A051006 -id:A051006 - cp's OEIS frontend

A132817 Decimal expansion of Sum_{n >= 1} 1/6^prime(n).

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

Offset: 0

Cino Hilliard, Nov 17 2007

Equivalently, the real number in (0,1) having the characteristic function of the primes, A010051, as its base-6 expansion. - M. F. Hasler, Jul 05 2017

			0.032539583308525544049260050781274181192986076617578098887664610099...

Vincenzo Librandi, Table of n, a(n) for n = 0..1110

Cf. A000720, A051006 (analog for base 2), A132800 (analog for base 3), A132806 (analog for base 4), A132797 (analog for base 5), A132822 (analog for base 7), A010051 (characteristic function of the primes), A000040 (the primes).

Join[{0}, RealDigits[FromDigits[{{Table[If[PrimeQ[n], 1, 0], {n, 370}]}, 0}, 6], 10, 111][[1]]] (* Vincenzo Librandi, Jul 05 2017 *)

/* Sum of 1/m^p for primes p */ sumnp(n,m) = { local(s=0,a,j); for(x=1,n, s+=1./m^prime(x); ); a=Vec(Str(s)); for(j=3,n, print1(eval(a[j])",") ) }

suminf(n=1, 1/6^prime(n)) \\ Then: digits(%\.1^default(realprecision))[1..-3] to remove the last 2 digits. N.B.: Functions sumpos() and sumnum() yield much less accurate results. - M. F. Hasler, Jul 04 2017

Equals 5 * Sum_{k>=1} pi(k)/6^(k+1), where pi(k) = A000720(k). - Amiram Eldar, Aug 11 2020

Offset corrected R. J. Mathar, Jan 26 2009

Edited by M. F. Hasler, Jul 05 2017

A132822 Decimal expansion of Sum_{n >= 1} 1/7^prime(n).

Original entry on oeis.org

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

Offset: 0

Cino Hilliard, Nov 17 2007

Equivalently, the real number in (0,1) having the characteristic function of the primes, A010051, as its base-7 expansion. - M. F. Hasler, Jul 05 2017

			0.023384328960353739909859822495912373489340935935944869619982884656523568...

Cf. A000720, A051006 (analog for base 2), A132800 (analog for base 3), A132806 (analog for base 4), A132797 (analog for base 5), A132817 (analog for base 6), A010051 (characteristic function of the primes), A132799 (base 8), A269327.

/* Sum of 1/m^p for primes p */ sumnp(n,m) = { local(s=0,a,j); for(x=1,n, s+=1./m^prime(x); ); a=Vec(Str(s)); for(j=3,n, print1(eval(a[j])",") ) }

suminf(n=1, 1/7^prime(n)) \\ Then: digits(%\.1^default(realprecision))[1..-3] to remove the last 2 digits. N.B.: Functions sumpos() and sumnum() yield much less accurate results. - M. F. Hasler, Jul 05 2017

From Amiram Eldar, Aug 11 2020: (Start)
Equals Sum_{k>=1} 1/A269327(k).
Equals 6 * Sum_{k>=1} pi(k)/7^(k+1), where pi(k) = A000720(k). (End)

Offset corrected by R. J. Mathar, Feb 05 2009

Edited by M. F. Hasler, Jul 05 2017

A121240 Numerator of sum_{k=1..n} 1/2^prime(k).

Original entry on oeis.org

1, 3, 13, 53, 849, 3397, 54353, 217413, 3478609, 222630977, 890523909, 56993530177, 911896482833, 3647585931333, 58361374901329, 3735127993685057, 239048191595843649, 956192766383374597, 61196337048535974209

Offset: 1

Alexander Adamchuk, Aug 22 2006

a(n) is prime for n = {2, 3, 4, 10, 21, 321,..} where it takes the values {3, 13, 53, 222630977, ...}.

The prime constant A051006 = 0.414682509.. is limit(n->infinity) a(n)/2^prime(n) .

Eric Weisstein's World of Mathematics, Prime Constant.

Cf. A034785 (denominators), A072762, A051006, A010051.

Table[Numerator[Sum[1/2^Prime[k],{k,1,n}]],{n,1,30}]
Accumulate[1/2^Prime[Range[30]]]//Numerator (* Harvey P. Dale, Aug 11 2021 *)

a(n) = Numerator[ Sum[ 1/2^Prime[k], {k,1,n} ] ]. a(n) = A072762[ Prime[n] ].

A275306 Decimal expansion of 1/2 - Sum_{k>=1} 1/2^prime(k).

