A069754 -id:A069754 - cp's OEIS frontend

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

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

A035026 Number of times that i and 2n-i are both prime, for i = 1, ..., 2n-1.

Original entry on oeis.org

0, 1, 1, 2, 3, 2, 3, 4, 4, 4, 5, 6, 5, 4, 6, 4, 7, 8, 3, 6, 8, 6, 7, 10, 8, 6, 10, 6, 7, 12, 5, 10, 12, 4, 10, 12, 9, 10, 14, 8, 9, 16, 9, 8, 18, 8, 9, 14, 6, 12, 16, 10, 11, 16, 12, 14, 20, 12, 11, 24, 7, 10, 20, 6, 14, 18, 11, 10, 16, 14, 15, 22, 11, 10, 24, 8, 16, 22, 9, 16, 20, 10

Offset: 1

Gordon R. Bower (siegmund(AT)mosquitonet.com)

a(n) is the convolution of terms 1 to 2n of the characteristic function of the primes, A010051, with itself. Related to Goldbach's conjecture that every even number can be expressed as the sum of two primes. - T. D. Noe, Aug 01 2002
The following sequences all appear to have the same parity (with an extra zero term at the start of A010051): A010051, A061007, A035026, A069754, A071574. - Jeremy Gardiner, Aug 09 2002
Total number of printer jobs in all possible schedules for n time slots in the first-come-first-served (FCFS) policy.
a(n) = Sum_{p prime < 2*n} A010051(2*n - p). - Reinhard Zumkeller, Oct 19 2011
For n > 1: length of n-th row of triangle A171637. - Reinhard Zumkeller, Mar 03 2014
a(n) = A001221(A238711(n)) = A238778(n) / n. - Reinhard Zumkeller, Mar 06 2014
From Robert G. Wilson v, Dec 15 2016: (Start)
First occurrence of k: 1, 2, 4, 5, 8, 11, 12, 17, 18, 37, 24, 53, 30, 89, 39, 71, 42, 101, 45, 179, 57, 137, 72, 193, 60, 233, ..., .
Conjectured last occurrence of k: 1, 3, 6, 19, 34, 31, 64, 61, 76, 79, 94, 83, 166, 199, 136, 181, 184, 229, 244, 271, 316, 277, 346, 313, 301, 293, ..., .
Conjectured number occurrences of k: 1, 2, 2, 3, 6, 3, 8, 4, 7, 5, 11, 5, 11, 8, 10, 3, 17, 7, 16, 3, 13, 8, 21, 4, 12, 3, 22, 7, 20, 8, 15, ..., .
Records: 0, 1, 2, 3, 4, 5, 6, 7, 8, 10, 12, 14, 16, 18, 20, 24, 26, 28, 38, 42, 48, 54, 60, 64, 82, 88, 102, 104, 114, 116, 136, 146, 152, 166, 182, ..., .
(End)

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Rémy Sigrist, Colored scatterplot of the first 500000 terms
Index entries for sequences related to Goldbach conjecture

Cf. A010051. Essentially the same as A002372.

Cf. A073610.

a035026 n = sum $ map (a010051 . (2 * n -)) $
   takeWhile (< 2 * n) a000040_list
-- Reinhard Zumkeller, Oct 19 2011

A035026 := proc(n)
    local a,i ;
    a := 0 ;
    for i from 1 to 2*n-1 do
        if isprime(i) and isprime(2*n-i) then
            a := a+1 ;
        end if;
    end do:
    a ;
end proc: # R. J. Mathar, Jul 01 2013

For[lst={}; n=1, n<=100, n++, For[cnt=0; i=1, i<=2n-1, i++ If[PrimeQ[i]&&PrimeQ[2n-i], cnt++ ]]; AppendTo[lst, cnt]]; lst
f[n_] := Block[{c = Boole@ PrimeQ[ n/2], p = 2}, While[ 2p < n, If[ PrimeQ[n - p], c += 2]; p = NextPrime@ p]; c];; Array[ f[ 2#] &, 90] (* Robert G. Wilson v, Dec 15 2016 *)

For n > 1, a(n) = 2*A045917(n) - A010051(n).
a(n) = A010051(n) + 2*A061357(n). - Wesley Ivan Hurt, Aug 21 2013
a(n) = A073610(2*n). - Ridouane Oudra, Sep 06 2023

Corrected by T. D. Noe, May 05 2002

A071574 If n = k-th prime, a(n) = 2a(k) + 1; if n = k-th nonprime, a(n) = 2a(k).

