cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-6 of 6 results.

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

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, 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

Views

Author

Eric W. Weisstein

Keywords

Comments

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

Examples

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

Links

Crossrefs

Cf. A000720, A010051, A034785, A051007, A132800, A092857, A092858, A092859, A092860, A092861, A092862, A092863, A092874, A119523, A370596.
Cf. A036680 (Egyptian fractions).

Programs

  • Maple
    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
  • Mathematica
    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 *)
  • PARI
    { 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
  • PARI
    { 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
  • PARI
    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

Formula

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)

A061286 Smallest integer for which the number of divisors is the n-th prime.

Original entry on oeis.org

2, 4, 16, 64, 1024, 4096, 65536, 262144, 4194304, 268435456, 1073741824, 68719476736, 1099511627776, 4398046511104, 70368744177664, 4503599627370496, 288230376151711744, 1152921504606846976
Offset: 1

Views

Author

Labos Elemer, May 22 2001

Keywords

Comments

Seems to be the same as "Even numbers with prime number of divisors" - Jason Earls, Jul 04 2001
Except for the first term, smallest number == 1 (mod prime(n)) having n divisors (by Fermat's little theorem). - Amarnath Murthy and Meenakshi Srikanth (menakan_s(AT)yahoo.com), Jun 20 2003

Links

Crossrefs

Cf. A000005, A005179, A003680, A061283, A061286, A006093, A005097, A006254, A119523, A196202.

Programs

  • Mathematica
    Table[2^(p-1),{p,Table[Prime[n],{n,1,18}]}] (* Geoffrey Critzer, May 26 2013 *)
  • PARI
    forstep(n=2,100000000,2,x=numdiv(n); if(isprime(x),print(n)))
  • PARI
    a(n)=2^(prime(n)-1) \\ Charles R Greathouse IV, Apr 08 2012
  • Python
    from sympy import isprime, divisor_count as tau
[2] + [2**(2*n) for n in range(1, 33) if isprime(tau(2**(2*n)))] # Karl V. Keller, Jr., Jul 10 2020

Formula

a(n) = 2^(prime(n)-1) = 2^A006093(n).
a(n) = A005179(prime(n)). - R. J. Mathar, Aug 09 2019
Sum_{n>=1} 1/a(n) = A119523. - Amiram Eldar, Aug 11 2020

A127197 a(n) = numerator( pi(n)/2^n ).

Original entry on oeis.org

1, 1, 1, 3, 3, 1, 1, 1, 1, 5, 5, 3, 3, 3, 3, 7, 7, 1, 1, 1, 1, 9, 9, 9, 9, 9, 9, 5, 5, 11, 11, 11, 11, 11, 11, 3, 3, 3, 3, 13, 13, 7, 7, 7, 7, 15, 15, 15, 15, 15, 15, 1, 1, 1, 1, 1, 1, 17, 17, 9, 9, 9, 9, 9, 9, 19, 19, 19, 19, 5, 5, 21, 21, 21, 21, 21, 21, 11, 11, 11, 11, 23, 23, 23, 23, 23, 23
Offset: 2

Views

Author

Artur Jasinski, Jan 08 2007

Keywords

Crossrefs

Cf. A000079 (2^n), A000720 (PrimePi), A119523.

Programs

  • Mathematica
    Table[Numerator[PrimePi[k]/2^k], {k, 2, 1000}]
  • PARI
    a(n) = numerator(primepi(n)/2^n); \\ Michel Marcus, Oct 28 2019

Extensions

Offset 2 from Michel Marcus, Oct 28 2019
New name using formula from Joerg Arndt, Jan 20 2024

A119524 Decimal expansion of Sum_{k >= 1} ((-1)^A010051(k))/2^k.

Original entry on oeis.org

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

Views

Author

Roger L. Bagula, May 27 2006

Keywords

Examples

			0.17063498029777667950378075569138458326845152372416604426350917102271807...

Crossrefs

Cf. A010051, A065855, A119523, A275306.

Programs

  • PARI
    suminf(k=1, (-1)^isprime(k)/2^k) \\ Michel Marcus, Jan 16 2024

Formula

Equals Sum_{k >= 1} (k - 1 - PrimePi(k))/2^k = Sum_{k >= 1} A065855(k)/2^k.
From Antonio Graciá Llorente, Jan 14 2024: (Start)
Equals Sum_{k >= 2} A005171(k)/2^(k-1).
Equals Sum_{k >= 1} ((-1)^A010051(k))/2^k.
Equals 2*A275306. (End)

Extensions

Edited and corrected by N. J. A. Sloane, Nov 17 2006
Better name from Joerg Arndt, Jan 16 2024

A127200 Numerator of n-th partial sum of the Van der Waerden-Ulam binary measure of the primes.

Original entry on oeis.org

1, 1, 5, 23, 49, 51, 13, 105, 211, 1693, 3391, 1697, 6791, 13585, 27173, 108699, 217405, 217409, 217411, 54353, 434825, 6957209, 13914427, 27828863, 55657735, 111315479, 222630967, 55657743, 445261949, 1781047807, 3562095625
Offset: 2

Views

Author

Artur Jasinski, Jan 08 2007

Keywords

Crossrefs

Cf. A119523, A127197, A127201.

Programs

  • Mathematica
    Table[Numerator[Sum[PrimePi[k]/2^k, {k, 2, x}]], {x, 2, 100}]
  • PARI
    a(n) = numerator(sum(k=2, n, primepi(k)/2^k)); \\ Michel Marcus, Oct 28 2019

A127201 Base-2 logarithms of denominators corresponding to A127200.

Original entry on oeis.org

2, 1, 3, 5, 6, 6, 4, 7, 8, 11, 12, 11, 13, 14, 15, 17, 18, 18, 18, 16, 19, 23, 24, 25, 26, 27, 28, 26, 29, 31, 32, 33, 34, 35, 36, 36, 35, 37, 38, 41, 42, 41, 43, 44, 45, 47, 48, 49, 50, 51, 52, 52, 52, 52, 46, 53, 54, 59, 60, 58, 61, 62, 63, 64, 65, 67, 68, 69, 70, 70, 68, 73
Offset: 2

Views

Author

Artur Jasinski, Jan 08 2007

Keywords

Crossrefs

Cf. A119523, A127197, A127200.

Programs

  • Mathematica
    Table[Round[(1/Log[2])Log[Denominator[Sum[PrimePi[k]/2^k, {k, 2, x}]]]], {x, 2, 100}]
  • PARI
    a(n) = logint(denominator(sum(k=2, n, primepi(k)/2^k)), 2); \\ Michel Marcus, Oct 28 2019
Showing 1-6 of 6 results.

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