A051154 -id:A051154 - cp's OEIS frontend

A001576 a(n) = 1^n + 2^n + 4^n.

3, 7, 21, 73, 273, 1057, 4161, 16513, 65793, 262657, 1049601, 4196353, 16781313, 67117057, 268451841, 1073774593, 4295032833, 17180000257, 68719738881, 274878431233, 1099512676353, 4398048608257, 17592190238721, 70368752566273, 281474993487873, 1125899940397057

Offset: 0

N. J. A. Sloane

Equals A135576, except for the first term. - Omar E. Pol, Nov 18 2008
Conjecture: For n > 1, if a(n) = 1^n + 2^n + 4^n is a prime number then n is of the form 3^h. For example, for h=1, n=3, a(n) = 1^3 + 2^3 + 4^3 = 73 (prime); for h=2, n=9, a(n) = 1^9 + 2^9 + 4^9 = 262657 (prime); for h=3, n=27, a(n) is not prime. - Vincenzo Librandi, Aug 03 2010
The previous conjecture was proved by Golomb in 1978. See A051154. - T. D. Noe, Aug 15 2010
Another more elementary proof can be found in Liu link. - Bernard Schott, Mar 08 2019
Fills in one quarter section of the figurate form of the Sierpinski square curve. See illustration in links and A141725. - John Elias, Mar 29 2023

T. D. Noe, Table of n, a(n) for n = 0..200
John Elias, Illustration of Initial Terms: 1/4 Sierpinski Square Curve
Andy Liu, West German Mathematical Olympiad 1982 - Second round, Problem 4, Crux Mathematicorum, p. 105, Vol. 12, May. 86.
Index entries for linear recurrences with constant coefficients, signature (7,-14,8).

Subsequence of A002061.
Cf. A001550, A034513, A001579, A074501-A074580, A135576, A135577.
See also comments in A051154.
Cf. A006095, A022166.

Mathematica
```
Table[1^n + 2^n + 4^n, {n, 0, 24}]
```

a(n)=1+2^n+4^n \\ Charles R Greathouse IV, Jun 10 2011

[sigma(4,n)for n in range(0,23)] # Zerinvary Lajos, Jun 04 2009

a(n) = 6*a(n-1) - 8*a(n-2) + 3.
O.g.f.: -1/(-1+x) - 1/(-1+2*x) - 1/(-1+4*x) = ( -3+14*x-14*x^2 ) / ( (x-1)*(2*x-1)*(4*x-1) ). - R. J. Mathar, Feb 29 2008
E.g.f.: e^x + e^(2*x) + e^(4*x). - Mohammad K. Azarian, Dec 26 2008
a(n) = A024088(n)/A000225(n). - Reinhard Zumkeller, Feb 15 2009
Exp( Sum_{n >= 1} a(n)*x^n/n ) = 1 + 7*x + 35*x^2 + 155*x^3 + ... is the o.g.f. for the 2nd subdiagonal of triangle A022166, essentially A006095. - Peter Bala, Apr 07 2015

A051156 a(n) = (2^p^2 - 1)/(2^p - 1) where p is the n-th prime.

Original entry on oeis.org

5, 73, 1082401, 4432676798593, 1298708349570020393652962442872833, 91355004067076339167413824240109498970069278721, 7588608256743087977590500540116743445925584618982806531428980886590618779326218241

Offset: 1

N. J. A. Sloane

Note that a(n) = Phi(p,2^p) or a(n) = Phi(p^2,2), where Phi(m,x) is the m-th cyclotomic polynomial and p is the n-th prime. - Thomas Ordowski, Feb 18 2014

Cf. A051154, A051155, A051157.

Table[Cyclotomic[Prime[n]^2, 2], {n, 7}] (* Arkadiusz Wesolowski, May 13 2012 *)
Table[(2^Prime[n]^2-1)/(2^Prime[n]-1),{n,10}] (* Harvey P. Dale, Apr 06 2019 *)

a(n) = A070526(prime(n)), a(n) = A019320(prime(n)^2). - Thomas Ordowski, Feb 18 2014

A143027 Sturdy prime numbers: p such that in binary notation k*p has at least as many 1-bits as p for all k > 0.

