A002235 -id:A002235 - cp's OEIS frontend

A079907 Numbers n such that 11*12^n -1 is prime.

Original entry on oeis.org

1, 2, 21, 25, 33, 54, 78, 235, 1566, 2273, 2310, 4121, 7775, 42249, 105974, 138961

Offset: 1

Robert G. Wilson v, Jan 16 2003

a(17) > 2*10^5. - Robert Price, Mar 20 2015

R. K. Guy, Unsolved Problems in Theory of Numbers, 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, A046865, A079906, A046866, A001771, A005541, A056725, A046867.

[n: n in [1..600]| IsPrime(11*12^n - 1)]; // Vincenzo Librandi, Mar 21 2015

Do[ If[ PrimeQ[11*12^n - 1], Print[n]], {n, 1, 2000}]
Select[Range[10000], PrimeQ[(11 12^# - 1)] &] (* Vincenzo Librandi, Mar 21 2015 *)

for(n=1,2000, if(isprime(11*12^n-1),print1(n, ", ")))

More terms from Herman Jamke (hermanjamke(AT)fastmail.fm), Jan 05 2008

a(13)-a(16) from Robert Price, Mar 20 2015

A046866 Numbers k such that 6*7^k - 1 is prime.

Original entry on oeis.org

0, 1, 2, 7, 18, 55, 69, 87, 119, 141, 189, 249, 354, 1586, 2135, 2865, 2930, 4214, 7167, 67485, 74402, 79326, 231349

Offset: 1

N. J. A. Sloane

a(23) > 2*10^5. - Robert Price, Nov 13 2015

R. K. Guy, Unsolved Problems in Number Theory, 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, A001771, A005541, A056725, A046867, A079907.

Do[ If[ PrimeQ[6*7^n - 1], Print[n]], {n, 0, 5650}]

for(n=0,2000, if(isprime(6*7^n-1),print1(n, ", ")))

One more term from Jason Earls, Jul 21 2001
More terms from Robert G. Wilson v, Jan 17 2003
One more term from Ryan Propper, Jun 05 2006
a(20)-a(22) from Donovan Johnson, Nov 26 2008
First term 0 inserted by Georg Fischer, Aug 01 2019
a(23) from Riley Fisher, Dec 02 2024

A005541 Numbers k such that 8*3^k - 1 is prime.

Original entry on oeis.org

0, 1, 2, 4, 10, 17, 50, 170, 184, 194, 209, 641, 1298, 4034, 5956, 7154, 9970, 35956, 42730, 132004, 190610

Offset: 1

N. J. A. Sloane

a(22) > 2*10^5. - Robert Price, Mar 16 2014
All terms are verified primes (i.e., not probable primes). - Robert Price, Mar 16 2014
896701 is a term, found in 2010 (see link). - Jeppe Stig Nielsen, Jul 31 2020

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

The Prime Pages, 8*3^896701 - 1.
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. A003307, A002235, A046865, A079906, A046866, A001771, A056725, A046867, A079907.

lst={};Do[If[PrimeQ[8*3^n-1], AppendTo[lst, n]], {n, 13^3}];lst (* Vladimir Joseph Stephan Orlovsky, Sep 08 2008 *)

is(n)=isprime(8*3^n-1) \\ Charles R Greathouse IV, Feb 17 2017

More terms from Douglas Burke (dburke(AT)nevada.edu)
0 prepended by Vincenzo Librandi, Sep 26 2012
a(18)-a(21) from Robert Price, Mar 16 2014

A181490 Numbers k such that 32^k-1 and 32^k+1 are twin primes (A001097).

Original entry on oeis.org

1, 2, 6, 18

Offset: 1

M. F. Hasler, Oct 30 2010

Sequences A181491 and A181492 list the corresponding primes.
No more terms below three million. - Charles R Greathouse IV, Mar 14 2011
Intersection of A002235 and A002253. - Jeppe Stig Nielsen, Mar 05 2018

Cf. A001097, A001359, A002235, A002253, A006512, A014574, A294730.

Filtered([1..300],k->IsPrime(3*2^k-1) and IsPrime(3*2^k+1)); # Muniru A Asiru, Mar 11 2018

a:=k->`if`(isprime(3*2^k-1) and isprime(3*2^k+1),k,NULL); seq(a(k),k=1..1000); # Muniru A Asiru, Mar 11 2018

fQ[n_] := PrimeQ[3*2^n - 1] && PrimeQ[3*2^n + 1]; k = 1; lst= {}; While[k < 15001, If[fQ@k, AppendTo[lst, k]; Print@k]; k++ ] (* Robert G. Wilson v, Nov 05 2010 *)
Select[Range[20],AllTrue[3*2^#+{1,-1},PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Sep 24 2014 *)

PARI
```
for( k=1,999, ispseudoprime(3<
				
```

Equals { k | A007283(k) in A014574 } = { k | A153893(k) in A001359 }.

Pari program repaired by Charles R Greathouse IV, Mar 14 2011

A066466 Numbers having just one anti-divisor.

