A002235 -id:A002235 - cp's OEIS frontend

A076078 a(n) is the number of nonempty sets of distinct positive integers that have a least common multiple of n.

1, 2, 2, 4, 2, 10, 2, 8, 4, 10, 2, 44, 2, 10, 10, 16, 2, 44, 2, 44, 10, 10, 2, 184, 4, 10, 8, 44, 2, 218, 2, 32, 10, 10, 10, 400, 2, 10, 10, 184, 2, 218, 2, 44, 44, 10, 2, 752, 4, 44, 10, 44, 2, 184, 10, 184, 10, 10, 2, 3748, 2, 10, 44, 64, 10, 218, 2, 44, 10, 218, 2, 3392, 2, 10

Offset: 1

Amarnath Murthy, Oct 05 2002

a(n)=1 iff n=1, a(p^k)=2^k, a(p*q)=10; where p & q are unique primes. a(n) cannot equal an odd number >1. - Robert G. Wilson v
If m has more divisors than n, then a(m) > a(n). - Matthew Vandermast, Aug 22 2004
If n is of the form p^r*q^s where p & q are distinct primes and r & s are nonnegative integers then a(n)=2^(rs)*(2^(r+s+1) -2^r-2^s+1); for example f(1400846643)=f(3^5*7^8)=2^(5*8)*(2^ (5+8+1)-2^5-2^8+1)=17698838672310272. Also if n=p_1^r_1*p_2^r_2*...*p_k^r_k where p_1,p_2,...,p_k are distinct primes and r_1,r_2,...,r_k are natural numbers then 2^(r_1*r_2*...*r_k)||a(n). - Farideh Firoozbakht, Aug 06 2005
None of terms is divisible by Mersenne numbers 3 or 7. For any n, a(n) is congruent to A008836(n) mod 3. Since A008836(n) is always 1 or -1, this implies that A000225(2)=3 never divides a(n). - Matthew Vandermast, Oct 12 2010
There are terms divisible by larger Mersenne numbers. For example, a(2*3*5*7*11*13*19*23^3) is divisible by 31. - Max Alekseyev, Nov 18 2010

			a(6) = 10. The sets with LCM 6 are {6}, {1,6}, {2,3}, {2,6}, {3,6}, {1,2,3}, {1,2,6}, {1,3,6}, {2,3,6}, {1,2,3,6}.

David Wasserman, Table of n, a(n) for n = 1..1000
Index entries for sequences related to prime signature

Cf. A076413, A097210-A097218, A097416, A002235.

Cf. A069626.

with(numtheory): seq(add(mobius(n/d)*(2^tau(d)-1), d in divisors(n)), n=1..80); # Ridouane Oudra, Mar 12 2024

f[n_] := Block[{d = Divisors[n]}, Plus @@ (MoebiusMu[n/d](2^DivisorSigma[0, d] - 1))]; Table[ f[n], {n, 75}] (* Robert G. Wilson v *)

a(n) = local(f, l, s, t, q); f = factor(n); l = matsize(f)[1]; s = 0; forvec(v = vector(l, i, [0, 1]), q = sum(i = 1, l, v[i]); t = (-1)^(l - q)*2^prod(i = 1, l, f[i, 2] + v[i]); s += t); s; \\ Definition corrected by David Wasserman, Dec 26 2007

2^d(n) - 1 = Sum_{m|n} a(m), where d(n) = A000005(n) is the number of divisors of n, so a(n) = Sum_{m|n} mu(n/m)*(2^d(m) - 1).

a(n) = 2*A069626(n), for n > 1. - Ridouane Oudra, Mar 12 2024

Edited by Dean Hickerson, Oct 08 2002
Definition corrected by David Wasserman, Dec 26 2007
Edited by Charles R Greathouse IV, Aug 02 2010
Edited by Max Alekseyev, Nov 18 2010

A003307 Numbers k such that 2*3^k - 1 is prime.

