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 1-4 of 4 results.

A095758 Number of A095748-primes in range ]2^n,2^(n+1)].

Original entry on oeis.org

0, 0, 0, 2, 5, 4, 4, 15, 16, 24, 33, 56, 72, 95, 149, 219, 322, 537, 655, 998, 1309, 1859, 2784, 3886, 5340, 8091, 10718, 16191, 22316, 30372, 43425, 63699, 88186
Offset: 1

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Author

Antti Karttunen, Jun 12 2004

Keywords

Comments

Ratio a(n)/A036378(n) converges as follows: 0, 0, 0, 0.4, 0.714286, 0.307692, 0.173913, 0.348837, 0.213333, 0.175182, 0.129412, 0.12069, 0.082569, 0.058933, 0.049175, 0.03836, 0.029956, 0.026336, 0.016954, 0.013562, 0.009328, 0.006931, 0.005419, 0.003942, 0.002819, 0.002219, 0.001525, 0.001194, 0.000852, 0.000599, 0.000442, 0.000335, 0.000239
Ratio a(n)/A095753(n) converges as follows: 1, 1, 0, 0.666667, 1, 1, 0.266667, 0.833333, 0.5, 0.727273, 0.52381, 0.691358, 0.605042, 0.659722, 0.582031, 0.688679, 0.611006, 0.839063, 0.63654, 0.779079, 0.58542, 0.724474, 0.651533, 0.718299, 0.646411, 0.762582, 0.635404, 0.767928, 0.657455, 0.704621, 0.636562, 0.71982, 0.646795

Crossrefs

The penultimate nonzero terms from each row of triangle A095759. Cf. A095757, A095742.

A095749 Square array A(row>=1, col>=1) by antidiagonals: A(r,c) contains the c:th prime p for which A037888(p)=(r-1).

Original entry on oeis.org

3, 5, 2, 7, 11, 43, 17, 13, 53, 151, 31, 19, 71, 179, 599, 73, 23, 79, 233, 683, 2111, 107, 29, 83, 241, 739, 2143, 8543, 127, 37, 101, 271, 797, 2503, 9103, 33023, 257, 41, 109, 311, 853, 2731, 9623, 33151, 131839, 313, 47, 113, 331, 937, 3011, 10427, 33599, 135647, 531071
Offset: 1

Views

Author

Antti Karttunen, Jun 12 2004

Keywords

Examples

			a(1) = A(1,1) = 3 (11 in binary) as it is the first prime whose binary expansion is palindromic. a(2) = A(1,2) = 5 (101 in binary) as it is the second prime whose binexp is palindromic. a(3) = A(2,1) = 2 (10 in binary) as it is the first prime whose binexp needs a flip of just one bit to become palindrome. a(4) = A(1,3) = 7 (111 in binary) as it is the third prime whose binexp is palindromic. a(5) = A(2,2) = 11 (1011 in binary) as it is the second prime whose binexp needs a flip of just one bit to become palindrome.
		

Crossrefs

Row 1: A016041, 2: A095743, 3: A095744, 4: A095745, 5: A095746. Cf. also A095759. A095747-A095748. Permutation of primes (A000040).

A095743 Primes p for which A037888(p) = 1, i.e., primes whose binary expansion is almost symmetric, needing just a one-bit flip to become palindrome.

Original entry on oeis.org

2, 11, 13, 19, 23, 29, 37, 41, 47, 59, 61, 67, 89, 97, 103, 131, 137, 157, 167, 173, 181, 191, 193, 199, 211, 223, 227, 229, 239, 251, 277, 281, 317, 337, 349, 367, 373, 383, 401, 419, 431, 463, 467, 479, 487, 491, 503, 509, 521, 563, 569, 577
Offset: 1

Views

Author

Antti Karttunen, Jun 12 2004

Keywords

Crossrefs

The second row of array A095749. Cf. A095753, A095748.

Programs

  • Maple
    f:= proc(n) local L,i;
      L:= convert(n,base,2);
      add(abs(L[i]-L[-i]),i=1..floor(nops(L)/2))
    end proc:
    select(t -> f(t) = 1, [seq(ithprime(i),i=1..1000)]); # Robert Israel, Dec 04 2023

A095747 Maximally asymmetric odd primes in binary.

Original entry on oeis.org

3, 5, 7, 11, 13, 19, 23, 29, 43, 53, 71, 79, 83, 101, 109, 113, 151, 179, 233, 241, 271, 311, 331, 347, 397, 421, 457, 599, 683, 739, 797, 853, 937, 977, 1087, 1103, 1223, 1307, 1427, 1459, 1597, 1613, 1733, 2017, 2111, 2143, 2503, 2731, 3011
Offset: 1

Views

Author

Antti Karttunen, Jun 12 2004

Keywords

Comments

Primes p for which A037888(p) = floor((A070939(p)-2)/2). Those numbers contain just the first and last bits mirroring each other. Hence all the odd primes without leading zeros begin and end in 1 bits, the unique totally asymmetric prime being (10)_2 = 2.

Examples

			a(10)=(110101)2 since the symmetry is limited to the first and last bits. The number 47=(101111)2 is not a term because from left to right, the third bit matches with the fourth.
		

Crossrefs

Programs

  • PARI
    A070939(p)={return(floor(log(p)/log(2))+1)};
    A037888(p)={v=binary(p);s=0;j=#v;for(k=1,#v,s+=abs(v[k]-v[j]);j--);return(s/2);}; forprime(p=3,3011, if(A037888(p) ==floor((A070939(p)-2)/2),print1(p,", ")))

Extensions

Edited by Washington Bomfim, Jan 13 2011
Showing 1-4 of 4 results.