A010051 -id:A010051 - cp's OEIS frontend

A051006 Prime constant: decimal value of (A010051 interpreted as a binary number).

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

Offset: 0

Eric W. Weisstein

From Ferenc Adorjan (fadorjan(AT)freemail.hu): (Start)
Decimal expansion of the representation of the sequence of primes by a single real in (0,1).
Any monotonic integer sequence can be represented by a real number in (0, 1) in such a way that in the binary representation of the real, the n-th digit of the fractional part is 1 if and only if n is in the sequence.
Examples of the inverse mapping are A092855 and A092857. (End)
Is the prime constant an EL number? See Chow's 1999 article. - Lorenzo Sauras Altuzarra, Oct 05 2020
The asymptotic density of numbers with a prime number of trailing 0's in their binary representation (A370596), or a prime number of trailing 1's. - Amiram Eldar, Feb 23 2024

			0.414682509851111660... (base 10) = .01101010001010001010001... (base 2).

Harry J. Smith, Table of n, a(n) for n = 0..20000
Ferenc Adorjan, Binary mapping of monotonic sequences and the Aronson function.
Timothy Y. Chow, What is a Closed-Form Number?, The American Mathematical Monthly, Vol. 106, No. 5. (May, 1999), pp. 440-448.
Igor Pak, Complexity problems in enumerative combinatorics, arXiv:1803.06636 [math.CO], 2018.
Michael Penn, What is the prime constant and is it irrational?, YouTube video, 2022.
Simon Plouffe, Primes coded in binary to 1000 digits.
Eric Weisstein's World of Mathematics, Prime Constant.

Cf. A000720, A010051, A034785, A051007, A132800, A092857, A092858, A092859, A092860, A092861, A092862, A092863, A092874, A119523, A370596.

Cf. A036680 (Egyptian fractions).

a := n -> ListTools:-Reverse(convert(floor(evalf[1000](sum(1/2^ithprime(k), k = 1 .. infinity)*10^(n+1))), base, 10))[n+1]: - Lorenzo Sauras Altuzarra, Oct 05 2020

RealDigits[ FromDigits[ {{Table[ If[ PrimeQ[n], 1, 0], {n, 370}]}, 0}, 2], 10, 111][[1]] (* Robert G. Wilson v, Jan 15 2005 *)
RealDigits[Sum[1/2^Prime[k], {k, 1000}], 10, 100][[1]] (* Alexander Adamchuk, Aug 22 2006 *)

{ 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)} \\ Ferenc Adorjan

{ default(realprecision, 20080); x=0; m=67000; for (n=1, m, if (isprime(n), a=1, a=0); x=2*x+a; ); x=10*x/2^m; for (n=0, 20000, d=floor(x); x=(x-d)*10; write("b051006.txt", n, " ", d)); } \\ Harry J. Smith, Jun 15 2009

suminf(n=1,.5^prime(n)) \\ Then: digits(%\.1^default(realprecision)) to get seq. of digits. N.B.: Functions sumpos() and sumnum() yield much less accurate results. - M. F. Hasler, Jul 04 2017

Prime constant C = Sum_{k>=1} 1/2^prime(k), where prime(k) is the k-th prime. - Alexander Adamchuk, Aug 22 2006
From Amiram Eldar, Aug 11 2020: (Start)
Equals Sum_{k>=1} A010051(k)/2^k.
Equals Sum_{k>=1} 1/A034785(k).
Equals (1/2) * A119523.
Equals Sum_{k>=1} pi(k)/2^(k+1), where pi(k) = A000720(k). (End)

A143519 Moebius transform of A010051, the characteristic function of the primes: a(n) = Sum_{d|n} mu(n/d)A010051(d); A054525 A010051.

Original entry on oeis.org

0, 1, 1, -1, 1, -2, 1, 0, -1, -2, 1, 1, 1, -2, -2, 0, 1, 1, 1, 1, -2, -2, 1, 0, -1, -2, 0, 1, 1, 3, 1, 0, -2, -2, -2, 0, 1, -2, -2, 0, 1, 3, 1, 1, 1, -2, 1, 0, -1, 1, -2, 1, 1, 0, -2, 0, -2, -2, 1, -1, 1, -2, 1, 0, -2, 3, 1, 1, -2, 3, 1, 0, 1, -2, 1, 1, -2, 3, 1, 0, 0, -2, 1, -1, -2, -2, -2, 0, 1

Offset: 1

Gary W. Adamson, Aug 22 2008

sign

A010051 = A051731 * A143519 (since A051731 = the inverse Mobius transform).