Original entry on oeis.org

0, 1, 3, 2, 7, 6, 5, 4, 14, 12, 15, 10, 13, 8, 28, 24, 11, 30, 9, 20, 26, 16, 29, 56, 48, 22, 60, 18, 25, 40, 31, 52, 32, 58, 112, 96, 21, 44, 120, 36, 27, 50, 17, 80, 62, 104, 57, 64, 116, 224, 192, 42, 49, 88, 240, 72, 54, 100, 23, 34, 61, 160, 124, 208, 114, 128, 19

Offset: 1

Christopher Eltschka (celtschk(AT)web.de), May 31 2002

The recursion start is implicit in the rule, since the rule demands that a(1)=2*a(1). All other terms are defined through terms for smaller indices until a(1) is reached.
a(n) is a bijective mapping from the positive integers to the nonnegative integers. Given the value of a(n), you can get back to n using the following algorithm:
Start with an initial value of k=1 and write a(n) in binary representation. Then for each bit, starting with the most significant one, do the following: - if the bit is 1, replace k by the k-th prime - if the bit is 0, replace k by the k-th nonprime. After you processed the last (i.e. least significant) bit of a(n), you've got n=k.
Example: From a(n) = 12 = 1100_2, you get 1->2->3=>6=>10; a(10)=12. Here each "->" is a step due to binary digit 1; each "=>" is a step due to binary digit 0.
The following sequences all appear to have the same parity (with an extra zero term at the start of A010051): A010051, A061007, A035026, A069754, A071574. - Jeremy Gardiner, Aug 09 2002. (At least with this sequence the identity a(n) = A010051(n) mod 2 is obvious, because each prime is mapped to an odd number and each composite to an even number. - Antti Karttunen, Apr 04 2015)
For n > 1: a(n) = 2 * a(if i > 0 then i else A066246(n) + 1) + A057427(i) with i = A049084(n). - Reinhard Zumkeller, Feb 12 2014
A237739(a(n)) = n; a(A237739(n)) = n. - Reinhard Zumkeller, Apr 30 2014

			1 is the 1st nonprime, so a(1) = 2*a(1), therefore a(1) = 0.
2 is the 1st prime, so a(2) = 2*a(1)+1 = 2*0+1 = 1.
4 is the 2nd nonprime, so a(4) = 2*a(2) = 2*1 = 2.

Inverse: A237739.
Cf. A000720 (pi), A049084, A065855, A066246.
Compare also to the permutation A246377.
Same parity: A010051, A061007, A035026, A069754.

a071574 1 = 0
a071574 n = 2 * a071574 (if j > 0 then j + 1 else a049084 n) + 1 - signum j
            where j = a066246 n
-- Reinhard Zumkeller, Feb 12 2014

a[1]=0; a[n_]:=If[PrimeQ[n],2*a[PrimePi[n]]+1,2*a[n-PrimePi[n]]];Table[a[n],{n,100}]

first(n) = my(res = vector(n), p); for(x=2, n, p=isprime(x); res[x]=2*res[x*!p-(-1)^p*primepi(x)]+p); res \\ Iain Fox, Oct 19 2018

;; With memoizing definec-macro.
(definec (A071574 n) (cond ((= 1 n) 0) ((= 1 (A010051 n)) (+ 1 (* 2 (A071574 (A000720 n))))) (else (* 2 (A071574 (+ 1 (A065855 n)))))))
;; Antti Karttunen, Apr 04 2015

a(1) = 0, and for n > 1, if A010051(n) = 1 [when n is a prime], a(n) = 1 + 2*a(A000720(n)), otherwise a(n) = 2*a(1 + A065855(n)). - Antti Karttunen, Apr 04 2015

Mathematica program completed by Harvey P. Dale, Nov 28 2024

A061007 a(n) = -(n-1)! mod n.