Original entry on oeis.org

1, 2, 3, 7, 8, 12, 20, 23, 27, 35, 56, 62, 68, 131, 222, 384, 387, 579, 644, 1772, 3751, 5270, 6335, 8544, 9204, 12312, 18806, 21114, 49340, 75551, 90012, 128295, 143552, 147488, 1010743, 1063844, 1360104

Offset: 1

N. J. A. Sloane

R. K. Guy, Unsolved Problems in Theory of Numbers, Section A3.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Kevin A. Broughan and Qizhi Zhou, On the Ratio of the Sum of Divisors and Euler's Totient Function II, Journal of Integer Sequences, Vol. 17 (2014), Article 14.9.2.
Steven Harvey, Williams Primes
W. Keller and J. Richstein, Solutions of the congruence a^(p-1) == 1 (mod p^r), Math. Comp. 74 (2005), 927-936.
H. C. Williams, The primality of certain integers of the form 2Ar^n - 1, Acta Arith. 39 (1981), 7-17.
H. C. Williams and C. R. Zarnke, Some prime numbers of the forms 2*3^n+1 and 2*3^n-1, Math. Comp., 26 (1972), 995-998.

Cf. A002235, A046865, A079906, A046866, A001771, A005541, A056725, A046867, A079907.

Cf. A079363 (primes of the form 2*3^k - 1), A003306 (k such that 2*3^k + 1 is prime).

for(n=1,1e4,if(ispseudoprime(2*3^n-1),print1(n", "))) \\ Charles R Greathouse IV, Jul 16 2011

More terms from Douglas Burke (dburke(AT)nevada.edu)
More terms from T. D. Noe, Aug 24 2005
Corrected and extended by Herman Jamke (hermanjamke(AT)fastmail.fm), Jan 05 2008
a(35) from Borys Jaworski, Sep 02 2011
a(36) from Borys Jaworski, Feb 13 2012
a(37) from Jeppe Stig Nielsen, Sep 28 2018

A002253 Numbers k such that 3*2^k + 1 is prime.

Original entry on oeis.org

1, 2, 5, 6, 8, 12, 18, 30, 36, 41, 66, 189, 201, 209, 276, 353, 408, 438, 534, 2208, 2816, 3168, 3189, 3912, 20909, 34350, 42294, 42665, 44685, 48150, 54792, 55182, 59973, 80190, 157169, 213321, 303093, 362765, 382449, 709968, 801978, 916773, 1832496, 2145353

Offset: 1

N. J. A. Sloane

From Zak Seidov, Mar 08 2009: (Start)
List is complete up to 3941000 according to the list of RB & WK.
So far there are only 4 primes: 2, 5, 41, 353. (End)

D. E. Knuth, The Art of Computer Programming. Addison-Wesley, Reading, MA, Vol. 2, p. 614.
H. Riesel, "Prime numbers and computer methods for factorization", Progress in Mathematics, Vol. 57, Birkhauser, Boston, 1985, Chap. 4, see pp. 381-384.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Jeppe Stig Nielsen, Table of n, a(n) for n = 1..50
Ray Ballinger, Proth Search Page
Ray Ballinger and Wilfrid Keller, List of primes k.2^n + 1 for k < 300
C. K. Caldwell, The Prime Pages
Y. Gallot, Proth.exe: Windows Program for Finding Large Primes
S. W. Golomb, Properties of the sequence 3.2^n+1, Math. Comp., 30 (1976), 657-663.
S. W. Golomb, Properties of the sequence 3.2^n+1, Math. Comp., 30 (1976), 657-663. [Annotated scanned copy]
Wilfrid Keller, List of primes k.2^n - 1 for k < 300
PrimeGrid, 321 Prime Search statistics
R. M. Robinson, A report on primes of the form k.2^n+1 and on factors of Fermat numbers, Proc. Amer. Math. Soc., 9 (1958), 673-681. [Annotated scanned copy of selected pages. The first page is (accidentally) included with the scan of the Wilson letter below.]
R. G. Wilson v, Letter (FAX) to N. J. A. Sloane, May 20 1994, concerning A002253-A002274, A000043, A002235-A002240, A001770-A001775, A007117, and many other sequences
Eric Weisstein's World of Mathematics, Proth Prime
R. G. Wilson, V, Letter to N. J. A. Sloane, Jan. 1989
Index entries for sequences of n such that k*2^n-1 (or k*2^n+1) is prime

Cf. A002254, A004119, A181565.

