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.
%I A134969 #20 Jul 12 2023 11:02:48 %S A134969 3,11,5,17,7,19,13,29,17,37,23,43,29,53,43,71,67,103,71,107,73,109,97, %T A134969 137 %N A134969 List of pairs of primes that are separated by the equivalent of 2 quadratic intervals. Both primes are greater than their preceding perfect squares by the same amount, or offset. The respective perfect squares can be both odd, in which case the offset is even, or both even, in which case the offset is odd. %H A134969 Michael M. Ross, <a href="http://www.naturalnumbers.org/quadprimes.html">What Are Odd Squared, or Biquadratic, Paired Primes?</a> %H A134969 Michael M. Ross, <a href="http://www.naturalnumbers.org/images/psline.gif">Diagram of paired primes</a> showing how two perfect pairs partially overlap: 23 and 43 span perfect squares 25 and 36; 29 and 53 span perfect squares 36 and 49. %H A134969 Michael M. Ross, <a href="http://www.naturalnumbers.org/quadprimes.html#visualizing">VBA source code</a> providing an Excel visualization of the odd-offset prime pairs and colocation with twin primes. %F A134969 PS1 + OfN = P1, PS3 + OfN = P2, where %F A134969 PS1 = first perfect square, %F A134969 PS3 = 3rd perfect square, %F A134969 OfN = an equal positive offset from preceding perfect square, and %F A134969 P1 and P2 = the prime pair. %e A134969 Prime pair 71 and 107 have an odd offset of 7 from 64 and 100: %e A134969 Interval 1: 8*8 = 64 + 7 = 71. %e A134969 Interval 2: 9*9 = 81 + 7 = 88. %e A134969 Interval 3: 10*10 = 100 + 7 = 107. %e A134969 Prime pair 97 and 137 have an even offset of 16 from 81 and 121: %e A134969 Interval 1: 9*9 = 81 + 16 = 97. %e A134969 Interval 2: 10*10 = 100 + 16 = 116. %e A134969 Interval 3: 11*11 = 121 + 16 = 137. %e A134969 In all cases there is a complete intermediate quadratic interval (#2). %e A134969 The pair (11,83), 11-9 = 83-81 = 2, does not work because there are 6 squares in between: 16,25,36,49,64,81. %o A134969 Excel VBA available at http://www.naturalnumbers.org/quadprimes.html#code Demonstrates odd-offset prime-pair distribution with associated twin primes. Limitations: -Only the subset of paired primes for even perfect squares with odd offsets are demonstrated. -Because of the 256-column limitation for an Excel worksheet, data in rows after perfect square 484 are not complete. -Only twin primes that are also paired primes are outlined (in red). %Y A134969 Cf. A056892. %K A134969 nonn,tabf,more,less %O A134969 1,1 %A A134969 _Michael M. Ross_, Feb 04 2008