Based on the 1,513,118 Ulam numbers <= 20,447,213, the frequency of gaps is: The density is holding up at about 7.4%, contrary to Ulam's expectation that the density goes to zero. gap freq percent 1 4 0.00 2 558959 36.94 3 207138 13.69 4 2 0.00 5 16811 1.11 7 3245 0.21 8 18431 1.22 9 3 0.00 10 1426 0.09 12 75087 4.96 13 2 0.00 15 51331 3.39 17 129419 8.55 19 3130 0.21 20 116950 7.73 22 59663 3.94 24 3066 0.20 25 82142 5.43 27 5511 0.36 29 1764 0.12 30 9868 0.65 32 265 0.02 34 18297 1.21 37 4988 0.33 39 33475 2.21 41 1898 0.13 42 49025 3.24 (omitted terms <= 42 did not occur) The largest gap found is 587 after 933709. Record-setting gaps are in A080287 and A80288. No larger gaps were found, even though the search went more than 20 times farther. --------------------------- I looked at which terms in the sequence are later used in the sum of other terms. The percentage here is the percentage of the terms generated that have this Ulam number as a term in the sum. Indices <= 59 that are omitted did not occur. Note that 2 or 3 are used in 50.631% of the terms. It could be conjectured that 50% of the terms use 2 or 3 in the sum. Index Ulam Freq Percent 1 1 3 0.000 2 2 558960 36.941 3 3 207139 13.690 4 4 3 0.000 5 6 2 0.000 6 8 24947 1.649 7 11 2 0.000 8 13 659 0.044 9 16 2 0.000 10 18 3 0.000 11 26 4 0.000 12 28 2 0.000 13 36 73144 4.834 15 47 177377 11.723 16 48 1 0.000 17 53 3342 0.221 18 57 19 0.001 19 62 1 0.000 20 69 153174 10.123 24 87 1 0.000 25 97 7767 0.513 26 99 1 0.000 27 102 127232 8.409 28 106 2 0.000 33 145 1 0.000 36 175 1 0.000 37 177 1 0.000 38 180 5 0.000 40 189 3 0.000 42 206 3411 0.225 44 219 2 0.000 46 236 1 0.000 47 238 1 0.000 48 241 1 0.000 51 258 1 0.000 52 260 1 0.000 53 273 44036 2.910 54 282 1 0.000 55 309 1 0.000 56 316 72 0.005 59 339 90452 5.978