A049479 -id:A049479 - cp's OEIS frontend

A309130 Smallest prime factor of A077586(n).

7, 127, 2147483647, 170141183460469231731687303715884105727, 47, 338193759479, 231733529, 62914441, 2351, 1399, 295257526626031, 18287, 106937, 863, 4703, 138863, 22590223644617

Offset: 1

Richard N. Smith, Jul 13 2019

A263686 is a subsequence.
Agrees with A263686 in the first four terms, but then the two sequences differ for the first time at n = 5, because prime(5) = 11 is not in A000043.
a(18) = A263686(9) is greater than 1.56*10^17*(2^61-1), see link.
a(n) = A077586(n) iff A077586(n) is prime, A077586(n) is prime for 1 <= n <= 4, but composite for 5 <= n <= 17. The status of A077586(18) = 2^(2^61-1)-1 is unknown. It is conjectured that A077586(n) is composite for all n >= 5.
a(20) = 456959, a(21) = 18384329, a(22) = 198839, a(23) = 2349023, a(24) = A263686(10) is greater than 1.25*10^16*(2^89-1).
Conjecture: All terms are in A122094 (all terms in A263686 are in A122094).
For examples related to that conjecture, see A322568. - Jeppe Stig Nielsen, Aug 29 2019
a(30) = 46559, a(32) = 23671, a(36) = 7151489, a(39) = 4698047, a(41) = 719, a(43) = 1440847, a(45) = 179689, a(47) = 11759383, a(48) = 23602441, a(50) = 9024439, a(51) = 28875361, a(52) = 6301423, a(54) = 2493983, a(56) = 33518137, a(59) = 6727783, a(66) = 95111, a(72) = 1439, a(73) = 99833, a(78) = 38119, a(81) = 26849, a(83) = 8258911, a(86) = 16173559, a(89) = 625343, a(93) = 9743. - Chai Wah Wu, Oct 16 2019

Double Mersennes Prime Search, History.
Eric Weisstein's World of Mathematics, Double Mersenne Number.

Cf. A077586, A263686, A001348, A049479.

A309130(n)=A020639(2^(2^prime(n)-1)-1) \\ For efficiency, use addprimes([large terms of this sequence]). - M. F. Hasler, Mar 01 2025

a(n) = A020639(A077586(n)).

a(n) = A049479(A001348(n)). - M. F. Hasler, Mar 01 2025

A367002 a(n) is the smallest prime factor of n*2^n-1.

Original entry on oeis.org

7, 23, 3, 3, 383, 5, 23, 17, 3, 3, 23, 5, 5, 7, 3, 3, 79, 13, 1879, 13, 3, 3, 47, 7, 229, 5, 3, 3, 32212254719, 263, 223, 5, 3, 3, 5, 73, 17, 1217, 3, 3, 6709, 29, 7, 71, 3, 3, 11, 97, 47, 228713, 3, 3, 5, 37, 5, 7, 3, 3, 9377, 11, 13, 479, 3, 3, 41, 5, 13, 137

Offset: 2

Sean A. Irvine, Oct 31 2023

nonn

Amiram Eldar, Table of n, a(n) for n = 2..884 (terms 2..325 from Robert Israel)

Cf. A003261, A020639, A049479, A367003, A367004, A366899.

f:= n -> min(numtheory:-factorset(n*2^n-1)):
map(f, [$2..100]); # Robert Israel, Nov 08 2023

Table[FactorInteger[n*2^n-1][[1,1]], {n,2,69}] (* Paul F. Marrero Romero, Dec 17 2023 *)

a(n) = A020639(A003261(n)).

a(n) = 3 iff n == 4 or 5 (mod 6). - Robert Israel, Nov 08 2023

A136033 a(n) = smallest number k such that number of prime factors of 2^k-1 is exactly n (counted with multiplicity).

Original entry on oeis.org

2, 4, 6, 16, 12, 18, 24, 40, 54, 36, 102, 110, 60, 72, 108, 140, 120, 156, 144, 200, 216, 210, 240, 180, 456, 288, 336, 300, 396, 480, 882, 360, 468, 700

Offset: 1

Artur Jasinski, Dec 11 2007

Cf. A000225, A003260, A016047, A046051, A049479, A088863, A136030, A136031.

