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Start with A001844 in Sloane. These numbers have a lot of nice properties. They are also quite rare - only 70 in the first 10000 natural numbers.
Occasionally, the prime numbers in A001844 are 4 less than another 'cousin' prime, the first such example being 13, which is 4 less than 17. 13 = 2^2+3^2. 17 = 1^2+4^2. Since (a+2)^2+(a-1)^2-(a^2+(a+1)^2) = 4, if:
(i) P is in A001844; and
(ii) P+4 is also prime; then the unique representation of P, P+4 as sums of squares will be as sums of 4 consecutive squares (a^2+(a+1)^2) , (a+2)^2+(a-1)^2 . Call such pairs 'Kissing Cousins'.
{13,17}, {313,317}, {613,617}, {3613,3617}, {4513,4517} are the only Kissing Cousins less than 10000, in fact the only five pairs below 20000. The smaller of each pair are listed at A217674, which includes a table of the first 1000 entries. The proportion of Kissing Cousins to all primes decreases quite rapidly: 1/25. 3/168, 5/1229, 9/9552, 17/78498, 36/664579, 77/5761455, 181/50847534, 437/455052511... As simple fractions, 1/25/1/56,1/246, 1/1066, 1/4618, 1/18461, 1/74824,1/280926,... but that is not the end of it.
Consider those denominators. Call, say, 246/56 the 'attenuation' (1 to 10000)/(1 to 1000) i.e. the rate of change of the decrease. The attenuation is at a remarkably consistent and above all a reducing, rate: 1066/246, 4618/1066,18461/4618, 74824/18461, 28096/74824, 1041310/280926,..decreasing from around 4.38 to 3.7 over the first 10^10 natural numbers.
It seems that we are guaranteed an infinite number of Kissing Cousins.
Which is really quite remarkable.
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