A000720(n) = Sum_{k=1..n} a(k) floor(n/k) where A000720(n) is the number of primes <= n. - Steven Foster Clark, May 25 2018

			a(4) = -1 since row 4 of triangle A043518 = (0, -1, 0, 0).
a(4) = -1 = (0, -1, 0, 1) dot (0, 1, 1, 0), where (0, -1, 0, 1) = row 4 of A054525 and A010051 = (0, 1, 1, 0, 1, 0, 1, 0, ...).

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

Cf. A008683, A010051, A143518, A054525, A137851.

Table[Sum[MoebiusMu[n/d] Boole[PrimeQ@ d], {d, Divisors@ n}], {n, 89}] (* Michael De Vlieger, Jul 19 2017 *)

A143519(n) = sumdiv(n,d,isprime(d)*moebius(n/d)); \\ (After Luschny's Sage-code) - Antti Karttunen, Jul 19 2017

def A143519(n) :
    D = filter(is_prime, divisors(n))
    return add(moebius(n/d) for d in D)
[A143519(n) for n in (1..89)]   # Peter Luschny, Feb 01 2012

Mobius transform of A010051, the characteristic function of the primes.
Row sums of triangle A143518.
a(n) = Sum_{d|n} A010051(d)*A008683(n/d). - Antti Karttunen, Jul 19 2017
a(n) = Sum_{a*b*c=n} omega(a)*mu(b)*mu(c). - Benedict W. J. Irwin, Mar 02 2022

More terms from R. J. Mathar, Jan 19 2009

A329350 a(n) = Product_{d|n} A276086(d)^A010051(n/d).

Original entry on oeis.org

1, 2, 2, 3, 2, 18, 2, 9, 6, 54, 2, 45, 2, 30, 108, 15, 2, 150, 2, 405, 60, 270, 2, 375, 18, 150, 30, 675, 2, 33750, 2, 225, 540, 1350, 180, 3125, 2, 750, 300, 5625, 2, 281250, 2, 10125, 4500, 6750, 2, 140625, 10, 56250, 2700, 16875, 2, 468750, 1620, 84375, 1500, 33750, 2, 65625, 2, 42, 22500, 21, 900, 236250, 2, 567, 13500, 425250, 2, 21875

Offset: 1

Antti Karttunen, Nov 12 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 1..8192

Cf. A010051, A069359, A276085, A276086, A329351 (rgs-transform).

Cf. also A329352, A329380.

A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A329350(n) = { my(m=1); fordiv(n,d,if(isprime(n/d), m *= A276086(d))); (m); };

a(n) = Product_{d|n} A276086(d)^A010051(n/d).

A276085(a(n)) = A069359(n).

A121497 Binomial transform of the characteristic function of the prime numbers (A010051).

Original entry on oeis.org

0, 0, 1, 4, 10, 21, 41, 78, 148, 282, 537, 1013, 1882, 3446, 6267, 11468, 21416, 41209, 81771, 166042, 340994, 700570, 1429375, 2886777, 5771828, 11453105, 22638215, 44742141, 88681674, 176545766, 352992931, 707922077, 1421120880, 2849433326

Offset: 0

T. D. Noe, Aug 03 2006

nonn

This is the binomial transform of the sequence {0,0,1,1,0,1,0,1,...}. Sequence A052467, the binomial transform of the sequence {0,1,1,0,1,0,1,...} is very similar. In fact, the first differences of this sequence yields A052467.
The number of pernicious numbers (A052294) less than 2^n. Although the graph looks almost like 2^n, the graph of a(n)/2^n has quite a bit of variation. - T. D. Noe, Mar 14 2009
a(n)/2^n is the probability that a series of Bernoulli trials with probability of success equal to 1/2 will result in a prime number of successes. Cf. A178851. - Eric M. Schmidt, Jul 13 2012
a(n) equals the number of subsets of [n] whose cardinalities are prime. - Ivan N. Ianakiev, Jul 14 2019
Upper and lower bounds are provided by Kim and Sinha (see links). - Jeffrey Shallit, Nov 14 2024

