A051006 -id:A051006 - cp's OEIS frontend

A051007 Continued fraction for prime constant A051006.

0, 2, 2, 2, 3, 12, 131, 1, 7, 1, 2, 1, 3, 3, 1, 2, 5, 39, 2, 1, 169, 2, 2, 2, 1, 1, 2, 5, 1, 2, 1, 199, 24, 7, 7, 1, 163, 1, 3, 2, 1, 2, 14, 1, 3, 5, 1, 1, 1, 1, 3, 3, 2, 1, 279, 1, 4, 1, 1, 4, 1, 92, 1, 16, 2, 3, 1, 5, 2, 25, 9, 1, 1, 2, 8, 1, 2, 5, 1, 1, 3, 1, 1, 2, 1, 7, 1, 1, 1, 5, 1, 2, 2, 5, 6, 1

Offset: 0

Eric W. Weisstein

			0.414682509851111660248109622... = 0 + 1/(2 + 1/(2 + 1/(2 + 1/(3 + ...)))). - _Harry J. Smith_, Jun 15 2009

Harry J. Smith, Table of n, a(n) for n = 0..20000
Eric Weisstein's World of Mathematics, Prime Constant.
G. Xiao, Contfrac
Index entries for continued fractions for constants

Cf. A010051, A051006. Increasing partial quotients are in A102878.

ContinuedFraction[ FromDigits[{{Table[ If[ PrimeQ[n], 1, 0], {n, 370}]}, 0}, 2], 95] (* Robert G. Wilson v, Jan 15 2005 *)

{ default(realprecision, 20080); x=0; m=67000; for (n=1, m, if (isprime(n), a=1, a=0); x=2*x+a; ); x=contfrac(x/2^m); for (n=0, 20000, write("b051007.txt", n, " ", x[n+1])); } \\ Harry J. Smith, Jun 15 2009

A102878 Increasing partial quotients in the continued fraction expansion of the prime constant (A051006).

Original entry on oeis.org

0, 2, 3, 12, 131, 169, 199, 279, 413, 851, 2771, 18514, 20740, 20780, 147756, 207783, 312134, 361393, 6931243

Offset: 0

Robert G. Wilson v, Jan 15 2005

nonn

a(19) is a 3010301-digit number (1.71206300894551448591713167863057582188628436436...*10^3010300) with digit distribution 301276, 301983, 300946, 301660, 301631, 301181, 299864, 300633, 300456 & 300671.

Cf. A051006, A051007, A103518.

cf = ContinuedFraction[ FromDigits[ {{Table[ If[ PrimeQ[n], 1, 0], {n, 10^7}]}, 0}, 2]]; a = 0; Do[ If[ cf[[n]] > a, a = cf[[n]]; Print[a]], {n, 5842783}]

A103518 Indices of increasing partial quotients PQ_i of the continued fraction for the prime constant (A051006).

Original entry on oeis.org

0, 1, 4, 5, 6, 20, 31, 54, 306, 356, 762, 3174, 20240, 22693, 35793, 58491, 81251, 206410, 228533, 2921406

Offset: 0

Robert G. Wilson v and Eric W. Weisstein, Feb 07 2005

nonn

Cf. A051006, A051007. A102878.

cf = ContinuedFraction[ FromDigits[ {{Table[ If[ PrimeQ[n], 1, 0], {n, 10^7}]}, 0}, 2]]; a = -1; Do[ If[ cf[[n]] > a, a = cf[[n]]; Print[n-1]], {n, 5842783}]

A092856 Duplicate of A051006.

Original entry on oeis.org

4, 1, 4, 6, 8, 2, 5, 0, 9, 8, 5, 1, 1, 1, 1, 6, 6, 0, 2, 4, 8, 1, 0, 9, 6, 2, 2, 1, 5, 4, 3, 0, 7, 7, 0

Offset: 1

dead

A010051 Characteristic function of primes: 1 if n is prime, else 0.

