A027866: Numbers k such that k^2 + (k+1)^2 + (k+2)^2 + (k+3)^2 + (k+4)^2 + (k+5)^2 is prime. Medium-pitched noise that sounds relatively softer when compared to the primes. Residue counts: 2 [44034, 44231] 3 [29416, 29481, 29368] 4 [22006, 22174, 22028, 22057] 5 [0, 22200, 22087, 22086, 21892] 6 [14653, 14840, 14740, 14763, 14641, 14628] 7 [14675, 0, 14776, 14701, 14824, 14621, 14668] 8 [11033, 11097, 10954, 11037, 10973, 11077, 11074, 11020] 9 [9767, 9798, 9906, 9844, 9843, 9770, 9805, 9840, 9692] 10 [0, 11100, 11059, 11052, 10841, 0, 11100, 11028, 11034, 11051] 11 [0, 9899, 9779, 9698, 9765, 9710, 0, 9967, 9833, 9795, 9819] 12 [7289, 7498, 7335, 7385, 7312, 7298, 7364, 7342, 7405, 7378, 7329, 7330] Density information: A(x) at x=10,100,1000,...: [4, 25, 184, 1376, 10683, 88265] A(x)/x at x=10,100,1000,...: 0.4,0.25,0.184,0.1376,0.10683,0.088265, First 100 first differences: [1, 1, 5, 3, 1, 5, 1, 2, 2, 1, 8, 2, 4, 10, 3, 1, 4, 6, 1, 18, 1, 5, 9, 1, 5, 2, 5, 10, 9, 14, 4, 6, 6, 1, 5, 2, 1, 32, 13, 2, 3, 2, 2, 2, 1, 5, 4, 3, 2, 11, 2, 7, 11, 4, 4, 7, 7, 5, 15, 1, 10, 2, 3, 1, 1, 2, 7, 8, 2, 5, 1, 12, 15, 1, 1, 4, 16, 5, 1, 2, 11, 4, 1, 14, 3, 3, 1, 10, 1, 7, 1, 4, 7, 18, 1, 2, 4, 4, 2, 4] First 99 second differences: [0,4,-2,-2,4,-4,1,0,-1,7,-6,2,6,-7,-2,3,2,-5,17,-17,4,4,-8,4,-3,3,5,-1,5,-10,2, 0,-5,4,-3,-1,31,-19,-11,1,-1,0,0,-1,4,-1,-1,-1,9,-9,5,4,-7,0,3,0,-2,10,-14,9,-8, 1,-2,0,1,5,1,-6,3,-4,11,3,-14,0,3,12,-11,-4,1,9,-7,-3,13,-11,0,-2,9,-9,6,-6,3,3, 11,-17,1,2,0,-2,2] There is a spectrum line at about 8820Hz which corresponds to 44100/5. In the residue classes we notice that there are no numbers in the sequence congruent to 0 mod 5. If we expand the polynomial we get the expression 6k^2 + 30k + 55 which is divisible by 5 when k is divisible 5 and so cannot be prime. When k is congruent to 1 mod 7, we have that the expression 6k^2 + 30k + 55 is divisible by 7 and so cannot be prime. We have another spectrum line at around 8000Hz corresponding to 44100/5.5, but interestingly none at 44100/11. --David Oh