See A039687 for the actual primes.

is(n)=isprime(3*2^n+1) \\ Charles R Greathouse IV, Feb 17 2017

A2253=[1]; A002253(n)=for(k=#A2253, n-1, my(m=A2253[k]); until(ispseudoprime(3<M. F. Hasler, Mar 03 2023

a(n) = log_2((A039687(n)-1)/3) = floor(log_2(A039687(n)/3)). - M. F. Hasler, Mar 03 2023

Corrected and extended according to the list of Ray Ballinger and Wilfrid Keller by Zak Seidov, Mar 08 2009
Edited by N. J. A. Sloane, Mar 13 2009
a(47) and a(48) from the Ballinger & Keller web page, Joerg Arndt, Apr 07 2013
a(49) from https://t5k.org/primes/page.php?id=116922, Fabrice Le Foulher, Mar 09 2014
Terms moved from Data to b-file (Links), and additional term appended to b-file, by Jeppe Stig Nielsen, Oct 30 2020

A001771 Numbers k such that 7*2^k - 1 is prime.

Original entry on oeis.org

1, 5, 9, 17, 21, 29, 45, 177, 18381, 22529, 24557, 26109, 34857, 41957, 67421, 70209, 169085, 173489, 177977, 363929, 372897

Offset: 1

N. J. A. Sloane

k is always of the form 4*j + 1.

If k is in the sequence and m=2^(k+2)*3*(7*2^k-1) then phi(m)+sigma(m)=3m (m is in the sequence A011251). The proof is easy. - Farideh Firoozbakht, Mar 04 2005

H. Riesel, "Prime numbers and computer methods for factorization", Progress in Mathematics, Vol. 57, Birkhäuser, Boston, 1985, Chap. 4, see pp. 381-384.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Wilfrid Keller, List of primes k.2^n - 1 for k < 300
H. C. Williams and C. R. Zarnke, A report on prime numbers of the forms M = (6a+1)*2^(2m-1)-1 and (6a-1)*2^(2m)-1, Math. Comp., 22 (1968), 420-422.
Index entries for sequences of n such that k*2^n-1 (or k*2^n+1) is prime

Cf. A050523, A003307, A002235, A046865, A079906, A046866, A005541, A056725, A046867, A079907.

Cf. A032353 (7*2^k+1 is prime).

Do[ If[ PrimeQ[7*2^n - 1], Print[n]], {n, 1, 2500}]

v=[ ]; for(n=0,2000, if(isprime(7*2^n-1),v=concat(v,n),)); v

More terms from Douglas Burke (dburke(AT)nevada.edu).

More terms from Hugo Pfoertner, Jun 23 2004

A002254 Numbers k such that 5*2^k + 1 is prime.

Original entry on oeis.org

1, 3, 7, 13, 15, 25, 39, 55, 75, 85, 127, 1947, 3313, 4687, 5947, 13165, 23473, 26607, 125413, 209787, 240937, 819739, 1282755, 1320487, 1777515

Offset: 1

N. J. A. Sloane

H. Riesel, "Prime numbers and computer methods for factorization," Progress in Mathematics, Vol. 57, Birkhauser, Boston, 1985, Chap. 4, see pp. 381-384.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Ray Ballinger, Proth Search Page
Ray Ballinger and Wilfrid Keller, List of primes k.2^n + 1 for k < 300
C. K. Caldwell, The Prime Pages
Y. Gallot, Proth.exe: Windows Program for Finding Large Primes
Wilfrid Keller, List of primes k.2^n - 1 for k < 300
R. M. Robinson, A report on primes of the form k.2^n+1 and on factors of Fermat numbers, Proc. Amer. Math. Soc., 9 (1958), 673-681.
R. M. Robinson, A report on primes of the form k.2^n+1 and on factors of Fermat numbers, Proc. Amer. Math. Soc., 9 (1958), 673-681. [Annotated scanned copy of selected pages. The first page is (accidentally) included with the scan of the Wilson letter below.]
Eric Weisstein's World of Mathematics, Proth Prime
R. G. Wilson v, Letter (FAX) to N. J. A. Sloane, May 20 1994, concerning A002253-A002274, A000043, A002235-A002240, A001770-A001775, A007117, and many other sequences
Index entries for sequences of n such that k*2^n-1 (or k*2^n+1) is prime

Cf. A050526.

lst={};Do[If[PrimeQ[5*2^n+1], AppendTo[lst, n]], {n, 0, 2*10^3}];lst (* Vladimir Joseph Stephan Orlovsky, Aug 19 2008 *)

is(n)=isprime(5*2^n+1) \\ Charles R Greathouse IV, Feb 17 2017

Corrected (removed incorrect term 40937) and added more terms (from http://web.archive.org/web/20161028080239/http://www.prothsearch.net/riesel.html), Joerg Arndt, Apr 07 2013

A046865 Numbers k such that 4*5^k - 1 is prime.

