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.
Paul Tek has authored 101 sequences. Here are the ten most recent ones:
The following table depicts the first few terms: +----+--------+-----------------------------------+ | n | a(n) | a(n) in situation with leading 0s | +----+--------+-----------------------------------+ | 1 | 1 | 1 | | 2 | 2 | 2 | | 3 | 3 | 3 | | 4 | 4 | 4 | | 5 | 5 | 5 | | 6 | 6 | 6 | | 7 | 7 | 7 | | 8 | 8 | 8 | | 9 | 9 | 9 | | 10 | 10 | 10 | | 11 | 11 | 011 | | 12 | 110 | 110 | | 13 | 100 | 100 | | 14 | 12 | 0012 | | 15 | 120 | 0120 | | 16 | 1200 | 1200 | | 17 | 20 | 200 | | 18 | 13 | 0013 | | 19 | 130 | 0130 | | 20 | 1300 | 1300 | | 21 | 30 | 300 | | 22 | 14 | 0014 | | 23 | 140 | 0140 | | 24 | 1400 | 1400 | | 25 | 40 | 400 | | 26 | 15 | 0015 | | 27 | 150 | 0150 | | 28 | 1500 | 1500 | | 29 | 50 | 500 | | 30 | 16 | 0016 | +----+--------+-----------------------------------+ Comments from _N. J. A. Sloane_, Jan 18 2016 (Start): After a(9)=9, the smallest possible choice for a(10) is the first number that has not yet appeared, which is 10. There is no contradiction, so we take a(10)=10. Now the smallest number that has not yet appeared is 11, and we can achieve a(11)=11 by making the string of digits starting at the 11th place read 011. Now the string of digits starting at the 12th pace is 11..., and the smallest candidate of that form is 110, which gives a(12)=110. And so on. (End)
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The following table lists few first terms, with the corresponding digits induced in the overall sequence: +----+------+------------------------------------------------------------+ | n | a(n) | New known digits | +----+------+------------------------------------------------------------+ | 1 | 1 | 1 | | 2 | 2 | 2 | | 3 | 3 | 3 | | 4 | 4 | 4 | | 5 | 5 | 5 | | 6 | 6 | 6 | | 7 | 7 | 7 | | 8 | 8 | 8 | | 9 | 9 | 9 | | 10 | 10 | 10 | | 11 | 14 | 1411 | | 12 | 1 | | | 13 | 17 | 713 | | 14 | 130 | 0 ... 14 | | 15 | 21 | 215 | | 16 | 50 | 0 16 | | 17 | 15 | 15 | | 18 | 28 | 2818 | +----+------+------------------------------------------------------------+
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Z(1) = {1};
Z(2) = {1} (+) {2} = {1,2};
Z(3) = {1,2} (+) {2,3,4} = {1,3,4};
Z(4) = {1,3,4} (+) {2,4,5,6,7,8} = {1,2,3,5,6,7,8};
Hence: a(1) = 1, a(2) = 2, a(3) = 3 and a(4) = 7.
