A138290 -id:A138290 - cp's OEIS frontend

A278740 Primes in A138290.

577, 5569, 29251

Offset: 1

Hans Havermann, Nov 27 2016

A208083 Number of primes of the form 2^n - 2^k - 1, 1 <= k < n.

0, 0, 2, 3, 2, 4, 0, 5, 4, 3, 1, 5, 1, 5, 0, 3, 2, 9, 1, 12, 4, 5, 0, 7, 1, 2, 0, 1, 5, 4, 0, 8, 5, 1, 1, 9, 0, 6, 0, 7, 1, 6, 0, 4, 7, 2, 1, 10, 3, 3, 1, 2, 1, 6, 0, 4, 3, 0, 1, 8, 3, 4, 0, 3, 1, 8, 1, 2, 2, 3, 0, 9, 1, 5, 2, 5, 8, 3, 0, 10, 3, 0, 2, 4, 4, 6

Offset: 1

Reinhard Zumkeller, Feb 23 2012

nonn

Number of primes in (n-1)-st row of the triangle in A081118;
a(A138290(n)+1) = 0;
for n >= 0: a(A208091(n)) = n and a(m) <> n for m < A208091(n).

			n _ A208083(n) ________________ (n-1)-st row of A081118 _________
5   #{23,29} = 2                [15,23,27,29]
6   #{31,47,59,61} = 4          [31,47,55,59,61]
7   #{} = 0                     [63,95,111,119,123,125]
8   #{127,191,223,239,251} = 5  [127,191,223,239,247,251,253]
9   #{383,479,503,509} = 4      [255,383,447,479,495,503,507,509]

Robert Israel, Table of n, a(n) for n = 1..1500

Cf. A010051, A081118, A138290, A181741, A208091.

a208083 = sum . map a010051 . a081118_row

f:= n -> nops(select(k -> isprime(2^n-2^k-1),[$1..n-1])):
map(f, [$1..100]); # Robert Israel, Jun 12 2018

a[n_] := Module[{m = 2^n - 1, cnt = 0}, For[ k = 1, k < n, k++, If[PrimeQ[m - 2^k], cnt++]]; cnt]; Table[a[n], {n, 2, 86}] (* Jean-François Alcover, Sep 12 2013 *)

a(n)=sum(k=1,n-1,ispseudoprime(2^n-2^k-1)) \\ Charles R Greathouse IV, Sep 12 2013

a(n) = Sum_{k=1..n-1} A010051(A081118(n-1,k)).

A095058 Number of primes with a single 0-bit (A095078) in range ]2^n,2^(n+1)].

Original entry on oeis.org

0, 1, 2, 2, 3, 0, 4, 4, 3, 1, 5, 1, 4, 0, 3, 2, 8, 1, 11, 4, 5, 0, 7, 1, 2, 0, 1, 5, 4, 0, 7, 5, 1, 1, 9, 0, 6, 0, 7, 1, 6, 0, 4, 7, 2, 1, 10, 3, 3, 1, 2, 1, 6, 0, 4, 3, 0, 1, 8, 3, 3, 0, 3, 1, 8, 1, 2, 2, 3, 0, 9, 1, 5, 2, 5, 8, 3, 0, 10, 3, 0, 2, 4, 4, 6, 1

Offset: 1

Antti Karttunen, Jun 01 2004

nonn

For large n, the average value of a(n) is about 4. See A138290 for the n such that a(n)=0. - T. D. Noe, Mar 14 2008

T. D. Noe, Table of n, a(n) for n=1..2048
A. Karttunen and J. Moyer, C-program for computing the initial terms of this sequence
Samuel S. Wagstaff, Jr., Prime Numbers with a fixed number of one bits or zero bits in their binary representation, Exp. Math. vol. 10, issue 2 (2001) 267, Table 3. - From _N. J. A. Sloane_, Jun 19 2011.
Samuel S. Wagstaff, Jr., Prime Numbers with a fixed number of one bits or zero bits in their binary representation, Exp. Math. vol. 10, issue 2 (2001) 267, Table 3.
Index entries for sequences related to occurrences of various subsets of primes in range ]2^n,2^(n+1)]

Cf. A095018.

a(n) = sum(k=2^n+1, 2^(n+1), isprime(k) && (#select(x->x==0, binary(k))==1)); \\ Michel Marcus, Sep 11 2015

A369375 Numbers m such that the Mersenne number 2^m - 1 is a de Polignac number (A006285).