N:= 24: # to get a(1) to a(N)
unknown:= N:
for k from 2 while unknown > 0 do
  q:= numtheory:-bigomega(2^k-1);
  if q <= N and not assigned(A[q]) then
     A[q]:= k;
     unknown:= unknown - 1;
  fi
od:
seq(A[i],i=1..N); # Robert Israel, Oct 24 2014

Module[{nn=250,tbl},tbl=Table[{k,PrimeOmega[2^k-1]},{k,nn}];Table[SelectFirst[tbl,#[[2]]==n&],{n,24}]][[;;,1]] (* The program generates the first 24 terms of the sequence. *)  (* Harvey P. Dale, May 25 2025 *)

a(n) = {k = 1; while(bigomega(2^k-1) != n, k++); k;} \\ Michel Marcus, Nov 04 2013

a(15)-a(20) from Michel Marcus, Nov 04 2013
a(21)-a(24) from Derek Orr, Oct 23 2014
a(25)-a(34) from Jinyuan Wang, Jun 07 2019

A249780 Product of lowest and highest prime factors of 2^n-1.

Original entry on oeis.org

9, 49, 15, 961, 21, 16129, 51, 511, 93, 2047, 39, 67092481, 381, 1057, 771, 17179607041, 219, 274876858369, 123, 2359, 2049, 8388607, 723, 55831, 24573, 1838599, 381, 486737, 993, 4611686014132420609, 196611, 4196353, 393213, 3810551, 327, 137438953471, 1572861, 849583, 185043

Offset: 2

Jacob Vecht, Nov 05 2014

nonn

			The lowest and higest prime factors of 2^6-1 are 3 and 7, so A(6) = 21

Chai Wah Wu, Table of n, a(n) for n = 2..200

a:= proc(n) local F; F:= numtheory:-factorset(2^n-1); min(F)*max(F) end proc:
seq(a(n),n=2..50); # Robert Israel, Nov 05 2014

plhpf[n_]:=Module[{fn=FactorInteger[n]},fn[[1,1]]fn[[-1,1]]]; Table[plhpf [2^n-1],{n,2,40}] (* Harvey P. Dale, May 23 2020 *)

for(n=2, 50, p=2^n-1; print1(factor(p)[1, 1]*factor(p)[#factor(p)[, 1], 1], ", ")) \\ Derek Orr, Nov 05 2014

from sympy import primefactors
A249780_list, x = [], 1
for n in range(2,10):
    x = 2*x + 1
    p = primefactors(x)
    A249780_list.append(max(p)*min(p)) # Chai Wah Wu, Nov 05 2014

a(n) = A005420(n) * A049479(n)

More terms from Derek Orr, Nov 05 2014

A140452 2^(a(n))-1 contains an overpseudoprime divisor.

Original entry on oeis.org

11, 22, 23, 25, 28, 29, 33, 35, 36, 37, 39, 41, 43, 44, 45, 46, 47, 48, 50, 51, 52, 53, 55, 56, 57, 58, 59, 60, 63, 64, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 81, 82, 83, 84, 86, 87, 88, 90, 91, 92, 94, 95, 96, 97, 99, 100, 101, 102, 103, 104, 105, 106, 108, 109

Offset: 1

Vladimir Shevelev, Jun 26 2008

nonn

If p is a prime then p is in the sequence iff 2^p-1 is a composite number.

V. Shevelev, Overpseudoprimes, Mersenne Numbers and Wieferich Primes, arXiv:0806.3412 [math.NT], 2008-2012.

Cf. A141232, A137576, A005420, A049479.

f(n) = my(t); sumdiv(2*n+1, d, eulerphi(d)/(t=znorder(Mod(2, d))))*t-t+1; \\ A137576
isopp(n) = (n>1) && !isprime(n) && (n == f((n-1)/2)); \\ A141232
isok(n) = {fordiv(2^n-1, d, if (isopp(d), return (1));); return (0);} \\ Michel Marcus, Dec 09 2018

More terms from Michel Marcus, Dec 09 2018

A239638 Numbers n such that the semiprime 2^n-1 is divisible by 2n+1.

