nov 07 2017

nonn

The eighteenth composite term is 3481. No other composite terms up to 10000.

			a(1) = 6 because 2^6 - 1 = 63 = 3^2*7 is a 3-almost prime.
a(2) = 8 because 2^8 - 1 = 255 = 3*5*17 is a 3-almost prime.

Dario Alejandro Alpern, Integer factorization calculator
John Brillhart et al., Cunningham Project [Factorizations of b^n +- 1, b = 2, 3, 5, 6, 7, 10, 11, 12 up to high powers]
S. S. Wagstaff, Jr., The Cunningham Project

Cf. A000043 (product of one prime), A000225, A085724 (product of two primes), A135977.

lst:=[]; factors:=func; bigomega:=func; IsCube:=func; for n in [2..254] do if IsPrime(n) then if bigomega(2^n-1) eq 3 then Append(~lst, n); end if; else f:=factors(n); a:=f[1,1]; if IsPrime(2^a-1) then if IsSquarefree(n) then if bigomega(n) eq 2 then b:=f[2,1]; if IsPrime(2^b-1) and IsPrime(Truncate((2^n-1)/((2^a-1)*(2^b-1)))) then Append(~lst, n); end if; end if; end if; if IsSquare(n) or IsCube(n) then if bigomega(Truncate((2^n-1)/(2^a-1))) eq 2 then Append(~lst, n); end if; end if; end if; end if; end for; lst;

ParallelMap[ If[ PrimeOmega[2^# - 1] == 3, #, Nothing] &, Range@1250] (* Robert G. Wilson v, Nov 28 2017 *)

PARI
```
is(n)=bigomega(2^n-1)==3
```

a(28)-a(56) added from the Cunningham project

A363491 Numbers k such that 2^k - 5 is a semiprime.

Original entry on oeis.org

7, 13, 14, 16, 19, 28, 30, 31, 40, 42, 51, 54, 55, 58, 62, 68, 85, 88, 96, 111, 112, 116, 128, 148, 160, 162, 188, 192, 198, 220, 222, 236, 242, 276, 300, 318, 319, 320, 332, 372, 373, 398, 420, 428, 432, 458, 460, 482, 505, 520, 532, 542, 546, 556, 650, 692, 714

Offset: 1

Kevin P. Thompson, Jun 05 2023

nonn

The numbers 806 and 811 are also terms with 770 being the only remaining unknown below them.

			13 is a term because 2^13 - 5 = 8187 = 3 * 2729 is a semiprime.

Cf. A001358, A085724, A363374.

Select[Range[715],PrimeOmega[2^#-5]==2&] (* Harvey P. Dale, Apr 22 2025 *)

isA363491(k) = bigomega(2^k-5) == 2; \\ Hugo Pfoertner, Jun 25 2023

cp's OEIS Frontend

A250203 Numbers n such that the Phi_n(2) is the product of exactly two primes and is divisible by 2n+1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A283455 Numbers m such that 2^m - 1 has at most 2 distinct prime factors.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A294729 Numbers n such that 2^n - 1 is the product of three primes.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Extensions

A363491 Numbers k such that 2^k - 5 is a semiprime.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

PARI