Original entry on oeis.org

3, 4, 6, 96, 393216

Offset: 1

Robert G. Wilson v, Jan 02 2002

See A066272 for definition of anti-divisor.
Jon Perry calls these anti-primes.
A066272(a(n)) = 1.
From Max Alekseyev, Jul 23 2007; updated Jun 25 2025: (Start)
Except for a(2) = 4, the terms of A066466 have form 2^k*p where p is odd prime and both 2^(k+1)*p-1, 2^(k+1)*p+1 are prime (i.e., twin primes). In other words, this sequence, omitting 4, is a subsequence of A040040 containing elements of the form 2^k*p with prime p.
Furthermore, since 2^(k+1)*p-1, 2^(k+1)*p+1 must equal -1 and +1 modulo 3, the number 2^(k+1)*p must be 0 modulo 3, implying that p=3. Therefore every term, except 4, must be of the form 3*2^k such that 3*2^(k+1)-1, 3*2^(k+1)+1 are twin primes. In other words, k+1 belongs to the intersection of A002253 and A002235.
According to Ballinger and Keller's lists, there are no other such k up to 22*10^6. Therefore a(6) (if it exists) is greater than 3*2^(22*10^6) ~= 10^6622660. (End)
From Daniel Forgues, Nov 23 2009: (Start)
The 2 last known anti-primes seem to relate to the Fermat primes (coincidence?):
96 = 3 * 2^5 = 3 * 2^F_1 = 3 * 2^[2^(2^1) + 1] and
393216 = 3 * 2^17 = 3 * 2^F_2 = 3 * 2^[2^(2^2) + 1],
where F_k is the k-th Fermat prime. (End)

Ray Ballinger and Wilfrid Keller, List of primes k·2^n + 1 for k < 300
Wilfrid Keller, List of primes k·2^n - 1 for k < 300
Jon Perry, The Anti-divisor [Cached copy]
Jon Perry, The Anti-divisor: Even More Anti-Divisors [Cached copy]

Cf. A000215, A002253, A002235, A066272, A019434.

antid[n_] := Select[ Union[ Join[ Select[ Divisors[2n - 1], OddQ[ # ] && # != 1 & ], Select[ Divisors[2n + 1], OddQ[ # ] && # != 1 & ], 2n/Select[ Divisors[ 2n], OddQ[ # ] && # != 1 &]]], # < n & ]; Select[ Range[10^5], Length[ antid[ # ]] == 1 & ]

Edited by Max Alekseyev, Oct 13 2009

A268061 Numbers k such that 7*8^k - 1 is prime.

Original entry on oeis.org

3, 7, 15, 59, 6127, 8703, 11619, 23403, 124299

Offset: 1

Robert Price, Jan 25 2016

a(10) > 2*10^5.

Terms are A001771(n)/3 that are integers.

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. similar sequences of the form k*(k+1)^n-1: A003307 (k=2), ... (k=3), A046865 (k=4), A079906 (k=5), A046866 (k=6), this sequence (k=7), ... (k=8), A056725 (k=9), A046867 (k=10), A079907 (k=11).

Cf. A001771, A002235, A005541, A247260.

Select[Range[0, 200000], PrimeQ[7*8^# - 1] &]

lista(nn) = for(n=1, nn, if(ispseudoprime(7*8^n-1), print1(n, ", "))) \\ Altug Alkan, Jan 25 2016

A281993 Integers m such that sigma(m) + sigma(2m) = 6m.

Original entry on oeis.org

10, 44, 184, 752, 12224, 49024, 12580864, 206158168064, 885443715520878608384, 226673591177468092350464, 232113757366000005450563584, 3894222643901120685369075227951104, 1020847100762815390371677078221595082752, 17126972312471518572699356075530215722269540352

Offset: 1

Michel Marcus, Feb 04 2017

nonn

This is the case h = 2 of the h-perfect numbers as defined in the Harborth link.

			10 is a term since sigma(10) + sigma(20) = 60, that is 6*10.

Heiko Harborth, On h-perfect numbers, Annales Mathematicae et Informaticae, 41 (2013) pp. 57-62.

Cf. A000203, A097215.

Select[Range[10^7], DivisorSigma[1, #] + DivisorSigma[1, 2 #] == 6 # &] (* Michael De Vlieger, Feb 04 2017 *)

isok(n, h=2) = sigma(n) + sigma(h*n) == 2*n*(h+1);

a(n) = 2^A002235(n+1) * A007505(n+1). - Daniel Suteu, Feb 08 2017 [See Harborth link for a proof.]

More terms from Jinyuan Wang, Feb 11 2020

A097214 Numbers m such that A076078(m) = m, where A076078(m) equals the number of sets of distinct positive integers with a least common multiple of m.

Original entry on oeis.org

1, 2, 4, 8, 10, 16, 32, 44, 64, 128, 184, 256, 512, 752, 1024, 2048, 4096, 8192, 12224, 16384, 32768, 49024, 61064, 65536, 131072, 262144, 524288, 981520, 1048576, 2097152, 4194304, 8388608, 12580864, 16777216, 33554432, 67108864, 134217728

Offset: 1

Matthew Vandermast, Aug 12 2004

nonn

Contains all powers of 2 (A000079). Union of A000079 and A097215.

