A051006 -id:A051006 - cp's OEIS frontend

A164293 Decimal expansion of the twin prime constant associated with the binary constant in A164292.

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

Offset: 1

Carlos Alves, Aug 12 2009

Twin prime constant decimals (A164292)_2 = 0.1646823906345389353962381...
Similar: prime constant decimals A051006 = 0.4146825098511116602481096...
If twin prime conjecture is false then this twin prime constant is rational! As a counterpart if it is not rational, the conjecture is true.
If the conjecture is proved by other means, it will remain to see if it is irrational or transcendental.

Cf. A164292, A051006.

A217054 Odd number version of the prime constant (A101264 interpreted as a binary number).

Original entry on oeis.org

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

Offset: 0

Alonso del Arte, Sep 25 2012

The prime constant (A051006) is essentially a set of flags that tell us whether a given integer is prime. But since all even numbers (except 2) are composite, every other bit is guaranteed to be 0.

Depending on the algorithm for which this is used, it may be more efficient to store a set of flags for just the odd numbers (and handle 2 as a special case). Lehmer (1969) suggests using about 64 kilobytes for the storage of this "characteristic number."

			1/2 + 1/4 + 1/8 + 1/32 + 1/64 + ... = 0.928319591...

D. H. Lehmer, "Computer Technology Applied to the Theory of Numbers," from Studies in Number Theory, ed. William J. LeVeque. Englewood Cliffs, New Jersey: Prentice Hall (1969): 138.

RealDigits[Sum[1/2^((Prime[k] - 1)/2), {k, 2, 1000}], 10, 100][[1]]

s=0; forprime(p=3, default(realprecision)*log(100)\log(2)+9, s += 1.>>(p\2)); s \\ Charles R Greathouse IV, Sep 26 2012

sum(k = 1 .. infinity, chi(2k + 1)/2^k), where chi(n) is the characteristic function of the prime numbers (A010051).

sum(k = 2 .. infinity, 1/2^((p(k) - 1)/2)), where p(k) is the k-th prime number.

A262153 Decimal expansion of Sum_{p prime} 1/(2^p-1), a prime equivalent of the Erdős-Borwein constant.

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Sep 13 2015

Erdős was interested in the question whether this constant is irrational. - Amiram Eldar, Apr 30 2020

Pratt gives a conditional proof that this constant is irrational. - Charles R Greathouse IV, Sep 26 2024

			0.51694281980564038424051660847985627797854694791309124165028...

Paul Erdős, Some of my favourite unsolved problems, in A. Baker, B. Bollobás and A. Hajnal (eds.), A Tribute to Paul Erdős, Cambridge University Press, 1990, p. 470.

Thomas Bloom, Is Sum+{n>=2} omega(n)/2^n irrational? (Here omega(n) counts the number of distinct prime divisors of n), Erdős Problems.
Paul Erdős, On arithmetical properties of Lambert series, J. Indian Math. Soc. (N.S.), Vol. 12 (1948), pp. 63-66.
Paul Erdős, On the irrationality of certain series, Math. Student, Vol. 36 (1969), pp. 222-226.
Kyle Pratt, The irrationality of a prime factor series under a prime tuples conjecture, arXiv:2409.15185 [math.NT], 2024.
Terence Tao, Erdős problem database, see no. 69.
Eric Weisstein's World of Mathematics, Erdős-Borwein Constant.
Eric Weisstein's World of Mathematics, Prime Constant.

Cf. A051006, A065442.

digits = 100; m0 = 100; dm = 100; s[m_] := s[m] = N[Sum[1/(2^Prime[n]-1), {n, 1, m}], digits+10]; s[m = m0]; Print[{m, s[m]}]; s[m = m + dm]; While[ Print[{m, s[m]}]; RealDigits[ s[m], 10, digits+5] != RealDigits[ s[m - dm], 10, digits+5], m = m + dm]; RealDigits[ s[m], 10, digits] // First

suminf(k=1, omega(k)/2^k) \\ Michel Marcus, Apr 30 2020

s=0.; forprime(p=2,bitprecision(1.)+1,s+=1./(2^p-1)); s \\ Charles R Greathouse IV, Sep 26 2024

Equals Sum_{i>=1} 1/A001348(i). - R. J. Mathar, Feb 17 2016

Equals Sum_{k>=1} omega(k)/2^k, where omega(k) is the number of distinct primes dividing k (A001221). - Amiram Eldar, Apr 30 2020

A280609 Odd prime powers with prime exponents.

Original entry on oeis.org

9, 25, 27, 49, 121, 125, 169, 243, 289, 343, 361, 529, 841, 961, 1331, 1369, 1681, 1849, 2187, 2197, 2209, 2809, 3125, 3481, 3721, 4489, 4913, 5041, 5329, 6241, 6859, 6889, 7921, 9409, 10201, 10609, 11449, 11881, 12167, 12769, 16129, 16807, 17161, 18769, 19321, 22201, 22801, 24389, 24649, 26569, 27889, 29791, 29929

Offset: 1

Ilya Gutkovskiy, Jan 06 2017

Intersection of A053810 and A061345.

