Here are some experimental results, especially on density. I used names with no "and"; length counts letters only; 0 is "zero"; N < 0 is "minus" N. I investigated only up to the neighborhood of abs(N) = 10^18. I agree with Bill Gosper that such problems are less interesting when they depend on decimal; and Victor Miller's original post noted that (his modification) can be generalized to other bases. That said, it's a cute problem. -- Mike Beeler // cases in range 1..10^0: 0 // cases in range 1..10^1: 2 // cases in range 1..10^2: 17 (the Puzzler) // cases in range 1..10^3: 56 // cases in range 1..10^4: 359 // cases in range 1..10^5: 2981 // cases in range 1..10^6: 22826 // cases in range 1..10^7: 194682 // cases in range 1..10^8: 1739872 // cases in range -(1..10^0): 0 // cases in range -(1..10^1): 1 // cases in range -(1..10^2): 7 (-9, -11, -13, -40, -50, -52, -60) // cases in range -(1..10^3): 42 // cases in range -(1..10^4): 314 // cases in range -(1..10^5): 2586 // cases in range -(1..10^6): 20759 // cases in range -(1..10^7): 177174 // cases in range -(1..10^8): 1575817 // count of both N and -N in list, N = 1..10^0: 0 // count of both N and -N in list, N = 1..10^1: 0 // count of both N and -N in list, N = 1..10^2: 3 (last = 60) // count of both N and -N in list, N = 1..10^3: 7 (last = 660) // count of both N and -N in list, N = 1..10^4: 27 (last = 9860) // count of both N and -N in list, N = 1..10^5: 156 (last = 99456) // count of both N and -N in list, N = 1..10^6: 1006 (last = 999450) // count of both N and -N in list, N = 1..10^7: 7934 (last = 9999660) // count of both N and -N in list, N = 1..10^8: 56435 (last = 99996600) // the 7 through 1000 are: 40, 50, 60, 240, 456, 600, 660 // count of pairs of consecutive numbers in list, N = 0..10^x: // for 0..10^0: 0 positive, 0 negative // for 0..10^1: 0 positive, 0 negative // for 0..10^2: 0 positive, 0 negative // for 0..10^3: 2 positive, 2 negative // for 0..10^4: 7 positive, 8 negative // for 0..10^5: 44 positive, 40 negative // for 0..10^6: 271 positive, 238 negative // for 0..10^7: 2020 positive, 1509 negative // for 0..10^8: 15063 positive, 12709 negative // First 10 positive (N, of N and N+1): // 405, 665, 2262, 3135, 4508, 5082, 8903, 10503, 11865, 13283 // First 10 negative (N, of N and N-1): // -160, -728, -1155, -1368, -2379, -3626, -6303, -6623, -16769, -17080 // count of triplets of consecutive numbers in list, N = 0..10^x: // for 0..10^0: 0 positive, 0 negative // for 0..10^1: 0 positive, 0 negative // for 0..10^2: 0 positive, 0 negative // for 0..10^3: 0 positive, 0 negative // for 0..10^4: 0 positive, 0 negative // for 0..10^5: 2 positive, 0 negative // for 0..10^6: 4 positive, 3 negative // for 0..10^7: 17 positive, 8 negative // for 0..10^8: 89 positive, 66 negative // First 15 positive (N, of N and N+1 and N+2): // 44608, 64638, 390382, 774639, 1062528, // 1882399, 2048542, 3408428, 3819879, 6731898, // 6731899, 6912788, 7514652, 8225972, 8622898 // First 15 negative (N, of N and N-1 and N-2): // -706102, -932958, -943102, -1304728, -5556042, // -7378239, -7394522, -8420048, -11374208, -13295679, // -14779798, -17226168, -19023398, -20187568, -21017202 // There is 1 positive quadruplet of consecutive numbers in the // list, in the range 1..10^8. It begins at 6731898. // There are no negative quadruplets of consecutive numbers in the // list, in the range -1..