T. D. Noe and Charles R Greathouse IV, Table of n, a(n) for n = 0..3324 (terms up to 1000 from Noe)
Sungjin Kim and Nilotpal Kanti Sinha, Binomial probability of prime number of successes, INTEGERS 20 (2020), #A99.
Vaclav Kotesovec, Plot of a(n) / (2^n/log(n/2)) for n = 2..10000

Cf. A000720, A010051, A052294, A052467, A178851

Primes:= select(isprime, [2,seq(i,i=3..100,2)]):
G:= add((z/(1-z))^p/(1-z),p=Primes):
S:= series(G,z,101):
seq(coeff(S,z,i),i=0..100); # Robert Israel, Sep 27 2018

Table[Sum[Binomial[n,Prime[i]], {i,PrimePi[n]}], {n,40}]

a(n)=my(s);forprime(p=2,n,s+=binomial(n,p));s \\ Charles R Greathouse IV, Mar 22 2013

a(n) = Sum_{i=1..pi(n)} binomial(n,prime(i)), where pi(n) is A000720(n), the number of primes <= n.
E.g.f.: exp(x) * (x^2/2! + x^3/3! + x^5/5! + ...) - Eric M. Schmidt, Jul 14 2012
G.f.: Sum_{p prime} x^p/(1-x)^(p+1). - Robert Israel, Sep 27 2018

a(0) inserted by Franklin T. Adams-Watters, Jul 13 2012

A320000 Square array A(n, k) read by descending antidiagonals: A(1, 1) = 2, A(1, k) = 1 for k > 1, and for n > 1, A(n, k) = Sum_{d|n, d>=k} A010051(1+d)*[Sum_{i=0..valuation(n,1+d)} A((n/d)/((1+d)^i), 1+d)].

Original entry on oeis.org

2, 1, 3, 1, 1, 0, 1, 0, 0, 4, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 4, 1, 0, 0, 1, 0, 2, 0, 1, 0, 0, 0, 0, 1, 0, 5, 1, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 2, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 6, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0

Offset: 1

Antti Karttunen, Dec 03 2018

This square array gives the values obtained from the recursive PARI-program that M. F. Hasler has provided Oct 05 2009 for A014197, in its two-argument form.

			Array begins as:
n  | k=1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13,14,15,16, ...
---+------------------------------------------------
1  |   2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
2  |   3, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...
3  |   0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...
4  |   4, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...
5  |   0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...
6  |   4, 2, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...
7  |   0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...
8  |   5, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...
9  |   0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...
10 |   2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, ...
11 |   0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...
12 |   6, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, ...
13 |   0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...
14 |   0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...
15 |   0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...
16 |   6, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...

Antti Karttunen, Table of n, a(n) for n = 1..10585; the first 145 antidiagonals of the array

Cf. A014197 (column 1).

Cf. A000010, A322310.

up_to = 120;
A320000sq(n, k) = if(1==n, if(1==k,2,1), sumdiv(n, d, if(d>=k && isprime(d+1), my(p=d+1, q=n/d); sum(i=0, valuation(n, p), A320000sq(q/(p^i), p))))); \\ After M. F. Hasler's code in A014197
A320000list(up_to) = { my(v = vector(up_to), i=0); for(a=1,oo, for(col=1,a, i++; if(i > up_to, return(v)); v[i] = A320000sq(col,(a-(col-1))))); (v); };
v320000 = A320000list(up_to);
A320000(n) = v320000[n];

A329352 a(n) = Product_{d|n} A019565(d)^A010051(n/d).