Original entry on oeis.org

0, 1, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0

Offset: 1

N. J. A. Sloane

The following sequences all have the same parity (with an extra zero term at the start of a(n)): a(n), A061007, A035026, A069754, A071574. - Jeremy Gardiner, Aug 09 2002
Hardy and Wright prove that the real number 0.011010100010... is irrational. See Nasehpour link. - Michel Marcus, Jun 21 2018
The spectral components (excluding the zero frequency) of the Fourier transform of the partial sequences {a(j)} with j=1..n and n an even number, exhibit a remarkable symmetry with respect to the central frequency component at position 1 + n/4. See the Fourier spectrum of the first 2^20 terms in Links, Comments in A289777, and Conjectures in A001223 of Sep 01 2019. It also appears that the symmetry grows with n. - Andres Cicuttin, Aug 23 2020

J.-P. Allouche and J. Shallit, Automatic Sequences, Cambridge Univ. Press, 2003, p. 3.
V. Brun, Über das Goldbachsche Gesetz und die Anzahl der Primzahlpaare, Arch. Mat. Natur. B, 34, no. 8, 1915.
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, Oxford University Press, London, 1975.
Clifford A. Pickover, A Passion for Mathematics, Wiley, 2005; see p. 65.
Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See p. 132.

Daniel Forgues, Table of n, a(n) for n = 1..100000 (first 10000 terms from N. J. A. Sloane)
Andres Cicuttin, Fourier spectrum of the first 2^20 terms of the characteristic function of primes.
William Craig, Jan-Willem van Ittersum and Ken Ono, Integer partitions detect the primes, PNAS, Vol. 121, No. 39 (2024), e2409417121.
Yoichi Motohashi, An overview of the Sieve Method and its History, arXiv:math/0505521 [math.NT], 2005-2006.
Peyman Nasehpour, A Computational Criterion for the Irrationality of Some Real Numbers, Journal of Algorithms and Computation, Vol. 52, No. 1 (2020), pp. 97-104, preprint, arXiv:1806.07560 [math.AC], 2018.
José L. Ramírez and Gustavo N. Rubiano, Properties and Generalizations of the Fibonacci Word Fractal, The Mathematica Journal, Vol. 16 (2014).
Eric Weisstein's World of Mathematics, Prime Number.
Eric Weisstein's World of Mathematics, Prime Constant.
Eric Weisstein's World of Mathematics, Prime zeta function primezeta(s).
Index entries for characteristic functions

Cf. A051006 (constant 0.4146825... (base 10) = 0.01101010001010001010... (base 2)), A001221 (inverse Moebius transform), A143519, A156660, A156659, A156657, A059500, A053176, A059456, A072762.
First differences of A000720, so A000720 gives partial sums.
Column k=1 of A117278.
Characteristic function of A000040.
Cf. A008683.

import Data.List (unfoldr)
a010051 :: Integer -> Int
a010051 n = a010051_list !! (fromInteger n-1)
a010051_list = unfoldr ch (1, a000040_list) where
   ch (i, ps'@(p:ps)) = Just (fromEnum (i == p),
                              (i + 1, if i == p then ps else ps'))
-- Reinhard Zumkeller, Apr 17 2012, Sep 15 2011

s:=[]; for n in [1..100] do if IsPrime(n) then s:=Append(s,1); else s:=Append(s,0); end if; end for; s;

[IsPrime(n) select 1 else 0: n in [1..100]];  // Bruno Berselli, Mar 02 2011

A010051:= n -> if isprime(n) then 1 else 0 fi;

Table[ If[ PrimeQ[n], 1, 0], {n, 105}] (* Robert G. Wilson v, Jan 15 2005 *)
Table[Boole[PrimeQ[n]], {n, 105}] (* Alonso del Arte, Aug 09 2011 *)
Table[PrimePi[n] - PrimePi[n-1], {n,50}] (* G. C. Greubel, Jan 05 2017 *)

a(n)=isprime(n) \\ Charles R Greathouse IV, Apr 16 2011

from sympy import isprime
def A010051(n): return int(isprime(n)) # Chai Wah Wu, Jan 20 2022