Original entry on oeis.org

2, 3, 5, 7, 17, 31, 73, 89, 127, 257, 1801, 2089, 8191, 65537, 131071, 178481, 262657, 524287, 2099863, 616318177, 2147483647, 4432676798593

Offset: 1

T. D. Noe, Jul 17 2008, Jul 21 2008

The primes in A125121. This sequence includes the Fermat primes (A019434), Mersenne primes (A000668) and the three known primes in A051154. It appears that almost all primes are flimsy numbers, A005360.
Odd sturdy primes appear to be the largest primitive prime factor of 2^q-1 for q a prime or prime power. The values of q for the current terms: 2, 4, 3, 8, 5, 9, 11, 16, 25, 29, 13, 32, 17, 23, 27 and 19. The sequence probably continues with 2099863, 6700417, 13264529, 20394401, 97685839.
From T. D. Noe, Mar 01 2010: (Start)
Max Alekseyev reports that 6700417, 13264529, 20394401, and 97685839 are not sturdy because each number divides a number having fewer 1-bits: 6700417 divides 2^32 + 1, 13264529 divides 331613225, 20394401 divides 1611157679, and 97685839 divides 18014398643699713. He conjectures that 616318177 is the next term.
If q is a prime power, q = r^s, then the primitive part of 2^q-1 is (2^r^s-1)/(2^r^(s-1)-1). According to Stolarsky's Theorem 2.1, this primitive part is sturdy. If the primitive part is prime, then it is in this sequence. Hence 7^2 produces the sturdy prime 4432676798593 and 59^2 produces a 1031-digit sturdy prime. (End)
Clokie et al. verify that the next two sturdy primes after 2099863 are 616318177 and 2147483647.  These are all up to 2^32.  Two additional sturdy primes are 57912614113275649087721 = (2^83 - 1)/167 and 10350794431055162386718619237468234569 = (2^131 - 1)/263, but probably there are some in between these and 2147483647. Jeffrey Shallit, Feb 10 2020
From Jason Yuen, Mar 30 2024: (Start)
For all x>log_2(p), 1+A000120(p-(2^x mod p)) >= A000120(p). This follows from the fact that 2^x+p-(2^x mod p) is a multiple of p.
a(23) > 5*10^12. See a143027_5e12.txt for more details. (End)

Trevor Clokie et al., Computational Aspects of Sturdy and Flimsy Numbers, arxiv preprint arXiv:2002.02731 [cs.DS], February 7 2020.
K. B. Stolarsky, Integers whose multiples have anomalous digital frequencies, Acta Arithmetica, 38 (1980), 117-128.
Jason Yuen, a143027_5e12.txt. This file shows that a(23) > 5*10^12.

Cf. A125121, A181863, A000120.

2089 and 8191 were found by Ray Chandler
2099863 added by T. D. Noe, Mar 01 2010
616318177, 2147483647 added by Jeffrey Shallit, Feb 10 2020
4432676798593 added by Jason Yuen, Mar 30 2024

A051155 a(n) = (2^5^(n+1) - 1)/(2^5^n - 1).

Original entry on oeis.org

31, 1082401, 1267650638007162390353805312001, 3273390607896141870013189696827599152293599089395397756694773868291726792119530172040230983402733964346814858022765439290901496446006940490331586560001

Offset: 0

N. J. A. Sloane

Cf. A051154, A051156, A051157.

Table[(2^5^(n+1) - 1)/(2^5^n - 1),{n,0,3}] (* Stefano Spezia, Dec 23 2022 *)

A051157 a(n) = (2^p^3 - 1)/(2^p^2 - 1) where p = n-th prime.

Original entry on oeis.org

17, 262657, 1267650638007162390353805312001, 31828687130226401637050536789795514059715404495050094614691019248562308626412218127220737

Offset: 1

N. J. A. Sloane

Cf. A051154, A051155, A051156.