Original entry on oeis.org

1, 2, 2, 3, 2, 18, 2, 5, 6, 30, 2, 75, 2, 90, 60, 7, 2, 210, 2, 105, 180, 126, 2, 245, 10, 210, 14, 525, 2, 66150, 2, 11, 252, 66, 300, 1155, 2, 198, 420, 385, 2, 173250, 2, 825, 2940, 990, 2, 847, 30, 3234, 132, 1155, 2, 15246, 420, 2695, 396, 2310, 2, 2223375, 2, 6930, 1540, 13, 700, 64350, 2, 195, 1980, 171990, 2, 5005, 2, 390, 32340, 975, 1260

Offset: 1

Antti Karttunen, Nov 12 2019

nonn

			The divisors of 30 are [1, 2, 3, 5, 6, 10, 15, 30], of which only d = 6, 10 and 15 are such that 30/d is a prime, thus a(n) = A019565(6) * A019565(10) * A019565(15) = 15 * 21 * 210 = 66150.

Antti Karttunen, Table of n, a(n) for n = 1..8192

Cf. A010051, A019565, A048675, A069359, A329353 (rgs-transform).
Cf. also A329350.
Differs from A300832 for the first time at n=30, where a(30) = 66150, while A300832(30) = 132300.

A019565(n) = {my(j,v); factorback(Mat(vector(if(n, #n=vecextract(binary(n), "-1..1")), j, [prime(j), n[j]])~))}; \\ From A019565
A329352(n) = { my(m=1); fordiv(n,d,if(isprime(n/d), m *= A019565(d))); (m); };

a(n) = Product_{d|n} A019565(d)^A010051(n/d).

For all n, A048675(a(n)) = A069359(n).

A292596 a(1) = a(2) = 0; for n > 2, a(n) = A010051(n) + 2*a(floor(n/2)).

Original entry on oeis.org

0, 0, 1, 0, 1, 2, 3, 0, 0, 2, 3, 4, 5, 6, 6, 0, 1, 0, 1, 4, 4, 6, 7, 8, 8, 10, 10, 12, 13, 12, 13, 0, 0, 2, 2, 0, 1, 2, 2, 8, 9, 8, 9, 12, 12, 14, 15, 16, 16, 16, 16, 20, 21, 20, 20, 24, 24, 26, 27, 24, 25, 26, 26, 0, 0, 0, 1, 4, 4, 4, 5, 0, 1, 2, 2, 4, 4, 4, 5, 16, 16, 18, 19, 16, 16, 18, 18, 24, 25, 24, 24, 28, 28, 30, 30, 32, 33, 32, 32, 32, 33

Offset: 1

Antti Karttunen, Sep 27 2017

nonn

1-bits in base-2 expansion of a(n) indicate the positions of odd primes in the sequence [n, floor(n/2), floor(n/4), ..., 1].

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences related to binary expansion of n

Cf. A010051, A292597, A292598.

Cf. also A292599 (variant for all primes).

a(1) = a(2) = 0; for n > 2, a(n) = A010051(n) + 2*a(floor(n/2)).
Other identities. For all n >= 1:
a(n) + A292597(n) = n.
A000120(a(n)) = A292598(n).
A007814(1+a(n)) <= A007814(1+n).

A292599 a(1) = 0; for n > 1, a(n) = A010051(n) + 2*a(floor(n/2)).

Original entry on oeis.org

0, 1, 1, 2, 3, 2, 3, 4, 4, 6, 7, 4, 5, 6, 6, 8, 9, 8, 9, 12, 12, 14, 15, 8, 8, 10, 10, 12, 13, 12, 13, 16, 16, 18, 18, 16, 17, 18, 18, 24, 25, 24, 25, 28, 28, 30, 31, 16, 16, 16, 16, 20, 21, 20, 20, 24, 24, 26, 27, 24, 25, 26, 26, 32, 32, 32, 33, 36, 36, 36, 37, 32, 33, 34, 34, 36, 36, 36, 37, 48, 48, 50, 51, 48, 48, 50, 50, 56, 57, 56, 56, 60, 60, 62, 62, 32

Offset: 1

Antti Karttunen, Sep 27 2017

nonn

1-bits in base-2 expansion of a(n) indicate the positions of primes in the sequence [n, floor(n/2), floor(n/4), ..., 1].

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences related to binary expansion of n

Cf. A010051, A078349, A292258, A292259, A292936.

Cf. also A292596 (variant for odd primes).