-10^8. // Above results suggest numbers in list become sparser as the numbers // get larger (either positive or negative). That requires the gap // between numbers in list grows. The results below show that the // max gap size does grow. The gap size is the count of consecutive // numbers not in the list, between two numbers that are in the list. // // gap search going positive // gap of 3 after 0 // gap of 5 after 6 // gap of 17 after 12 // gap of 37 after 112 // gap of 39 after 200 // gap of 47 after 252 // gap of 64 after 340 // gap of 75 after 1071 // gap of 83 after 2178 // gap of 105 after 4048 // gap of 141 after 5536 // gap of 169 after 17610 // gap of 194 after 29202 // gap of 233 after 38532 // gap of 245 after 50904 // gap of 289 after 84490 // gap of 310 after 185856 // gap of 327 after 198842 // gap of 332 after 261712 // gap of 390 after 378400 // gap of 401 after 535950 // gap of 433 after 1205946 // gap of 530 after 1552320 // gap of 549 after 3622960 // gap of 570 after 7523191 // gap of 583 after 17188600 // gap of 602 after 22357364 // gap of 602 after 24674883 // gap of 653 after 27346200 // gap of 709 after 28373233 // gap of 712 after 36238357 // gap of 716 after 39391176 // gap of 724 after 49277495 // gap of 830 after 65840289 // gap analysis done through 10^8 // // gap search going negative // gap of 8 after 0 // gap of 26 after -13 // gap of 64 after -60 // gap of 70 after -832 // gap of 86 after -903 // gap of 184 after -1443 // gap of 199 after -8050 // gap of 208 after -17081 // gap of 209 after -17880 // gap of 261 after -36320 // gap of 275 after -79684 // gap of 411 after -118512 // gap of 476 after -149354 // gap of 534 after -716375 // gap of 551 after -4824320 // gap of 625 after -5583370 // gap of 698 after -11658024 // gap of 710 after -35366383 // gap of 758 after -50834025 // gap of 860 after -61538100 // gap analysis done through -10^8 // For various powers of 10, what is the number in the list just // less than the 10^x, and just greatert than the 10^x? Also note // whether the power of 10 itself is in the list. // // for positive N: // bracketing 10^0 (N not in list): 10^0-1, 10^0+3 // bracketing 10^1 (N not in list): 10^1-4, 10^1+2 // bracketing 10^2 (N is in list): 10^2-10, 10^2+12 // bracketing 10^3 (N not in list): 10^3-40, 10^3+5 // bracketing 10^4 (N not in list): 10^4-1, 10^4+5 // bracketing 10^5 (N not in list): 10^5-40, 10^5+2 // bracketing 10^6 (N is in list): 10^6-8, 10^6+50 // bracketing 10^7 (N is in list): 10^7-206, 10^7+4 // bracketing 10^8 (N not in list): 10^8-152, 10^8+5 // bracketing 10^9 (N is in list): 10^9-67, 10^9+1 // bracketing 10^10 (N is in list): 10^10-56, 10^10+10 // bracketing 10^11 (N not in list): 10^11-102, 10^11+22 // bracketing 10^12 (N not in list): 10^12-16, 10^12+5 // bracketing 10^13 (N not in list): 10^13-70, 10^13+5 // bracketing 10^14 (N not in list): 10^14-28, 10^14+56 // bracketing 10^15 (N not in list): 10^15-158, 10^15+20 // bracketing 10^16 (N not in list): 10^16-137, 10^16+20 // bracketing 10^17 (N not in list): 10^17-54, 10^17+14 // bracketing 10^18 (N not in list): 10^18-180, 10^18+2 // // for negative N: // bracketing -10^0 (N not in list): -10^0-8, -10^0+1 // bracketing -10^1 (N not in list): -10^1-1, -10^1+1 // bracketing -10^2 (N not in list): -10^2-25, -10^2+40 // bracketing -10^3 (N not in list): -10^3-8, -10^3+10 // bracketing -10^4 (N is in list): -10^4-56, -10^4+6 // bracketing -10^5 (N not in list): -10^5-28, -10^5+32 // bracketing -10^6 (N not in list): -10^6-20, -10^6+49 // bracketing -10^7 (N not in list): -10^7-4, -10^7+20 // bracketing -10^8 (N not in list): -10^8-4, -10^8+136 // bracketing -10^9 (N not in list): -10^9-25, -10^9+78 // bracketing -10^10 (N not in list): -10^10-25, -10^10+10 // bracketing -10^11 (N not in list): -10^11-8, -10^11+100 // bracketing -10^12 (N is in list): -10^12-56, -10^12+1 // bracketing -10^13 (N is in list): -10^13-6, -10^13+16 // bracketing -10^14 (N not in list): -10^14-30, -10^14+125 // bracketing -10^15 (N not in list): -10^15-8, -10^15+80 // bracketing -10^16 (N not in list): -10^16-8, -10^16+144 // bracketing -10^17 (N not in list): -10^17-65, -10^17+362 // bracketing -10^18 (N not in list): -10^18-8, -10^18+343 // From various results above, we would expect the density of // numbers in list to drop as the size of the numbers increases. // Here, we start with 10^x, for x=6..18, and count how many // of the next million cases are in list. Positive and negative. // // sample block 10^ 6 + 1..10^6: 19562 in list // sample block 10^ 7 + 1..10^6: 19485 in list // sample block 10^ 8 + 1..10^6: 16628 in list // sample block 10^ 9 + 1..10^6: 19515 in list // sample block 10^10 + 1..10^6: 19737 in list // sample block 10^11 + 1..10^6: 16614 in list // sample block 10^12 + 1..10^6: 18859 in list // sample block 10^13 + 1..10^6: 19223 in list // sample block 10^14 + 1..10^6: 16387 in list // sample block 10^15 + 1..10^6: 18099 in list // sample block 10^16 + 1..10^6: 18037 in list // sample block 10^17 + 1..10^6: 15236 in list // sample block 10^18 + 1..10^6: 18038 in list // // sample block -10^ 6 - 1..10^6: 17681 in list // sample block -10^ 7 - 1..10^6: 17429 in list // sample block -10^ 8 - 1..10^6: 15580 in list // sample block -10^ 9 - 1..10^6: 17463 in list // sample block -10^10 - 1..10^6: 17442 in list // sample block -10^11 - 1..10^6: 15416 in list // sample block -10^12 - 1..10^6: 17342 in list // sample block -10^13 - 1..10^6: 17579 in list // sample block -10^14 - 1..10^6: 15305 in list // sample block -10^15 - 1..10^6: 16133 in list // sample block -10^16 - 1..10^6: 16145 in list // sample block -10^17 - 1..10^6: 14937 in list // sample block -10^18 - 1..10^6: 16269 in list // // If the density is decreasing, it is doing so very slowly. // Perhaps starting the sample block on a 10^x value skews // the results; that is, perhaps in-list cases are unevenly // distributed within each power of 10. To investigate, count // the in-list cases among each of the 100 samples of 1 million // numbers per sample, from 1 to 10^8. Those counts are: // // 22826, 19562, 19594, 18439, 19290, 19208, 19494, 18468, 18538, 19263, // 19485, 18004, 17864, 17954, 17870, 17868, 17635, 17304, 17783, 17732, // 17914, 17466, 17368, 16291, 16979, 16905, 17504, 16286, 16291, 16825, // 17866, 17303, 17451, 16543, 16895, 16943, 17355, 16439, 16650, 16811, // 18439, 17888, 17604, 16729, 17466, 17211, 17708, 16827, 16829, 17521, // 17953, 17682, 17638, 16824, 17449, 17273, 17585, 17059, 17057, 17399, // 17968, 17641, 17934, 16952, 17257, 17423, 17580, 16773, 16798, 17199, // 17583, 16743, 16832, 16257, 16213, 16142, 16680, 15908, 16219, 16248, // 17660, 17433, 17380, 16604, 16898, 16985, 17289, 16602, 16627, 17087, // 17569, 17203, 17360, 16494, 16685, 16855, 17013, 16283, 16403, 16685 // // There is a trend toward lower density, but it is very slow after the // initial million, and is quite noisy.