Original entry on oeis.org

0, 1, 1, 2, 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, 0

Offset: 1

Henry Bottomley, Apr 12 2001

The following sequences all appear to have the same parity (with an extra zero term at the start of A010051): A010051, A061007, A035026, A069754, A071574. - Jeremy Gardiner, Aug 09 2002

In particular, this is identical to the isprime function A010051 except for a(4) = 2 instead of 0. This is equivalent to Wilson's theorem, (n-1)! == -1 (mod n) iff n is prime. If n = p*q with p, q > 1, then p, q < n-1 and (n-1)! will contain the two factors p and q, unless p = q = 2 (if p = q > 2 then also 2p < n-1, so there are indeed two factors p in (n-1)!), whence (n-1)! == 0 (mod n). - M. F. Hasler, Jul 19 2024

			a(4) = 2 since -(4 - 1)! = -6 = 2 mod 4.
a(5) = 1 since -(5 - 1)! = -24 = 1 mod 5.
a(6) = 0 since -(6 - 1)! = -120 = 0 mod 6.

Antti Karttunen, Table of n, a(n) for n = 1..10000

Positive for all but the first term of A046022.

Cf. A000040 (the primes), A000142, A010051 (isprime function), A055976, A061006, A061008, A061009.

Table[Mod[-(n - 1)!, n], {n, 100}] (* Alonso del Arte, Mar 20 2014 *)

A061007(n) = ((-((n-1)!))%n); \\ Antti Karttunen, Aug 27 2017

apply( {A061007(n) = !(n-1)!%n}, [0..99]) \\ M. F. Hasler, Jul 19 2024

from sympy import isprime
def A061007(n): return 2 if n == 4 else int(isprime(n)) # Chai Wah Wu, Mar 22 2023

a(4) = 2, a(p) = 1 for p prime, a(n) = 0 otherwise. Apart from n = 4, a(n) = A010051(n) = A061006(n)/(n-1).

A211005 Pair (i, j) where i = number of adjacent nonprimes and j = number of adjacent primes.

Original entry on oeis.org

1, 2, 1, 1, 1, 1, 3, 1, 1, 1, 3, 1, 1, 1, 3, 1, 5, 1, 1, 1, 5, 1, 3, 1, 1, 1, 3, 1, 5, 1, 5, 1, 1, 1, 5, 1, 3, 1, 1, 1, 5, 1, 3, 1, 5, 1, 7, 1, 3, 1, 1, 1, 3, 1, 1, 1, 3, 1, 13, 1, 3, 1, 5, 1, 1, 1, 9, 1, 1, 1, 5, 1, 5, 1, 3, 1, 5, 1, 5, 1, 1, 1, 9, 1, 1, 1

Offset: 1

Omar E. Pol, Aug 11 2012

nonn

Also number of consecutive occurrences of n-1 in A069754. - Reinhard Zumkeller, Dec 04 2012

Run lengths of A010051. - Paolo Xausa, Jan 17 2023

			----------------------------------------------------------
.     Array from              Number of   Number of
n      A000027                nonprimes    primes    a(n)
----------------------------------------------------------
1         1;                      1          0        1
2         2, 3;                   0          2        2
3         4;                      1          0        1
4         5;                      0          1        1
5         6;                      1          0        1
6         7;                      0          1        1
7         8, 9, 10;               3          0        3
8        11;                      0          1        1
9        12;                      1          0        1
10       13;                      0          1        1
11       14, 15, 16;              3          0        3
12       17;                      0          1        1
13       18;                      1          0        1
14       19;                      0          1        1
15       20, 21, 22;              3          0        3
16       23;                      0          1        1
17       24, 25, 26, 27, 28;      5          0        5
18       29;                      0          1        1
19       30;                      1          0        1
20       31;                      0          1        1

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

1 together with A162154.

Cf. A000040, A001223, A010051, A018252, A046933, A211006, A211007.

import Data.List (group)
a211005 n = a211005_list !! (n-1)
a211005_list = map length $ group a069754_list
-- Reinhard Zumkeller, Dec 04 2012

A211005[upto_]:=Map[Length, Most[Split[PrimeQ[Range[upto]]]]];
A211005[500] (* Paolo Xausa, Jan 17 2023 *)

a(n) = A162154(n-1), n >= 2.

cp's OEIS Frontend

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

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A035026 Number of times that i and 2n-i are both prime, for i = 1, ..., 2n-1.

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A071574 If n = k-th prime, a(n) = 2*a(k) + 1; if n = k-th nonprime, a(n) = 2*a(k).

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A061007 a(n) = -(n-1)! mod n.

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A211005 Pair (i, j) where i = number of adjacent nonprimes and j = number of adjacent primes.

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A071574 If n = k-th prime, a(n) = 2a(k) + 1; if n = k-th nonprime, a(n) = 2a(k).