a(n) = floor(cos(Pi*((n-1)! + 1)/n)^2) for n >= 2. - Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Nov 07 2002
Let M(n) be the n X n matrix m(i, j) = 0 if n divides ij + 1, m(i, j) = 1 otherwise; then for n > 0 a(n) = -det(M(n)). - Benoit Cloitre, Jan 17 2003
n >= 2, a(n) = floor(phi(n)/(n - 1)) = floor(A000010(n)/(n - 1)). - Benoit Cloitre, Apr 11 2003
a(n) = Sum_{d|gcd(n, A034386(n))} mu(d). [Brun]
a(m*n) = a(m)*0^(n - 1) + a(n)*0^(m - 1). - Reinhard Zumkeller, Nov 25 2004
a(n) = 1 if n has no divisors other than 1 and n, and 0 otherwise. - Jon Perry, Jul 02 2005
Dirichlet generating function: Sum_{n >= 1} a(n)/n^s = primezeta(s), where primezeta is the prime zeta function. - Franklin T. Adams-Watters, Sep 11 2005
a(n) = (n-1)!^2 mod n. - Franz Vrabec, Jun 24 2006
a(n) = A047886(n, 1). - Reinhard Zumkeller, Apr 15 2008
Equals A051731 (the inverse Möbius transform) * A143519. - Gary W. Adamson, Aug 22 2008
a(n) = A051731((n + 1)! + 1, n) from Wilson's theorem: n is prime if and only if (n + 1)! is congruent to -1 mod n. - N-E. Fahssi, Jan 20 2009, Jan 29 2009
a(n) = A166260/A001477. - Mats Granvik, Oct 10 2009
a(n) = 0^A070824, where 0^0=1. - Mats Granvik, Gary W. Adamson, Feb 21 2010
It appears that a(n) = (H(n)*H(n + 1)) mod n, where H(n) = n!*Sum_{k=1..n} 1/k = A000254(n). - Gary Detlefs, Sep 12 2010
Dirichlet generating function: log( Sum_{n >= 1} 1/(A112624(n)*n^s) ). - Mats Granvik, Apr 13 2011
a(n) = A100995(n) - sqrt(A100995(n)*A193056(n)). - Mats Granvik, Jul 15 2011
a(n) * (2 - n mod 4) = A151763(n). - Reinhard Zumkeller, Oct 06 2011
(n - 1)*a(n) = ( (2*n + 1)!! * Sum_{k=1..n}(1/(2*k + 1))) mod n, n > 2. - Gary Detlefs, Oct 07 2011
For n > 1, a(n) = floor(1/A001222(n)). - Enrique Pérez Herrero, Feb 23 2012
a(n) = mu(n) * Sum_{d|n} mu(d)*omega(d), where mu is A008683 and omega A001222 or A001221 indistinctly. - Enrique Pérez Herrero, Jun 06 2012
a(n) = A003418(n+1)/A003418(n) - A217863(n+1)/A217863(n) = A014963(n) - A072211(n). - Eric Desbiaux, Nov 25 2012
For n > 1, a(n) = floor(A014963(n)/n). - Eric Desbiaux, Jan 08 2013
a(n) = ((abs(n-2))! mod n) mod 2. - Timothy Hopper, May 25 2015
a(n) = abs(F(n)) - abs(F(n)-1/2) - abs(F(n)-1) + abs(f(n)-3/2), where F(n) = Sum_{m=2..n+1} (abs(1 - (n mod m)) - abs(1/2 - (n mod m)) + 1/2), n > 0. F(n) = 1 if n is prime, > 1 otherwise, except F(1) = 0. a(n) = 1 if F(n) = 1, 0 otherwise. - Timothy Hopper, Jun 16 2015
For n > 4, a(n) = (n-2)! mod n. - Thomas Ordowski, Jul 24 2016
From Ilya Gutkovskiy, Jul 24 2016: (Start)
G.f.: A(x) = Sum_{n>=1} x^A000040(n) = B(x)*(1 - x), where B(x) is the g.f. for A000720.
a(n) = floor(2/A000005(n)), for n>1. (End)
a(n) = pi(n) - pi(n-1) = A000720(n) - A000720(n-1), for n>=1. - G. C. Greubel, Jan 05 2017
Decimal expansion of Sum_{k>=1} (1/10)^prime(k) = 9 * Sum_{k>=1} pi(k)/10^(k+1), where pi(k) = A000720(k). - Amiram Eldar, Aug 11 2020
a(n) = 1 - ceiling((2/n) * Sum_{k=2..floor(sqrt(n))} floor(n/k)-floor((n-1)/k)), n>1. - Gary Detlefs, Sep 08 2023
a(n) = Sum_{d|n} mu(d)*omega(n/d), where mu = A008683 and omega = A001221. - Ridouane Oudra, Apr 12 2025
a(n) = 0 if (n^2 - 3*n + 2) * A000203(n) - 8 * A002127(n) > 0 else 1 (n>2, see Craig link). - Bill McEachen, Jul 04 2025

A034785 a(n) = 2^(n-th prime).