A292599 := proc(n)
    option remember;
    if n = 1 then
        0 ;
    else
        A010051(n) + 2*procname(floor(n/2)) ;
    end if;
end proc:
seq(A292599(n),n=1..100) ; # R. J. Mathar, Sep 28 2017

a[1] = 0; a[n_] := a[n] = Boole[PrimeQ[n]] + 2*a[Floor[n/2]]; Array[a, 96] (* Jean-François Alcover, Sep 29 2017 *)

a(1) = 0; for n > 1, a(n) = A010051(n) + 2*a(floor(n/2)).
Other identities. For all n >= 1:
A000120(a(n)) = A078349(n).
A007814(1+a(n)) = A292936(n).

A317939 Numerators of sequence whose Dirichlet convolution with itself yields A080339 = A010051 (characteristic function of primes) + A063524 (1, 0, 0, 0, ...).

Original entry on oeis.org

1, 1, 1, -1, 1, -1, 1, 1, -1, -1, 1, 3, 1, -1, -1, -5, 1, 3, 1, 3, -1, -1, 1, -5, -1, -1, 1, 3, 1, 3, 1, 7, -1, -1, -1, -15, 1, -1, -1, -5, 1, 3, 1, 3, 3, -1, 1, 35, -1, 3, -1, 3, 1, -5, -1, -5, -1, -1, 1, -15, 1, -1, 3, -21, -1, 3, 1, 3, -1, 3, 1, 35, 1, -1, 3, 3, -1, 3, 1, 35, -5, -1, 1, -15, -1, -1, -1, -5, 1, -15, -1, 3, -1, -1, -1, -63, 1, 3, 3

Offset: 1

Antti Karttunen, Aug 14 2018

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A010051, A080339, A046644 (denominators).

Cf. also A257098, A317830, A317936, A317937.

up_to = 65537;
DirSqrt(v) = {my(n=#v, u=vector(n)); u[1]=1; for(n=2, n, u[n]=(v[n]/v[1] - sumdiv(n, d, if(d>1&&dA317937.
v317939aux = DirSqrt(vector(up_to, n, if(1==n,1,isprime(n))));
A317939(n) = numerator(v317939aux[n]);

a(n) = numerator of f(n), where f(1) = 1, f(n) = (1/2) * (A010051(n) - Sum_{d|n, d>1, d 1.

A280417 Number of distinct length-n blocks (a.k.a. subword complexity) of the characteristic sequence of the prime numbers A010051.

Original entry on oeis.org

1, 2, 4, 7, 9, 13, 16, 22, 28, 38, 48, 62, 76, 104, 132, 174, 216, 273, 330, 435, 540, 700, 860

Offset: 0

Jeffrey Shallit, Jan 02 2017

Unlike A023192, this sequence also counts blocks that occur finitely often in A010051. And unlike A023192, the correctness of the numbers provided here depend on no conjecture.

			For n = 4, the 9 blocks (in the order they occur in A010051) are 0110,1101,1010,0101,0100,1000,0001,0010,0000.

Cf. A010051, A023192. A280418 gives the length of shortest prefix needed to get all blocks that occur.

cp's OEIS Frontend

A051006 Prime constant: decimal value of (A010051 interpreted as a binary number).

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A143519 Moebius transform of A010051, the characteristic function of the primes: a(n) = Sum_{d|n} mu(n/d)*A010051(d); A054525 * A010051.

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A329350 a(n) = Product_{d|n} A276086(d)^A010051(n/d).

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A121497 Binomial transform of the characteristic function of the prime numbers (A010051).

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A320000 Square array A(n, k) read by descending antidiagonals: A(1, 1) = 2, A(1, k) = 1 for k > 1, and for n > 1, A(n, k) = Sum_{d|n, d>=k} A010051(1+d)*[Sum_{i=0..valuation(n,1+d)} A((n/d)/((1+d)^i), 1+d)].

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A329352 a(n) = Product_{d|n} A019565(d)^A010051(n/d).

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A292596 a(1) = a(2) = 0; for n > 2, a(n) = A010051(n) + 2*a(floor(n/2)).

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A143519 Moebius transform of A010051, the characteristic function of the primes: a(n) = Sum_{d|n} mu(n/d)A010051(d); A054525 A010051.