			9 is in the sequence because 9 = 3^2;
25 is in the sequence because 25 = 5^2;
27 is in the sequence because 27 = 3^3, etc.

Eric Weisstein's World of Mathematics, Prime Power.

Cf. A000961, A034785, A051006, A053810, A061345, A078422, A246547, A246551, A246655.

Select[Range[30000], PrimePowerQ[#1] && PrimeQ[PrimeOmega[#1]] && Mod[#1, 2] == 1 & ]

from sympy import primepi, integer_nthroot, primerange
def A280609(n):
    def f(x): return int(n+x-sum(primepi(integer_nthroot(x, p)[0])-1 for p in primerange(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

a(n) = p^q, where p, q are primes and p > 2.

Sum_{n>=1} 1/a(n) = Sum_{p prime} P(p) - A051006 = 0.25699271237062131298..., where P(s) is the prime zeta function. - Amiram Eldar, Sep 13 2024

A332905 Decimal expansion of Sum_{p prime, k>=1} 1 / 2^(p^k).

Original entry on oeis.org

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

Offset: 0

Ilya Gutkovskiy, Mar 10 2020

Prime power constant: binary expansion is the characteristic function of prime powers (A069513).

			0.48305718112590956493... = (0.0111101110...)_2.
                                |||| |||
                                2345 789

Cf. A051006, A069513, A246655, A333182.

A333392 a(0) = 1; thereafter a(n) = 2^(prime(n)-1) + Sum_{k=1..n} 2^(prime(n)-prime(k)).

Original entry on oeis.org

1, 3, 7, 29, 117, 1873, 7493, 119889, 479557, 7672913, 491066433, 1964265733, 125713006913, 2011408110609, 8045632442437, 128730119078993, 8238727621055553, 527278567747555393, 2109114270990221573, 134983313343374180673, 2159733013493986890769, 8638932053975947563077

Offset: 0

Ilya Gutkovskiy, Mar 18 2020

nonn

			a(7) = 119889 (in base 10) = 11101010001010001 (in base 2).
                             ||| | |   | |   |
                             123 5 7  1113  17

Cf. A000040, A008578, A010051, A034785, A051006, A072762, A076793, A080339, A080355, A121240, A139104, A333393.

a[0] = 1; a[n_] := 2^(Prime[n] - 1) + Sum[2^(Prime[n] - Prime[k]), {k, 1, n}]; Table[a[n], {n, 0, 21}]

a(n) = if (n==0, 1, 2^(prime(n)-1) + sum(k=1, n, 2^(prime(n)-prime(k)))); \\ Michel Marcus, Mar 18 2020

a(n) = floor(c * 2^prime(n)) for n > 0, where c = 0.91468250985... = 1/2 + A051006.

A072812 Take the binary number P = 1.01101010001... which has an 1 in each decimal position that is a prime (e.g. 2,3,5,7...) and convert this number to base 10.

Original entry on oeis.org

1, 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: 1

Andre Neumann Kauffman (andrekff(AT)hotmail.com), Aug 13 2002

			1.414682509851111660248109622154307708366...

Essentially the same as A051006, which is the main entry.

Cf. A000040, A010051.

a := 1; for i from 2 to 1000 do if isprime(i) then a := a+10^(-i); end if; end do; b := evalf(a,100); evalf(convert(b,decimal,binary),40);

More digits from Alois P. Heinz, Jan 28 2017

A103313 Positions of records in the continued fraction expansion of the prime constant.

Original entry on oeis.org

0, 1, 4, 5, 6, 20, 31, 54, 306, 356, 762, 3174, 20240, 22693, 35793, 58491, 81251, 206410, 228533, 3987683, 5635890

Offset: 0

Eric W. Weisstein, Jan 30 2005

Cf. A051006, A051007, A102878.

cp's OEIS Frontend

A164293 Decimal expansion of the twin prime constant associated with the binary constant in A164292.

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A217054 Odd number version of the prime constant (A101264 interpreted as a binary number).

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A262153 Decimal expansion of Sum_{p prime} 1/(2^p-1), a prime equivalent of the Erdős-Borwein constant.

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A280609 Odd prime powers with prime exponents.

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A332905 Decimal expansion of Sum_{p prime, k>=1} 1 / 2^(p^k).

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A333392 a(0) = 1; thereafter a(n) = 2^(prime(n)-1) + Sum_{k=1..n} 2^(prime(n)-prime(k)).

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A072812 Take the binary number P = 1.01101010001... which has an 1 in each decimal position that is a prime (e.g. 2,3,5,7...) and convert this number to base 10.

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A103313 Positions of records in the continued fraction expansion of the prime constant.

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