Original entry on oeis.org

4, 8, 32, 128, 2048, 8192, 131072, 524288, 8388608, 536870912, 2147483648, 137438953472, 2199023255552, 8796093022208, 140737488355328, 9007199254740992, 576460752303423488, 2305843009213693952

Offset: 1

Asher Auel

These are the "outputs" in Conway's PRIMEGAME (see A007542). - Alonso del Arte, Jan 03 2011

Multiplicative encoding of the n-th prime. - Daniel Forgues, Feb 26 2017

			a(4) = 128 because the fourth prime number is 7 and 2^7 = 128.

Vincenzo Librandi, Table of n, a(n) for n = 1..200
Index to divisibility sequences.

Cf. A000040, A000430, A051006, A073718 (2^(n-th composite)), A074736.

Cf. A184082, A184083.

a034785 = (2 ^) . a000040
-- Reinhard Zumkeller, Feb 07 2015, Jan 24 2012

[2^p: p in PrimesUpTo(100)]; // Vincenzo Librandi, Apr 29 2014

2^Prime@Range@40  (* Vladimir Joseph Stephan Orlovsky, Apr 11 2011 *)

a(n)=1<Charles R Greathouse IV, Apr 07 2012

from sympy import prime
def A034785(n): return 1<Chai Wah Wu, Aug 09 2024

From Amiram Eldar, Aug 11 2020: (Start)
a(n) = 2^A000040(n).
Sum_{n>=1} 1/a(n) = A051006. (End)
From Amiram Eldar, Nov 22 2022: (Start)
Product_{n>=1} (1 + 1/a(n)) = A184083.
Product_{n>=1} (1 - 1/a(n)) = A184082. (End)

More terms from James Sellers, Feb 04 2000

A092855 Numbers k such that the k-th bit in the binary expansion of sqrt(2) - 1 is 1: sqrt(2) - 1 = Sum_{n>=1} 1/2^a(n).

Original entry on oeis.org

2, 3, 5, 7, 13, 16, 17, 18, 19, 22, 23, 26, 27, 30, 31, 32, 33, 34, 35, 36, 39, 40, 41, 43, 44, 45, 46, 49, 50, 53, 56, 61, 65, 67, 68, 71, 73, 74, 75, 76, 77, 79, 80, 84, 87, 88, 90, 91, 94, 95, 97, 98, 99, 101, 103, 105, 108, 110, 112, 114, 115, 116, 117, 118, 120, 123, 124

Offset: 1

Ferenc Adorjan (fadorjan(AT)freemail.hu)

Previous name was: Representation of sqrt(2) - 1 by an infinite sequence.
Any real number in the range (0,1), having infinite number of nonzero binary digits, can be represented by a monotonic infinite sequence, such a way that n is in the sequence iff the n-th digit in the fraction part of the number is 1. See also A092857.
An example for the inverse mapping is A051006.
It is relatively rich in primes, but cf. A092875.

Paolo Xausa, Table of n, a(n) for n = 1..10000
Ferenc Adorjan, Binary mapping of monotonic sequences and the Aronson function.

Cf. A004539, A051006, A092857, A092875, A320985 (complement).

PositionIndex[First[RealDigits[Sqrt[2], 2, 200, -1]]][1] (* Paolo Xausa, Sep 01 2024 *)

v=binary(sqrt(2))[2]; for(i=1,#v,if(v[i],print1(i,","))) \\ Ralf Stephan, Mar 30 2014

New name from Joerg Arndt, Aug 26 2024

A093515 Numbers k such that either k or k-1 is a prime.

Original entry on oeis.org

2, 3, 4, 5, 6, 7, 8, 11, 12, 13, 14, 17, 18, 19, 20, 23, 24, 29, 30, 31, 32, 37, 38, 41, 42, 43, 44, 47, 48, 53, 54, 59, 60, 61, 62, 67, 68, 71, 72, 73, 74, 79, 80, 83, 84, 89, 90, 97, 98, 101, 102, 103, 104, 107, 108, 109, 110, 113, 114, 127, 128, 131, 132, 137, 138, 139

Offset: 1

Ferenc Adorjan (fadorjan(AT)freemail.hu)