Original entry on oeis.org

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

Offset: 0

Ilya Gutkovskiy, Jul 22 2016

Composite constant: decimal value of A066247 interpreted as a binary number.
The characteristic function of composite numbers (A066247) has values 0, 0, 0, 1, 0, 1, 0, 1, 1, ... for n = 1, 2, 3, ... The constant obtained by concatenating these digits and interpreting them as a binary fraction is therefore C = 0.0001010111010... (base 2) = 0.0853174901...(base 10).
Continued fraction [0; 11, 1, 2, 1, 1, 2, 1, 1, 131, 2, 1, 1, 2, 6, 4, 2, 21, ...].

			0.0853174901... = (0.00010101110...)_2.
                        | | |||
                        4 6 8910

Simon Plouffe, Primes coded in binary to 1000 digits.
Eric Weisstein's World of Mathematics, Composite Number.
Eric Weisstein's World of Mathematics, Prime Constant.

Cf. A002808, A051006, A005171, A062298, A066247, A073718.

nn = 121; Take[#, nn] &@ PadLeft[First@ #, Abs@ Last@ # + Length@ First@ #] &@ RealDigits@ N[1/2 - Sum[ 1/2^Prime[k], {k, 10^4}], nn + 2] (* Michael De Vlieger, Jul 22 2016 *)

s=.5; forprime(p=2,bitprecision(s)+2, s-=1.>>p); s \\ Charles R Greathouse IV, Jul 22 2016

Equals Sum_{k>=1} 1/2^A002808(k).
From Amiram Eldar, Aug 11 2020: (Start)
Equals Sum_{k>=1} 1/A073718(k).
Equals Sum_{k>=1} A066247(k)/2^k.
Equals -(1/2) + Sum_{k>=1} A062298(k)/2^(k+1). (End)
Equals Sum_{k >= 1} ((-1)^A010051(k))/2^(k+1). - Antonio Graciá Llorente, Jan 13 2024

A333182 Decimal expansion of Sum_{k>=1} mu(k)^2 / 2^k.

Original entry on oeis.org

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

Offset: 0

Ilya Gutkovskiy, Mar 10 2020

Squarefree constant: binary expansion is the characteristic function of squarefree numbers (A008966).

			0.93137635638313358... = (0.1110111001...)_2.
                            ||| |||  |
                            123 567 10

Cf. A005117, A008683, A008966, A051006, A332905.

RealDigits[Sum[MoebiusMu[n]^2/2^n, {n, 1, 500}], 10, 100][[1]] (* Amiram Eldar, Jun 22 2020 *)

Equals Sum_{k>=1} mu(k)/(2^(k^2) - 1). - Amiram Eldar, Jun 22 2020

A370596 Numbers k such that A007814(k) is a prime number.

Original entry on oeis.org

4, 8, 12, 20, 24, 28, 32, 36, 40, 44, 52, 56, 60, 68, 72, 76, 84, 88, 92, 96, 100, 104, 108, 116, 120, 124, 128, 132, 136, 140, 148, 152, 156, 160, 164, 168, 172, 180, 184, 188, 196, 200, 204, 212, 216, 220, 224, 228, 232, 236, 244, 248, 252, 260, 264, 268, 276

Offset: 1

Amiram Eldar, Feb 23 2024

Numbers whose binary representation has a prime number of trailing 0's.
a(n)-1 is the sequence of numbers whose binary representation has a prime number of trailing 1's.
Numbers of the form (2^(p+1))*k + 2^p = 2^p * (2*k + 1), where p is prime and k >= 0.
All the terms are divisible by 4.
The asymptotic density of this sequence is Sum_{p prime} 1/2^(p+1) = 0.20734125492555583012... = A051006 / 2.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A007814, A051006, A359794.

Subsequences: A017113, A051062.

Select[Range[300], PrimeQ[IntegerExponent[#, 2]] &]

PARI
```
is(n) = isprime(valuation(n, 2));
```

A036680 Expansion of C in Egyptian fractions, where C contains the primes in binary.

Original entry on oeis.org

3, 13, 226, 757098, 1493980747140, 3358884634272343840743139, 31207490927201886805011133752520957788381952996897, 1532767887228913328212783830676536430748824655051820601315140788525355884052980975649269521964006675

Offset: 0

Simon Plouffe

nonn

			C in binary = 0.011010100010100010100010000010100... = 1/3 + 1/13 + 1/226 + ...

Index entries for sequences related to Egyptian fractions

Cf. A010051 (binary), A051006 (decimal), A051007 (continued fraction).

A102913 Take characteristic function of the semiprimes A001358, interpret it as a binary fraction and convert to a decimal fraction.

Original entry on oeis.org

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

Offset: 0

Jonathan Vos Post, Jan 17 2005

Eric Weisstein's World of Mathematics, Prime Constant.
Eric Weisstein's World of Mathematics, Semiprime.
Eric Weisstein et al., Characteristic Function.

For the continued fraction form of the semiprime constant, see A102914. For the equivalent characteristic function for primes, see A010051; interpreted as a binary fraction see A051006; for the continued fraction form of that see A051007.

