Three sequences related to near-square primes p = k² + 1

  • A1912: Numbers k such that 4k² + 1 is prime.
    1, 2, 3, 5, 7, 8, 10, 12, 13, 18, 20, 27, 28, 33, 37, 42, 45, 47, 55, 58, 60, 62, 63, 65, 67, 73, 75, 78, 80, 85, 88, 90, 92, 102, 103, 105, 112, 115, 118, 120, 125, 128, 130, 132, 135, 140, 142, 150, 153, 157, 163, 170, 175, 192, 193, 198, 200, ...

  • A2496: Primes of the form k² + 1.
    2, 5, 17, 37, 101, 197, 257, 401, 577, 677, 1297, 1601, 2917, 3137, 4357, 5477, 7057, 8101, 8837, 12101, 13457, 14401, 15377, 15877, 16901, 17957, 21317, 22501, 24337, 25601, 28901, 30977, 32401, 33857, 41617, 42437, 44101, 50177, ...

  • A5574: Numbers k such that k² + 1 is prime.
    1, 2, 4, 6, 10, 14, 16, 20, 24, 26, 36, 40, 54, 56, 66, 74, 84, 90, 94, 110, 116, 120, 124, 126, 130, 134, 146, 150, 156, 160, 170, 176, 180, 184, 204, 206, 210, 224, 230, 236, 240, 250, 256, 260, 264, 270, 280, 284, 300, 306, 314, 326, 340, 350, 384, 386, 396, ...

FORMULA: A2496(n) = A5574(n)^2 + 1 = 4*A1912(n-1)^2 + 1

(The first p = k²+1 = 2 is the only with an odd k, not of the form (2m)²+1.)

Growth: A1912(1, 10, 100, 10^3, 10^4, ...) = 2, 18, 432, 6072, 79355, ...
Seems to follow roughly: $ a(10^k) = (2k-2)*10^k $.

The graph ressembles an almost straight line, but the slope appears to grow a little. We have investigated this a little more. The terms are somewhat “random”, heavily scattered. It’s not practical to plot the gaps a(n+1)-a(n) directly. We consider a smoothed curve averaged over, e.g., 50 or 100 terms. We note that the average slope seems to increase at a steady rate. Here’s a picture made from the first $10^4$ terms:

A005574 average slope

PARI code for plotting:

\r /temp/b2v.gp
#K = b2v("/temp/b005574.txt")
ploth(i=1, 9900,(K[100+i\=1]-K[i])/100, 128, 9900) \\ 128 = "Points_too" 

/* plot a vector w/ points only: */
pv(v) = ploth(i=.5,#v+.49,v[i\/1],"no_Lines",#v)