Original name: Transform of the prime sequence by the Rule 110 cellular automaton.
As described in A051006, a monotonic sequence can be mapped into a fractional real. Then the binary digits of that real can be treated (transformed) by an elementary cellular automaton. Taking the resulting sequence of binary digits as a fractional real, it can be mapped back into a sequence, as in A092855.
From M. F. Hasler, Mar 01 2008: (Start)
The "Rule110" transform as used here involves a right-shift of the sequence before applying the transform as described on the MathWorld page.
Robert G. Wilson v observed that this sequence contains exactly the indices for which A121561 equals 1. (End)
From M. F. Hasler, Jan 07 2019: (Start)
The correspondence of monotonic sequences with fractional reals mentioned in the first comment is not really relevant here: RuleX most naturally maps directly one characteristic sequence to another and thus one set (or increasing sequence) to another one. Interpreting the characteristic sequences as binary digits of a fractional real then yields a map from [0,1] into [0,1] rather as a consequence.
Antti Karttunen observed that this seems to be the complement of A005381 (k and k-1 are composite). This is indeed the case: The characteristic sequence of primes has no three subsequent 1's. In all other cases of the 8 possible inputs for Rule110, the output is 0 if and only if the cell itself and its neighbor to the right are zero, which here means "k and k+1 are composite", and with the right shift mentioned above, the complement of A005381, i.e., numbers such that k or k-1 is prime (or: primes U primes + 1). We have actually proved the more general
Theorem: Rule110 transforms any set S having no three consecutive integers into the set S' = {k | k or k-1 is in S} = S U (1 + S). (End)

M. F. Hasler, Table of n, a(n) for n = 1..19998 (using prime(1..10^4)).
Ferenc Adorjan, Binary mapping of monotonic sequences - the Aronson and the CA functions
Eric Weisstein's World of Mathematics, Elementary Cellular Automaton
Eric Weisstein's World of Mathematics, Rule110 Elementary Cellular Automaton

Cf. A092855, A051006, A093510, A093511, A093512, A093513, A093514, A093516, A093517, A161903.
Cf. A005381 (complement, apart from 1 which is in neither sequence), A323162.
Cf. A121561.

[n: n in [2..180] | not(not IsPrime(n) and not IsPrime(n-1))]; // Vincenzo Librandi, Jan 08 2019

Select[Range[2, 150], !(!PrimeQ[# - 1] && !PrimeQ[#]) &] (* Vincenzo Librandi, Jan 08 2019 *)

{ca_tr(ca,v)= /* Calculates the Cellular Automaton transform of the vector v by the rule ca */
local(cav=vector(8),a,r=[],i,j,k,l,po,p=vector(3));
a=binary(min(255,ca));k=matsize(a)[2];forstep(i=k,1,- 1,cav[k-i+1]=a[i]);
j=0;l=matsize(v)[2];k=v[l];po=1;
for(i=1,k+2,j*=2;po=isin(i,v,l,po);j=(j+max(0,sign(po)))% 8;if(cav[j+1],r=concat(r,i)));
return(r) /* See the function "isin" at A092875 */}

/* transform a sequence v by the rule r - Note: v could be replaced by a function, e.g. v[c] => prime(c) here */
seqruletrans(v,r)={my(c=1,L=List(),t=0); r=Vecrev(binary(r),8); for(i=1,v[#v], v[c]A093515=seqruletrans(primes(10^4),110) \\ M. F. Hasler, Mar 01 2008, updated Jan 07 2019

PARI

A121561_is_1(N,n=0)=vector(N,i, while(!isprime(n+=1)&&!isprime(n-1),);n) \\ M. F. Hasler, Mar 01 2008

PARI

is(n)=isprime(n)||isprime(n-1) \\ M. F. Hasler, Jan 07 2019

Python

from sympy import isprime def ok(n): return isprime(n) or isprime(n-1) print(list(filter(ok, range(140)))) # Michael S. Branicky, Aug 10 2021

Formula

{a(n)} = A000040 U (A000040 + 1), where A000040 are the primes. - M. F. Hasler, Jan 07 2019

a(1) = 2, a(n) = a(n-1) + 1 if a(n-1) is prime, a(n) is the next prime after a(n-1) otherwise. - Luca Armstrong, Aug 10 2021

Extensions

Name changed by Antti Karttunen, Jan 07 2019

A092874 Decimal expansion of the "binary" Liouville number.