Cf. A001358, A010051, A051006, A064911, A102914.

Semiprime[n_] := If[Plus @@ Last[ Transpose[ FactorInteger[n]]] == 2, 1, 0]; RealDigits[ FromDigits[{Table[ Semiprime[n], {n, 2, 350}], -2}, 2], 10, 111][[1]] (* Ed Pegg Jr *)

The characteristic function of the semiprimes is the function f(n) = 1 iff n is semiprime, 0 otherwise. This begins, for n = 0, 1, 2, 3, ... f(n) = 0, 0, 0, 0, 1, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1... If we concatenate these bits and interpret them as the binary fraction 0.0000101001100011000001... (base 2) we have, expressed as a decimal fraction, 0.0405735002013980686743112664235357506936275821940023586083340694633362...

The characteristic function of A001358 is A064911 (for n >= 1, starting with 0, 0, 0, 1 ...). The binary constant here has an additional 0 after the binary point. - Georg Fischer, Aug 04 2021

More terms from Robert G. Wilson v, Jan 24 2005

A102914 Continued fraction expansion of the number in A102913.

Original entry on oeis.org

0, 24, 1, 1, 1, 4, 1, 7, 4, 2, 8, 35, 1, 6, 3, 3, 3, 4, 1, 4, 3, 4, 1, 1, 1, 1, 8, 1, 1, 1, 3, 4, 1, 3, 1, 1, 1, 1, 9, 7, 4, 1, 24, 1, 5, 6, 1, 3, 1, 1, 10, 1, 237, 1, 12, 2, 1, 12, 1, 21, 1, 1, 1, 14, 1, 4, 14, 72, 29, 5, 1, 1, 2, 1, 8, 2, 1, 43, 1, 1, 8, 1, 1, 1, 2, 8, 2, 1, 12, 3, 2, 1, 10, 1, 5, 6, 1

Offset: 0

Jonathan Vos Post, Jan 17 2005

Cf. A001358, A010051, A051006, A102913.

SemiprimeQ[n_] := If[Plus @@ Last[ Transpose[ FactorInteger[n]]] == 2, 1, 0]; a = FromDigits[{Table[ SemiprimeQ[n], {n, 2, 10^3}], -2}, 2]; ContinuedFraction[ a, 100] (* Robert G. Wilson v, Jan 24 2005 *)

More terms from Robert G. Wilson v, Jan 24 2005

A118092 Odd primes raised to odd prime powers.

Original entry on oeis.org

27, 125, 243, 343, 1331, 2187, 2197, 3125, 4913, 6859, 12167, 16807, 24389, 29791, 50653, 68921, 78125, 79507, 103823, 148877, 161051, 177147, 205379, 226981, 300763, 357911, 371293, 389017, 493039, 571787, 704969, 823543, 912673, 1030301

Offset: 1

Jonathan Vos Post, May 11 2006

Subset of A053810 Prime powers of prime numbers. Subset of A000961 Prime powers. Subsets include A030078 Cubes of primes, A050997 Fifth powers of primes.

Cf. A000040, A000961, A030078, A050997, A051006, A053810, A065091.

With[{prs=Prime[Range[2,30]]},Take[Union[First[#]^Last[#]&/@ Tuples[prs,2]],40]] (* Harvey P. Dale, Dec 23 2011 *)

from sympy import primepi, integer_nthroot, primerange
def A118092(n):
    def f(x): return int(n+x-sum(primepi(integer_nthroot(x, p)[0])-1 for p in primerange(3,x.bit_length())))
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    return bisection(f,n,n) # Chai Wah Wu, Sep 12 2024

{p^q where p is in A065091 and q is in A065091}.

Sum_{n>=1} 1/a(n) = Sum_{p odd prime} P(p) - A051006 + 1/4 = 0.054745292329555814476..., where P(s) is the prime zeta function. - Amiram Eldar, Sep 13 2024

Extended by Ray Chandler, Oct 28 2008

cp's OEIS Frontend

A132817 Decimal expansion of Sum_{n >= 1} 1/6^prime(n).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

Extensions

A132822 Decimal expansion of Sum_{n >= 1} 1/7^prime(n).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI

PARI

Formula

Extensions

A121240 Numerator of sum_{k=1..n} 1/2^prime(k).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

A275306 Decimal expansion of 1/2 - Sum_{k>=1} 1/2^prime(k).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A333182 Decimal expansion of Sum_{k>=1} mu(k)^2 / 2^k.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

A370596 Numbers k such that A007814(k) is a prime number.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

A036680 Expansion of C in Egyptian fractions, where C contains the primes in binary.

Views

Author

Keywords

Examples

Links

Crossrefs

A102913 Take characteristic function of the semiprimes A001358, interpret it as a binary fraction and convert to a decimal fraction.

Views

Author

Keywords