Original entry on oeis.org

0, 1, 3, 9, 13, 15, 25, 39, 69, 165, 171, 209, 339, 2033, 6583, 15393, 282989, 498483, 504221, 754611, 864751

Offset: 1

N. J. A. Sloane

a(22) > 1000000. - Karsten Bonath, Apr 04 2019

R. K. Guy, Unsolved Problems in Number Theory, Section A3.

Steven Harvey, Williams Primes
H. C. Williams, The primality of certain integers of the form 2Ar^n - 1, Acta Arith. 39 (1981), 7-17.

Cf. A003307, A002235, A079906, A046866, A001771, A005541, A056725, A046867, A079907.

Print[0]; Do[ If[ PrimeQ[4*5^n - 1], Print[n]], {n, 1, 8100, 2}]

is(n)=isprime(4*5^n-1) \\ Charles R Greathouse IV, Feb 07 2017

Two more terms from Robert G. Wilson v, Jan 16 2003 and Jan 26 2003
a(16) from Herman Jamke (hermanjamke(AT)fastmail.fm), Jan 05 2008
a(1)=0 prepended by Robert Price, Feb 27 2015
a(17) from Karsten Bonath, Dec 07 2018
a(18)-a(19) from Karsten Bonath, Jan 17 2019
a(20)-a(21) from Karsten Bonath, Apr 04 2019

A056725 Numbers k such that 9*10^k - 1 is prime.

Original entry on oeis.org

1, 3, 7, 19, 29, 37, 93, 935, 8415, 9631, 11143, 41475, 41917, 48051, 107663, 212903, 223871, 260253, 364521, 383643, 1009567, 1762063

Offset: 1

Robert G. Wilson v, Aug 11 2000

Also numbers k such that 8*10^k + 9*R_k is prime, where R_k = 11...1 is the repunit (A002275) of length k.

1009567 is also a member of this sequence, but its position is presently undetermined: 9 * 10^1009567 - 1 is prime. - Predrag Kurtovic, Sep 19 2016

Makoto Kamada, Prime numbers of the form 899...999.
Index entries for primes involving repunits

Cf. A003307, A002235, A046865, A079906, A046866, A001771, A005541, A046867, A079907.

Do[ If[ PrimeQ[ 8*10^n + (10^n-1)], Print[n]], {n, 1, 6750, 2}]

is(n)=isprime(9*10^n-1) \\ Charles R Greathouse IV, Feb 17 2017

More terms from Herman Jamke (hermanjamke(AT)fastmail.fm), Jan 01 2008
a(16)-a(20) from Kamada data by Robert Price, Oct 19 2014
a(21) from Kamada data by Robert Price, Mar 10 2019
a(22) from Kamada data by Mohammed Yaseen, Jul 18 2021

A007505 Primes of form 3*2^n - 1.

Original entry on oeis.org

2, 5, 11, 23, 47, 191, 383, 6143, 786431, 51539607551, 824633720831, 26388279066623, 108086391056891903, 55340232221128654847, 226673591177742970257407, 59421121885698253195157962751, 30423614405477505635920876929023

Offset: 1

N. J. A. Sloane, Robert G. Wilson v

a(1) = 2, define f(k) = 2k+1, then a(n+1) = least prime fff...(a(n)). After 383 the next terem is 6143. We have f(383) = 767 (composite), f(767) = 1535 (composite), f(1565)=3071(composite), f(3071) = 6143 (prime), hence the next term is 6143= ffff(383). - Amarnath Murthy, Jul 13 2005

If n is in the sequence and m=(n+1)/3 then m is a solution of the equation, sigma(x+sigma(x))=3x (*). Is it true that there is no other solution of (*)? - Farideh Firoozbakht, Dec 05 2005

