A368735 Table read by ascending antidiagonals: A(n,m) is the smallest number k such that k and k+1 have the n-th and m-th prime signatures, respectively, or -1 if no such k exists.
-1, -1, 1, -1, 2, -1, -1, 4, 3, -1, -1, 6, -1, 5, -1, -1, -1, -1, 9, 7, -1, -1, 12, 8, 14, -1, 11, -1, -1, 16, -1, -1, 26, 49, -1, -1, -1, 40, -1, 20, -1, 51, -1, 23, -1, -1, 30, 24, 81, 124, 27, 15, -1, 29, -1, -1, -1, -1, 54, -1, 44, -1, 39, 169, 31, -1
Offset: 1
Examples
A(6,10) = 242 because 242 is the smallest number k of the form p^2 * q (the 6th prime signature; see A025487) such that k+1 is of the form r^5 (the 10th prime signature): 242 = 2 * 11^2 and 243 = 3^5. A(2,7) = -1 because there exists no number k such that k is a prime (the 2nd prime signature) and k+1 is the fourth power of a prime (the 7th prime signature). (If k+1 = q^4 for some prime q, then k = (q-1)*(q+1)*(q^2+1), which cannot be a prime.) The table below gives additional terms. . n\m| 1 2 3 4 5 6 7 8 9 10 11 12 ---+------------------------------------------------------------------- 1 | -1 1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 2 | -1 2 3 5 7 11 -1 23 29 31 -1 47 3 | -1 4 -1 9 -1 49 -1 -1 169 -1 -1 57121 4 | -1 6 -1 14 26 51 15 39 65 -1 35 111 5 | -1 -1 8 -1 -1 27 -1 343 2197 -1 -1 -1 6 | -1 12 -1 20 124 44 -1 188 153 242 99 175 7 | -1 16 -1 81 -1 -1 -1 -1 130321 -1 -1 -1 8 | -1 40 24 54 -1 152 -1 135 104 -1 -1 1647 9 | -1 30 -1 105 205378 170 -1 231 230 16806 195 890 10 | -1 -1 -1 32 -1 243 -1 -1 3125 -1 -1 -1 11 | -1 36 -1 225 -1 1444 -1 69189124 441 -1 -1 96393124 12 | -1 112 48 176 4912 368 80 567 272 1419856 6723 2511