Original entry on oeis.org

11, 23, 83, 131, 3359, 130439, 406583

Offset: 1

Zak Seidov, Mar 23 2014

All terms are primes == 5 modulo 6 (A005384 Sophie Germain primes).

a(8) >= 500000. - Max Alekseyev, May 28 2022

			n = 11, 2^n -1 = 2047 = 23*89,
n = 23, 8388607 = 47*178481,
n = 131, 2722258935367507707706996859454145691647 =  263*10350794431055162386718619237468234569.

Cf. A000043, A001348, A046051, A005384, A005420, A085724, A049479.

Select[Range[4000], PrimeQ[2*# + 1] && PowerMod[2, #, 2*# + 1] == 1 &&
PrimeQ[(2^# - 1)/(2*# + 1)] &] (* Giovanni Resta, Mar 23 2014 *)

is(n)=n%6==5 && Mod(2,2*n+1)^n==1 && isprime(2*n+1) && ispseudoprime((2^n-1)/(2*n+1)) \\ Charles R Greathouse IV, Aug 25 2016

from sympy import isprime, nextprime
A239638_list, p = [], 5
while p < 10**6:
    if (p % 6) == 5:
        n = (p-1)//2
        if pow(2,n,p) == 1 and isprime((2**n-1)//p):
            A239638_list.append(n)
    p = nextprime(p) # Chai Wah Wu, Jun 05 2019

a(5)-a(6) from Giovanni Resta, Mar 23 2014

a(7) from Eric Chen, added by Max Alekseyev, May 21 2022

A368811 a(n) = period length of the sequence A020639(n^k - 1), k >= 1.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 1, 12, 1, 10, 1, 1, 1, 60, 1, 10, 1, 1, 1, 18, 1, 2, 1, 1, 1, 660, 1, 66, 1, 1, 1, 1, 1, 10, 1, 1, 1, 4620, 1, 6, 1, 1, 1, 660, 1, 2, 1, 1, 1, 31878, 1, 2, 1, 1, 1, 197340, 1, 5742, 1, 1, 1, 1, 1, 52026, 1, 1, 1, 440220, 1, 28014, 1, 1, 1, 4, 1, 2610, 1, 1, 1, 28014, 1, 2, 1, 1, 1, 3693690, 1, 2, 1, 1, 1, 1, 1, 7590, 1, 1, 1, 1642460820

Offset: 3

Max Alekseyev, Jan 06 2024

nonn

For n = 2, the sequence A020639(n^k - 1) is not periodic (see A049479), but it is such for any n >= 3.

a(n) divides A058254(A000720(A020639(n-1))).

			a(8) = 2 is the period length of A010705.
a(12) = 12 is the period length of A366717.

Max Alekseyev, Table of n, a(n) for n = 3..10000

Cf. A010705, A020639, A049479, A058254, A366717.

{ a368811(n) = my(r=[], z); forprime(p=2, factor(n-1)[1, 1], if(n%p==0, next); z=znorder(Mod(n, p)); if(!#r || vecmin(apply(x->z%x,r)), r=concat(r,[z])) ); lcm(r); }

For odd n >= 3, a(n) = 1.

A136034 a(n) = smallest number k such that number of distinct prime factors of 2^k-1 is exactly n.

Original entry on oeis.org

1, 2, 4, 8, 12, 20, 24, 40, 36, 48, 88, 60, 72, 150, 132, 120, 156, 144, 200, 204, 210, 180, 324, 476, 288, 300, 432, 396, 480, 360, 468, 576, 700, 504, 420, 648, 540, 660, 792, 720

Offset: 0

Artur Jasinski, Dec 11 2007

First occurrence of n in A046800.

Cf. A000225, A003260, A016047, A046051, A046800, A049479, A088863, A136030, A136031, A136032, A136033.

