A095759 -id:A095759 - cp's OEIS frontend

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).

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

Antti Karttunen, Jun 12 2004

			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.

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

A095741 Number of base-2 palindromic primes (A016041) in range [2^2n,2^(2n+1)].

Original entry on oeis.org

2, 2, 3, 3, 7, 12, 23, 40, 94, 142, 271, 480, 856, 1721, 3099, 5572, 10799, 20782, 39468, 72672, 139867, 274480, 520376, 986318, 1914097, 3726617, 7107443, 13682325, 26430797, 51412565, 99204128, 190457946, 372035117, 727434192, 1407026351, 2724590109, 5315491839

Offset: 1

Antti Karttunen, Jun 12 2004

Note that there are no such primes in any range ]2^(2n-1),2^2n], as all even-length binary palindromes are divisible by 3 (cf. A048702).

The ratio a(n)/A036378(2n) converges as follows: 1, 0.4, 0.230769, 0.069767, 0.051095, 0.025862, 0.014268, 0.007006, 0.00461, 0.00193, 0.00101, 0.000487, 0.000235, 0.000127, 0.000061, 0.000029

Bisection of the first diagonal of triangle A095759.

Cf. A095731, A117773.

palindromicQ[n_, b_:10] := TrueQ[IntegerDigits[n, b] == Reverse[IntegerDigits[n, b]]]; Table[Length[Select[Range[2^(2n), 2^(2n + 1)], palindromicQ[#, 2] && PrimeQ[#] &]], {n, 10}] (* Alonso del Arte, Jan 13 2012 *)

m=vector(65536);u=vector(#m);u[1]=1;for(b=1,#m-1,c=b;e=2^floor(log(b+.5)/log(2));d=0;u[b+1]=e;while(c>0,d=d+e*(c%2);c=floor(c/2);e=e/2);m[b+1]=d);for(x=0,31,h=0;y=2^x;for(w=y,2*y-1,if(x<16,v1=4*y*w+m[w+1];v2=v1+2*y,w1=floor(w/65536);w2=w-65536*w1;v1=262144*y*w1+4*y*w2+65536*u[w1+1]/u[w2+1]*m[w2+1]+m[w1+1];v2=v1+2*y);if(isprime(v1),h++);if(isprime(v2),h++));print(2*x+3" bits: "h)) \\ Martin Raab, Jan 13 2012

a(n) = A117773(2*n+1). - Chai Wah Wu, Jul 05 2019

a(27)-a(32) from Martin Raab, Oct 20 2015
a(33)-a(35) from Chai Wah Wu, Jul 05 2019
a(36)-a(37) from Chai Wah Wu, Jul 11 2019

A095753 Number of almost base-2 palindromic primes (A095743) in range ]2^n,2^(n+1)].

Original entry on oeis.org

0, 0, 2, 3, 5, 4, 15, 18, 32, 33, 63, 81, 119, 144, 256, 318, 527, 640, 1029, 1281, 2236, 2566, 4273, 5410, 8261, 10610, 16868, 21084, 33943, 43104, 68218, 88493, 136343

Offset: 1

Antti Karttunen, Jun 12 2004

Ratio a(n)/A036378(n) converges as follows: 0, 0, 1, 0.6, 0.714286, 0.307692, 0.652174, 0.418605, 0.426667, 0.240876, 0.247059, 0.174569, 0.136468, 0.08933, 0.084488, 0.055702, 0.049028, 0.031388, 0.026634, 0.017408, 0.015933, 0.009567, 0.008318, 0.005488, 0.004361, 0.00291, 0.0024, 0.001555, 0.001295, 0.00085, 0.000695, 0.000465, 0.000369

Ratio a(n)/A095758(n) converges as follows: 1, 1, 0, 1.5, 1, 1, 3.75, 1.2, 2, 1.375, 1.909091, 1.446429, 1.652778, 1.515789, 1.718121, 1.452055, 1.636646, 1.191806, 1.570992, 1.283567, 1.708174, 1.380312, 1.534842, 1.392177, 1.547004, 1.311334, 1.573801, 1.302205, 1.521016, 1.419202, 1.570938, 1.389237, 1.546084

The second diagonal of triangle A095759. Cf. A095742.

A095757 Number of A095747-primes in range ]2^n,2^(n+1)].

Original entry on oeis.org

1, 2, 2, 3, 2, 6, 4, 7, 7, 10, 9, 26, 20, 43, 27, 74, 41, 112, 93, 181, 167, 495, 274, 796, 558, 1232, 935, 2602, 1512, 5164, 3275, 8689, 6309

Offset: 1

Antti Karttunen, Jun 12 2004

nonn

Ratio a(n)/A036378(n) converges as follows: 1, 1, 1, 0.6, 0.285714, 0.461538, 0.173913, 0.162791, 0.093333, 0.072993, 0.035294, 0.056034, 0.022936, 0.026675, 0.008911, 0.012962, 0.003814, 0.005493, 0.002407, 0.00246, 0.00119, 0.001846, 0.000533, 0.000807, 0.000295, 0.000338, 0.000133, 0.000192, 0.000058, 0.000102, 0.000033, 0.000046, 0.000017

The last nonzero terms from each row of triangle A095759. Bisection: A095760.