H. Riesel, Prime numbers and computer methods for factorization, Progress in Mathematics, Vol. 57, Birkhauser, Boston, 1985, Chap. 4, pp. 381-384.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 1..27
Heiko Harborth, On h-perfect numbers, Annales Mathematicae et Informaticae, 41 (2013) pp. 57-62.
Ernest G. Hibbs, Component Interactions of the Prime Numbers, Ph. D. Thesis, Capitol Technology Univ. (2022), see p. 33.
Wilfrid Keller, List of primes k*2^n - 1 for k < 300
Amelia Carolina Sparavigna, A recursive formula for Thabit numbers, Politecnico di Torino (Italy, 2019).
Amelia Carolina Sparavigna, Composition Operations of Generalized Entropies Applied to the Study of Numbers, International Journal of Sciences (2019) Vol. 8, No. 4, 87-92.
Eric Weisstein's World of Mathematics, Thabit ibn Kurrah Number
Index entries for sequences of n such that k*2^n-1 (or k*2^n+1) is prime

Subsequence of A083329.
See A002235 for more terms.
Cf. A039687 (primes of the form 3*2^n+1).
Cf. A010051.

a007505 n = a007505_list !! (n-1)
a007505_list = filter ((== 1) . a010051') a083329_list
-- Reinhard Zumkeller, Sep 10 2013

[a: n in [0..200] | IsPrime(a) where a is 3*2^n-1]; // Vincenzo Librandi, Mar 20 2013

Reap[For[n = 0, n <= 103, n++, If[PrimeQ[p = 3*2^n - 1], Sow[p]]]][[2, 1]] (* Jean-François Alcover, Dec 12 2012 *)
Select[Table[3 2^n - 1, {n, 0, 100}], PrimeQ] (* Vincenzo Librandi, Mar 20 2013 *)

for(n=0,100, if(isprime(t=3<Charles R Greathouse IV, Feb 07 2017

a(n) = 3*2^A002235(n)-1. - Zak Seidov, Jul 21 2016

A046867 Numbers n such that 10*11^n -1 is prime.

Original entry on oeis.org

1, 3, 37, 119, 255, 355, 371, 497, 1759, 34863, 50719, 147709

Offset: 1

N. J. A. Sloane

a(13) > 2*10^5. - Robert Price, Jan 19 2015

R. K. Guy, Unsolved Problems in Theory of Numbers, Section A3.

H. C. Williams, The primality of certain integers of the form 2Ar^n - 1, Acta Arith. 39 (1981), 7-17.

Cf. A003307, A002235, A046865, A079906, A046866, A001771, A005541, A056725, A079907.

Do[ If[ PrimeQ[10*11^n - 1], Print[n]], {n, 1, 2000}]

is(n)=isprime(10*11^n-1) \\ Charles R Greathouse IV, Feb 17 2017

One more term from Robert G. Wilson v, Jan 16 2003

a(10)-a(12) from Robert Price, Jan 19 2015

A079906 Numbers k such that 5*6^k - 1 is prime.

Original entry on oeis.org

1, 2, 6, 7, 11, 23, 33, 48, 68, 79, 116, 151, 205, 1016, 1332, 1448, 3481, 3566, 3665, 11233, 13363, 29166, 44358, 58530, 191706, 386450, 605168, 616879, 1204077

Offset: 1

Robert G. Wilson v, Jan 16 2003

a(29) > 618000. - Karsten Bonath, Nov 04 2019

R. K. Guy, Unsolved Problems in Theory of Numbers, Section A3.

H. C. Williams, The primality of certain integers of the form 2Ar^n - 1, Acta Arith. 39 (1981), 7-17.

Cf. A003307, A002235, A046865, A046866, A001771, A005541, A056725, A046867, A079907, A247260.

Do[ If[ PrimeQ[5*6^n - 1], Print[n]], {n, 1, 5000}]

for(n=1,2000, if(isprime(5*6^n-1),print1(n, ", ")))

a(20)-a(24) from Donovan Johnson, Nov 26 2008
a(25) from Robert Price, Jan 23 2016
a(26) from Karsten Bonath, Jul 01 2019
a(27) from Karsten Bonath, Oct 29 2019
a(28) from Karsten Bonath, Nov 04 2019
a(29) from Ryan Propper, Nov 21 2023

cp's OEIS Frontend

A076078 a(n) is the number of nonempty sets of distinct positive integers that have a least common multiple of n.

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A003307 Numbers k such that 2*3^k - 1 is prime.

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A002253 Numbers k such that 3*2^k + 1 is prime.

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A001771 Numbers k such that 7*2^k - 1 is prime.

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A002254 Numbers k such that 5*2^k + 1 is prime.

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A046865 Numbers k such that 4*5^k - 1 is prime.

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A056725 Numbers k such that 9*10^k - 1 is prime.

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