If 3*2^k - 1 is prime then 2^k*(3*2^k-1) is in the sequence. So 2^A002235*(3*2^A002235-1) is a subsequence of this sequence. - Farideh Firoozbakht, Aug 06 2005

			A total of 10 sets of distinct positive integers have a least common multiple of 10: {1,2,5}, {1,2,5,10}, {1,2,10}, {1,5,10}, {1,10}, {2,5}, {2,5,10}, {2,10}, {5,10} and {10}. Hence 10 is in the sequence.

Jinyuan Wang, Table of n, a(n) for n = 1..334

Cf. A076078, A097215.

Cf. A002235.

a(26) corrected by Jinyuan Wang, Feb 11 2020

A097215 Numbers m such that A076078(m) = m and bigomega(m) >= 2; or in other words, A097214, excluding powers of 2.

Original entry on oeis.org

10, 44, 184, 752, 12224, 49024, 61064, 981520, 12580864, 206158168064, 16492668126208, 1080863908958322688, 18374686467592175488, 885443715520878608384, 4703919738602662723328, 226673591177468092350464, 232113757366000005450563584, 3894222643901120685369075227951104

Offset: 1

Matthew Vandermast, Aug 12 2004

nonn

A076078(m) equals the number of sets of distinct positive integers with a least common multiple of m.
If 3*2^k - 1 is an odd prime then 2^k*(3*2^k-1) is in the sequence. - Farideh Firoozbakht, May 03 2009
For what seems to be an appearance of this sequence in a different context, see Harborth (2013). - N. J. A. Sloane, Jun 08 2013

			For example, there are 184 sets of distinct positive integers with a least common multiple of 184.

F. Firoozbakht, M. F. Hasler, Variations on Euclid's formula for Perfect Numbers, JIS 13 (2010) #10.3.1.
Heiko Harborth, On h-perfect numbers, Annales Mathematicae et Informaticae, 41 (2013) pp. 57-62.

Cf. A097214, A097416, A281993.

Cf. A002235. - Farideh Firoozbakht, May 03 2009

f[n_] := Block[{d = Divisors[n]}, Plus @@ (MoebiusMu[n/d](2^DivisorSigma[0, d] - 1))]; t = Union[ Table[ f[n], {n, 28000000}]]; Select[t, f[ # ] == # && !IntegerQ[ Log[2, # ]] &] (* Robert G. Wilson v, Aug 17 2004 *)

A076078(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; }
lista(nn) = {my(w=List([]), m=1, q=2, g); for(k=1, logint(nn, 2)-1, q=nextprime(q+1); m=m*q; for(r=1, nn\2^k-1, g=factor(A076078(m*2^r))[, 2]; if(#g==k+1&&g[2]==1, listput(w, A076078(m*2^r))))); Set(w); } \\ Jinyuan Wang, Feb 11 2020

More terms from Robert G. Wilson v, Aug 18 2004

More terms from Jinyuan Wang, Feb 11 2020

A230527 Numbers n such that 3^6*2^n - 1 is prime.

Original entry on oeis.org

5, 17, 23, 113, 125, 173, 199, 319, 377, 397, 785, 937, 2167, 3785, 3977, 5957, 7727, 8249, 14677, 19577, 20485, 36319, 57509, 60703, 66677, 76877, 77017, 83407, 229405, 1003373

Offset: 1

Lei Zhou, Oct 22 2013

Riesel Primes with k = 3^6 = 729.

Checked up to n = 1003600.

			729*2^5-1=23327 is a prime number.

K. Bonath, Riesel Prime Database
C. K. Caldwell, 729*2^229405-1
C. K. Caldwell, 729*2^1003373-1

Cf. A002235, A002236, A050539, A050566, A050880.

b=3^6; i=0; Table[While[i++; cp=b*2^i-1; !PrimeQ[cp]]; i, {j, 1, 15}]

is(n)=ispseudoprime(3^6*2^n-1) \\ Charles R Greathouse IV, May 22 2017

Lei Zhou, Nov 08 2013, added a Mathematica program for small elements.

cp's OEIS Frontend

A079907 Numbers n such that 11*12^n -1 is prime.

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Keywords

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Magma

Mathematica

PARI

Extensions

A046866 Numbers k such that 6*7^k - 1 is prime.

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Mathematica

PARI

Extensions

A005541 Numbers k such that 8*3^k - 1 is prime.

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Mathematica

PARI

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A181490 Numbers k such that 3*2^k-1 and 3*2^k+1 are twin primes (A001097).

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GAP

Maple

Mathematica

PARI

Formula

Extensions

A066466 Numbers having just one anti-divisor.

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Mathematica

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A268061 Numbers k such that 7*8^k - 1 is prime.

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Links

Crossrefs

Programs

Mathematica

PARI

A281993 Integers m such that sigma(m) + sigma(2*m) = 6*m.

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A181490 Numbers k such that 32^k-1 and 32^k+1 are twin primes (A001097).

A281993 Integers m such that sigma(m) + sigma(2m) = 6m.