With[{pn1=PrimeNu[2^Range[800]-1]},Table[Position[pn1,n,1,1],{n,0,40}]]//Flatten (* Harvey P. Dale, Jan 10 2025 *)

a(n) = my(k=1); while (omega(2^k-1) != n, k++); k; \\ Michel Marcus, Jan 09 2023

More terms from Julián Aguirre, Feb 04 2013
a(31)-a(39) from Chai Wah Wu, Oct 03 2019
a(0) = 1 inserted by Michel Marcus, Jan 09 2023

A347141 a(1) = 11; for n > 1, a(n) is the smallest prime factor of 2^a(n-1) - 1.

Original entry on oeis.org

11, 23, 47, 2351, 4703

Offset: 1

J. Lowell, Aug 19 2021

			2^11 - 1 = 23*89, so the next term after 11 is 23.

PrimeNet, M4703.
Wikipedia, Lenstra elliptic-curve factorization.
YAFU, Automated integer factorization.

Cf. A049479.

a[1] = 11; a[n_] := a[n] = Module[{p = 3}, While[PowerMod[2, a[n - 1], p] != 1, p = NextPrime[p]]; p]; Array[a, 5] (* Amiram Eldar, Aug 19 2021 *)

A350381 Composite numbers k such that the multiplicative order of 2 modulo lpf(2^k-1) is k, where lpf = least prime factor.

Original entry on oeis.org

169, 221, 323, 611, 779, 793, 923, 1121, 1159, 1271, 1273, 1349, 1513, 1717, 1829, 1919, 2033, 2077, 2201, 2413, 2533, 2603, 2759, 2951, 3097, 3131, 3173, 3193, 3281, 3379, 3599, 3721, 3791, 3937, 3953, 4043, 4223, 4309, 4331, 4607, 4619, 4867, 4883, 4981, 5111

Offset: 1

Jianing Song, Dec 28 2021

Obviously, if p is a prime, then the multiplicative order of 2 modulo lpf(2^p-1) is p.
It is easy to see that this is a subsequence of A292559 and A322568, so this sequence is included in the intersection of those two sequences. The inclusion is proper. 68231 is in A292559 and A322568 but not in this sequence: lpf(2^68231-1) = 136463 = 2*68231 + 1, the multiplicative order of 2 modulo 136463 is 2201 = 31 * 71 < 68231.
A semiprime in A322568 is in this sequence by definition. 20519, 48263, 63023, 138263, 216239, 341651, 421259, 480323 are examples of terms that are not semiprimes.
Every term is coprime to 2, 3, 5, 7, 11 and 23.

			169 is a term since the least prime factor of 2^169 - 1 is 4057, and the multiplicative order of 2 modulo 4057 is 169.
323 is a term since the least prime factor of 2^323 - 1 is 647, and the multiplicative order of 2 modulo 647 is 323.
1343 is not a term since the least prime factor of 2^1343 - 1 is 2687, and the multiplicative order of 2 modulo 2687 is 79 < 1343.

Cf. A049479 (lpf(2^n-1)), A292559, A322568.

b(n) = forprime(p=3, oo, if(n % znorder(Mod(2,p))==0, return(p)))
isA350381(n) = !isprime(n) && (n>1) && znorder(Mod(2,b(n)))==n \\ Warning: this program can only give the first 7 terms.

More terms from Jinyuan Wang, Jan 22 2025

cp's OEIS Frontend

A309130 Smallest prime factor of A077586(n).

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A367002 a(n) is the smallest prime factor of n*2^n-1.

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A136033 a(n) = smallest number k such that number of prime factors of 2^k-1 is exactly n (counted with multiplicity).

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Maple

Mathematica

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A249780 Product of lowest and highest prime factors of 2^n-1.

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A140452 2^(a(n))-1 contains an overpseudoprime divisor.

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PARI

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A239638 Numbers n such that the semiprime 2^n-1 is divisible by 2n+1.

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A368811 a(n) = period length of the sequence A020639(n^k - 1), k >= 1.

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PARI

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A136034 a(n) = smallest number k such that number of distinct prime factors of 2^k-1 is exactly n.

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