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

Antti Karttunen, Jun 12 2004

nonn

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

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

A095754 Number of A095744-primes in range ]2^n,2^(n+1)].

Original entry on oeis.org

0, 0, 0, 0, 2, 6, 4, 15, 20, 63, 62, 135, 150, 398, 347, 913, 895, 2196, 1907, 5151, 4483, 11932, 10033, 26498, 22645, 59155, 49968, 130032, 108271, 283447, 233865, 606296, 503884

Offset: 1

Antti Karttunen, Jun 12 2004

nonn

Ratio a(n)/A036378(n) converges as follows: 0, 0, 0, 0, 0.285714, 0.461538, 0.173913, 0.348837, 0.266667, 0.459854, 0.243137, 0.290948, 0.172018, 0.246898, 0.114521, 0.159923, 0.083264, 0.1077, 0.049359, 0.07, 0.031945, 0.044487, 0.019531, 0.026879, 0.011955, 0.016226, 0.007111, 0.009588, 0.004131, 0.005591, 0.002382, 0.003185, 0.001364

The third diagonal of triangle A095759.

A095755 Number of A095745-primes in range ]2^n,2^(n+1)].

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 4, 7, 16, 24, 88, 154, 314, 479, 959, 1568, 2620, 4394, 7110, 11987, 18434, 31444, 46474, 78790, 114464, 194921, 276169, 471971, 656213, 1123888, 1535212, 2648075, 3543260

Offset: 1

Antti Karttunen, Jun 12 2004

nonn

Ratio a(n)/A036378(n) converges as follows: 0, 0, 0, 0, 0, 0, 0.173913, 0.162791, 0.213333, 0.175182, 0.345098, 0.331897, 0.360092, 0.297146, 0.316502, 0.274654, 0.243744, 0.215498, 0.18403, 0.162898, 0.131356, 0.117234, 0.090468, 0.079923, 0.060431, 0.053465, 0.0393, 0.034801, 0.025039, 0.022168, 0.015636, 0.013913, 0.009594

The fourth diagonal of triangle A095759.

A095756 Number of A095746-primes in range ]2^n,2^(n+1)].

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 7, 10, 33, 56, 197, 430, 773, 1482, 2737, 5769, 8912, 17912, 26885, 55407, 77839, 158860, 214829, 441818, 575634, 1178349, 1499574, 3096644, 3836084, 7918328, 9615133

Offset: 1

Antti Karttunen, Jun 12 2004

nonn

Ratio a(n)/A036378(n) converges as follows: 0, 0, 0, 0, 0, 0, 0, 0, 0.093333, 0.072993, 0.129412, 0.12069, 0.225917, 0.266749, 0.255116, 0.25959, 0.254628, 0.282933, 0.230672, 0.243416, 0.191576, 0.206576, 0.151524, 0.161145, 0.113419, 0.121187, 0.081914, 0.086887, 0.05722, 0.061081, 0.039071, 0.041602, 0.026034

The fifth diagonal of triangle A095759.

A095760 Number of A095747-primes in range ]2^(2n-1),2^2n].

Original entry on oeis.org

1, 2, 2, 4, 7, 9, 20, 27, 41, 93, 167, 274, 558, 935, 1512, 3275, 6309

Offset: 1

Antti Karttunen, Jun 12 2004

nonn

Bisection of A095757, the central diagonal of triangle A095759.

cp's OEIS Frontend

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).

Views

Author

Keywords

Examples

Crossrefs

A095741 Number of base-2 palindromic primes (A016041) in range [2^2n,2^(2n+1)].

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A095753 Number of almost base-2 palindromic primes (A095743) in range ]2^n,2^(n+1)].

Views

Author

Keywords

Comments

Links

Crossrefs

A095757 Number of A095747-primes in range ]2^n,2^(n+1)].

Views

Author

Keywords

Comments

Links

Crossrefs

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

Views

Author

Keywords

Comments

Links

Crossrefs

A095754 Number of A095744-primes in range ]2^n,2^(n+1)].

Views

Author

Keywords

Comments

Links

Crossrefs

A095755 Number of A095745-primes in range ]2^n,2^(n+1)].

Views

Author

Keywords

Comments

Links

Crossrefs

A095756 Number of A095746-primes in range ]2^n,2^(n+1)].

Views

Author

Keywords

Comments

Links

Crossrefs

A095760 Number of A095747-primes in range ]2^(2n-1),2^2n].

Views

Author

Keywords

Links

Crossrefs