Original entry on oeis.org

7, 6, 5, 6, 2, 5, 0, 5, 9, 6, 0, 4, 6, 4, 4, 7, 7, 5, 3, 9, 0, 6, 2, 5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 7, 5, 2, 3, 1, 6, 3, 8, 4, 5, 2, 6, 2, 6, 4, 0, 0, 5, 0, 9, 9, 9, 9, 1, 3, 8, 3, 8, 2, 2, 2, 3, 7, 2, 3, 3, 8, 0, 3, 9, 4, 5, 9, 5, 6, 3, 3, 4, 1, 3, 6

Offset: 0

Ferenc Adorjan (fadorjan(AT)freemail.hu)

The famous Liouville number is defined so that its n-th fractional decimal digit is 1 if and only if there exists k, such that k! = n.
The binary Liouville number is defined similarly, but as a binary number: its n-th fractional binary digit is 1 if and only if there exists k, such that k! = n.
According to the definitions introduced in A092855 and A051006, this number is "the binary mapping" of the sequence of factorials (A000142).
For the numerators of the partial sums of B(n) := Sum_{j=1..n} 1/j^(n!) see A145572. - Wolfdieter Lang, Apr 10 2024

			0.7656250596046447753906250000... = 1/2^1 + 1/2^2 + 1/2^6 + 1/2^24 + 1/2^120 + ...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 2.22, p. 172.

G. C. Greubel, Table of n, a(n) for n = 0..5000
Ferenc Adorján, Binary mapping of monotonic sequences and the Aronson function.
Burkard Polster, Liouville's number, the easiest transcendental and its clones, Mathologer video (2017).
Fedoua Sghiouer, Kacem Belhroukia, and Ali Kacha, Transcendence of some infinite series, arXiv preprint (2023). arXiv:2301.06495 [math.NT].
Index entries for transcendental numbers.

Cf. A012245, A092855, A051006, A145572.

RealDigits[Sum[1/2^(n!), {n, Infinity}], 10, 105][[1]] (* Alonso del Arte, Dec 03 2012 *)

{ mt(v)= /*Returns the binary mapping of v monotonic sequence as a real in (0,1)*/
local(a=0.0,p=1,l);l=matsize(v)[2];
for(i=1,l,a+=2^(-v[i])); return(a)}

suminf(n=2,2^-gamma(n)) \\ Charles R Greathouse IV, Jun 14 2020

Offset corrected by Franklin T. Adams-Watters, Dec 14 2017

A164292 Binary sequence identifying the twin primes (characteristic function of twin primes: 1 if n is a twin prime else 0).

Original entry on oeis.org

0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0

Offset: 1

Carlos Alves, Aug 12 2009

nonn

Similar to prime binary digit sequence A010051.
In decimal notation A164292=0.1646823906345389353962381...
See also A164293 (similar to prime decimal sequence A051006).
a(A001097(n))=1; a(A001359(n))=1; a(A006512(n))=1. - Reinhard Zumkeller, Mar 29 2010
Characteristic function of A001097. - Georg Fischer, Aug 04 2021

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index entries for characteristic functions [From _Reinhard Zumkeller_, Mar 29 2010]

Cf. A001097, A010051, A051006, A057427, A129950, A164293.

a164292 1 = 0
a164292 2 = 0
a164292 n = signum (a010051' n * (a010051' (n - 2) + a010051' (n + 2)))
-- Reinhard Zumkeller, Feb 03 2014

Table[(PrimePi[n] - PrimePi[n - 1]) * Ceiling[(PrimePi[n + 2] - PrimePi[n + 1] + PrimePi[n - 2] - PrimePi[n - 3])/2], {n, 100}] (* Wesley Ivan Hurt, Jan 31 2014 *)
Table[If[PrimeQ[n]&&AnyTrue[n+{2,-2},PrimeQ],1,0],{n,120}] (* Harvey P. Dale, Jan 04 2025 *)

a(n) = A057427(A010051(n)*(A010051(n-2)+A010051(n+2))), for n>2. - Reinhard Zumkeller, Mar 29 2010

a(n) = c(n) * ceiling(( c(n+2) + c(n-2) )/2), where c is the prime characteristic. - Wesley Ivan Hurt, Jan 31 2014

cp's OEIS Frontend

A051007 Continued fraction for prime constant A051006.

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A102878 Increasing partial quotients in the continued fraction expansion of the prime constant (A051006).

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A103518 Indices of increasing partial quotients PQ_i of the continued fraction for the prime constant (A051006).

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A092856 Duplicate of A051006.

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A010051 Characteristic function of primes: 1 if n is prime, else 0.

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Haskell

Magma

Magma

Maple

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PARI

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A034785 a(n) = 2^(n-th prime).

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A092855 Numbers k such that the k-th bit in the binary expansion of sqrt(2) - 1 is 1: sqrt(2) - 1 = Sum_{n>=1} 1/2^a(n).

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Mathematica

PARI

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A093515 Numbers k such that either k or k-1 is a prime.

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