This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
1 = 3^2 - 2^3; 2 = 3^3 - 2^5; 3 = 2^2 - 1^2 = 2^7 - 5^3; 4 = 2^3 - 2^2 = 6^2 - 2^5 = 5^3 - 11^2.
nn = 10^12; pp = Join[{1}, Union[Flatten[Table[n^i, {i, 2, Log[2, nn]}, {n, 2, nn^(1/i)}]]]]; d = Select[Differences[pp], # <= 100 &]; Table[Count[d, n], {n, 100}]
a(1) = 1 since 2 - 1 = 1; a(2) = 5 since 7 - 5 = 2; a(3) = 13 since 16 - 13 = 3; a(4) = 19 since 23 - 19 = 4; a(5) = 32 since 37 - 32 = 5; etc.
nxt[n_] := nxt[n] = Block[{k = n + 1}, While[! PrimePowerQ@k, k++]; k]; prv[n_] := prv[n] = Block[{k = n - 1}, While[! PrimePowerQ@k, k--]; k]; f[n_] := Block[{d = 0, exp = 2, p, q}, While[d == 0, p = prv[2^exp]; q = nxt[2^exp]; If[n == 2^exp - p, d = p]; If[n == q - 2^exp, d = 2^exp]; exp++]; d]; Do[ t[n] = f[n], {n, 3, 99, 2}]; p = 1; q = 2; t[_] = 0; While[p < 1110000, d = q - p; If[t[d] == 0, t[d] = p]; p = q; q = nxt@ q]; t@# & /@ Range@ 100
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