Original entry on oeis.org

1, 7, 15, 23, 27, 31, 37, 39, 43, 55, 58, 63, 71, 79, 82, 91, 95, 111, 123, 127, 133, 135, 139, 143, 148, 151, 159, 167, 169, 172, 173, 175, 179, 183, 191, 195, 199, 207, 211, 223, 239, 255, 286, 295, 313, 316, 319, 335, 337, 351, 367, 373, 383, 406, 415, 417, 433, 435, 447, 455, 461, 463, 479

Offset: 1

Thomas Ordowski, Jan 22 2024

nonn

Integers m > 0 such that 2^m-1 - 2^n is not prime for every natural n < m.
For m > 2, a number m is a term of this sequence if and only if A208083(m) = 0.
All Mersenne number m = 2^k-1 for k > 2 are in this sequence. The proof is below.
Cf. A138290 (see Chai Wah Wu's conjecture in the third comment). By Crocker's (1971) theorem: if m > 2 and a <> b, then 2^(2^m)-1 - 2^a - 2^b is not prime.
If a = 2^m-1, then b < a, so for m > 2, 2^(2^m-1)-1 is a de Polignac number, QED.
Note that 2^(2^m-1)-1 - 2^n is divisible by some prime factor of 2^(2^m)-1.
Prime numbers of this sequence are Mersenne primes > 3, and many other primes.
Conjecture: if n > 5, then |2^(2^n-1)-1 - 2^m| is not prime for every m > 0.
If so, then by the dual Riesel conjecture, 2^(2^n-1)-1 is a (dual) Riesel number, i.e., if n > 5, then (2^(2^n-1)-1)2^m-1 is composite for every integer m > 0.
For example, the double Mersenne prime 2^(2^7-1)-1 may be a dual Riesel number.
It seems that the natural density of these numbers is about twice as high as the density of de Polignac numbers.
For many terms m, 2m+1 is also in this sequence. By iteration (x -> 2x+1), the subsequence b(n) = (m+1)2^n-1, for n >= 0, is infinite if m = 7 (which has already been proven) and probably if m = 27 (which is hard to prove).

			7 is a term since {2^7-1-2, 2^7-1-2^2, 2^7-1-2^3, 2^7-1-2^4, 2^7-1-2^5, 2^7-1-2^6} = {125, 123, 119, 111, 95, 63} and all six of these numbers are composite.
Note that both 2^148-1 and 2^148+1 are de Polignac numbers.

Cf. A000225, A006285, A000668, A101036, A138290, A208083, A232565, A369378.

fQ[n_] := Block[{k = n -1}, While[k > 1 && !PrimeQ[2^n -1 -2^k], k--]; k == 1]; Select[ Range[3, 450], fQ] (* Robert G. Wilson v, Jan 22 2024 *)

For n > 1, a(n) = A138290(n-1) + 1.

A208083(a(n)) = 0, for n > 0.

More terms from Robert G. Wilson v, Jan 22 2024

A368214 Primes with a single 0-bit in binary expansion such that changing the position of the 0-bit always gives a nonprime (including the one with a leading zero).

Original entry on oeis.org

2, 2039, 6143, 522239, 33546239, 260046847, 16911433727, 32212254719, 2196875771903, 140735340871679, 2251799813685119, 9005000231485439, 576460752169205759, 36893488147410714623, 147573811852188057599, 9444732965739282038783, 154742504910672534362390399

Offset: 1

Ya-Ping Lu, Dec 23 2023

It seems that most of the terms end with '9', followed by those ending with '3', '7', and '1'.

			2 is a term because 2 is a prime with one '0' in binary form ('10') and '01' is not a prime. 2039 is a term because 2039 is a prime with one '0' in binary form ('11111110111') and changing the position of the '0', for example, '11111111011' = 2043 and '01111111111' = 1023, always results in a composite.

Subsequence of A095078.

Cf. A039986, A081118, A095058, A138290, A208083.

from sympy import isprime
for n in range(1,100):
    s = n*'1'; c = 0
    for j in range(n+1):
        num = int(s[:j]+'0'+s[j:], 2)
        if isprime(num):
            c += 1
            if c == 1: r = num
            if c == 2: break
    if c == 1: print(r, end = ', ')

cp's OEIS Frontend

A278740 Primes in A138290.

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A208083 Number of primes of the form 2^n - 2^k - 1, 1 <= k < n.

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A095058 Number of primes with a single 0-bit (A095078) in range ]2^n,2^(n+1)].

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PARI

A369375 Numbers m such that the Mersenne number 2^m - 1 is a de Polignac number (A006285).

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Extensions

A368214 Primes with a single 0-bit in binary expansion such that changing the position of the 0-bit always gives a nonprime (including the one with a leading zero).

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Python