Table[(2^Prime[n]^3 - 1)/(2^Prime[n]^2 - 1),{n,4}] (* Stefano Spezia, Dec 23 2022 *)

A245730 Primes of the form 1+2^k+2^(2k)+...+2^((j-1)k) for some k>0, j>0.

Original entry on oeis.org

3, 5, 7, 17, 31, 73, 127, 257, 8191, 65537, 131071, 262657, 524287, 2147483647, 4432676798593, 2305843009213693951, 618970019642690137449562111, 162259276829213363391578010288127, 170141183460469231731687303715884105727

Offset: 1

Chai Wah Wu, Jul 30 2014

nonn

Contains the Mersenne primes A000668 which correspond to k=1. In base 2, primes with n 1's and k-1 0's between pairs of 1's.  Is a factor of 2^(n*k)-1.
Primes of the form (2^(n*k)-1)/(2^k-1). k=1 gives Mersenne primes 2^n-1 for n in A000043. n=2 and k=2^m gives Fermat primes 2^(2^m)+1 (A019434) for m = 0 to 4. k=n gives (2^(n^2)-1)/(2^n-1) which is prime for n = 2, 3, 7, 59 (A156585, n must be prime). The only other term below 2000 digits is 262657 for k=9 and n=3. - Jens Kruse Andersen, Aug 02 2014
The case n=3 gives the primes in A051154. - John Blythe Dobson
Wells Johnson (1977), 199, Corollary 6, proved that members of this sequence cannot be Wieferich primes (A001220). - John Blythe Dobson

			The number 4432676798593 is in the list as it is prime and it is equal to 1+2^7+2^(2*7)+2^(3*7)+2^(4*7)+2^(5*7)+2^(6*7).

Wells Johnson, On the nonvanishing of Fermat quotients (mod p), J. für Math. 292 (1977), 196-200.

Jens Kruse Andersen, Table of n, a(n) for n = 1..25

Cf. A000043, A000668, A019434, A156585.

from sympy import isprime
sorted([int(('0'*m+'1')*n,2) for m in range(50) for n in range(1,50) if isprime(int(('0'*m+'1')*n,2))])

A198915 a(n) = 1 - 2^k + 4^k where k = 3^n.

Original entry on oeis.org

3, 57, 261633, 18014398375264257, 5846006549323611672814736913013492849365380759553

Offset: 0

Artur Jasinski, Oct 31 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..6
Dario A. Alpern, Factors of Generalized Fermat Numbers

Cf. A051154.

[4^(3^n) - 2^(3^n) + 1: n in [0..7]]; // Vincenzo Librandi, Jun 29 2014

A198915:=n->4^(3^n) - 2^(3^n) + 1: seq(A198915(n), n=0..6); # Wesley Ivan Hurt, May 03 2017

Mathematica
```
Table[4^(3^n) - 2^(3^n) + 1, {n, 0, 5}]
```

a(n)=1<<(2*3^n)-1<<3^n+1 \\ Charles R Greathouse IV, Oct 31 2011

cp's OEIS Frontend

A001576 a(n) = 1^n + 2^n + 4^n.

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A051156 a(n) = (2^p^2 - 1)/(2^p - 1) where p is the n-th prime.

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A143027 Sturdy prime numbers: p such that in binary notation k*p has at least as many 1-bits as p for all k > 0.

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A051155 a(n) = (2^5^(n+1) - 1)/(2^5^n - 1).

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A051157 a(n) = (2^p^3 - 1)/(2^p^2 - 1) where p = n-th prime.

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A245730 Primes of the form 1+2^k+2^(2*k)+...+2^((j-1)*k) for some k>0, j>0.

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A198915 a(n) = 1 - 2^k + 4^k where k = 3^n.

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A245730 Primes of the form 1+2^k+2^(2k)+...+2^((j-1)k) for some k>0, j>0.