![Graphic representation of the elements in the range [1..300] of the first 300 sets Z(n)](/A263402/a263402.png){kind=link}
lista(nn) = {zprec = Set([1]); print1(#zprec, ", "); for (n=2, nn, zs = setbinop((x,y)->x+y, zprec); zn = setminus(setunion(zprec, zs), setintersect(zprec, zs)); print1(#zn, ", "); zprec = zn;);} \\ Michel Marcus, Oct 20 2015
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The first terms of the sequence are: +----+-------------+----------------------------+ | n | a(n) | a(n) in factorial base | +----+-------------+----------------------------+ | 1 | 1 | 1 | | 2 | 5 | 2_1 | | 3 | 21 | 3_1_1 | | 4 | 17 | 2_2_1 | | 5 | 633 | 5_1_1_1_1 | | 6 | 23 | 3_2_1 | | 7 | 36153 | 7_1_1_1_1_1_1 | | 8 | 65 | 2_2_2_1 | | 9 | 93 | 3_3_1_1 | | 10 | 635 | 5_1_1_2_1 | | 11 | 443122713 | 11_1_1_1_1_1_1_1_1_1_1 | | 12 | 71 | 2_3_2_1 | | 13 | 81474226713 | 13_1_1_1_1_1_1_1_1_1_1_1_1 | | 14 | 36155 | 7_1_1_1_1_2_1 | | 15 | 645 | 5_1_3_1_1 | | 16 | 113 | 4_2_2_1 | +----+-------------+----------------------------+
f[n_] := Block[{d = Divisors@ n, g, k, m = {1}}, g[x_] := Flatten[Table[#1, {#2}] & @@@ FactorInteger@ x]; Do[k = Max@ Select[d, # <= i &]; If[! IntegerQ@ k, AppendTo[m, 1], d = Divisors[Last[d]/k]; AppendTo[m, k]]; If[d == {1}, Break[]], {i, 2, n}]; Reverse@ m]; Table[FromDigits[#, MixedRadix[Reverse@ Range[2, Length@ #]]] &@ f@ n, {n, 30}] (* Michael De Vlieger, Oct 12 2015, Version 10.2 *)
a(n) = {j = sumdigits(n); v = vector(n, k, sumdigits(k)); i = #select(x->x==j, v); nb = 0; k = 0; while(nb != j, k++; if (sumdigits(k) == i, nb++)); k;} \\ Michel Marcus, Oct 16 2015
a(n) = {j = hammingweight(n); v = vector(n, k, hammingweight(k)); i = #select(x->x==j, v); nb = 0; k = 0; while(nb != j, k++; if (hammingweight(k) == i, nb++)); k;} \\ Michel Marcus, Oct 16 2015
After 0 generation: - We have a unique ON cell at position z=0, - Hence, a(0) = 1. After 1 generation: - ON cells appear at positions z=-1 and z=+1, - No ON cell dies, - Hence a(1) = a(0)+2-0 = 3. After 2 generations: - ON cells appears at positions z=-2 and z=+2, - No ON cell dies, - Hence a(2) = a(1)+2-0 = 5. After 3 generations: - ON cells appears at positions z=-3 and z=+3, - ON cells at positions z=-1 and z=+1 die (as they have 2 ON neighbors), - ON cells at positions z=-2 and z=+2 die (as they have 1 ON neighbor), - Hence a(3) = a(2)+2-4 = 3. Schematically: +-----+-----------+------+ | n | ON cells | a(n) | +-----+-----------+------+ | 0 | # | 1 | | 1 | ### | 3 | | 2 | ##### | 5 | | 3 | # # # | 3 | +=====+-----------+------+ | z%2 | 1010101 | +-----+-----------+
The numbers with sum of divisors 72 are: 30, 46, 51, 55, 71. Hence: a(30)=1, a(46)=2, a(51)=3, a(55)=4, a(71)=5. More generally: the terms of each row of A085790 (say of length i) map to 1, 2, ..., i. Also: for any n>0, the n terms of the n-th row of A201915 map to 1, 2, ..., n.
N:= 1000: # to get a(1) to a(N) Sigmas:= [seq(numtheory:-sigma(i),i=1..N)]: seq(numboccur(Sigmas[n], Sigmas[1..n]),n=1..N); # Robert Israel, Oct 09 2015
t = DivisorSigma[1, #] & /@ Range@ 10000; s = Position[t, #] & /@ Range@ Max@ t; Flatten[Position[s, #, {3}]][[2]] & /@ Range@ 87 (* Michael De Vlieger, Oct 09 2015 *)
cnt = vector(224); for (n=1, 87, s=sigma(n); cnt[s] = cnt[s]+1; print1(cnt[s] ", "))
The first terms of the sequence are: +----+--------+ | n | a(n) | +----+--------+ | 1 | 2 | | 2 | 23 | | 3 | 3 | | 4 | 13 | | 5 | 11 | | 6 | 17 | | 7 | 7 | | 8 | 37 | | 9 | 43 | | 10 | 31 | | 11 | 19 | | 12 | 41 | | 13 | 101 | | 14 | 61 | | 15 | 103 | | 16 | 71 | | 17 | 47 | | 18 | 73 | | 19 | 67 | | 20 | 79 | | 21 | 97 | | 22 | 29 | | 23 | 229 | | 24 | 293 | | 25 | 307 | +----+--------+
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