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.
The table begins: 1.2.4..8.16.32.64.128.256.512.1024 ..3.5..9.17.33.65.129.257.513.1025 .......6.10.18.34..66.130.258..514 ....7.11.19.35.67.131.259.515.1027 ............12.20..36..68.132..260 .........13.21.37..69.133.261..517 ............14.22..38..70.134..262 ......15.23.39.71.135.263.519.1031 ...................24..40..72..136 ...............25..41..73.137..265 ...................26..42..74..138 ............27.43..75.139.267..523 .......................28..44...76 ...............29..45..77.141..269 ...................30..46..78..142 .........31.47.79.143.271.527.1039 ...........................48...80 .......................49..81..145 ...........................50...82 ...................51..83.147..275 This can be viewed as an irregular table, where row r (>= 1) has A000041(r) elements, that is, as 1; 2,3; 4,5,7; 8,9,6,11,15; 16,17,10,19,13,23,31; etc. A125106 illustrates how each number is mapped to a partition.
columns = 9; row[n_] := n - 2^Floor[Log2[n]]; col[0] = 0; col[n_] := If[EvenQ[n], col[n/2] + DigitCount[n/2, 2, 1], col[(n - 1)/2] + 1]; Clear[T]; T[, ] = 0; Do[T[row[k], col[k]] = k, {k, 1, 2^columns}]; Table[DeleteCases[Table[T[n - 1, k], {n, 1, 2^(k - 1)}], 0], {k, 1, columns}] // Flatten (* Jean-François Alcover, Sep 09 2017 *)
a:= proc(n) local i, l, r; r, l:= 0, [0, sort(map(i->
numtheory[pi](i[1])$i[2], ifactors(n)[2]))[]];
for i to nops(l)-1 do
r:= 2*((x-> 2*x+1)@@(l[i+1]-l[i]))(r)
od; r/2
end:
seq(a(n), n=1..120); # Alois P. Heinz, Jul 21 2017
Table[Floor@ Total@ Flatten@ MapIndexed[#1 2^(#2 - 1) &, Flatten[ Table[ 2^(PrimePi@ #1 - 1), {#2}] & @@@ FactorInteger@ #]] &[If[n == 1, 1, Module[{l = #, m = 0}, Times @@ Power @@@ Table[l -= m; l = DeleteCases[l, 0]; {Prime@ Length@ l, m = Min@ l}, Length@ Union@ l]] &@ Catenate[ConstantArray[PrimePi[#1], #2] & @@@ FactorInteger@ n]]], {n, 57}] (* Michael De Vlieger, Sep 08 2016, after JungHwan Min at A122111 *)
(define (A253564 n) (A156552 (A122111 n)))
The values 8,9,6,11,15 map to 1111,211,22,31,4, respectively; so a(4) = 8+9+6+11+15 = 49.
with(combinat): a := proc (n) local ff, partovi: ff := proc (X) local s: s := [1, seq(0, j = 1 .. X[2])]: s := map(convert, s, string): return cat(op(s)) end proc: partovi := proc (P) local X, n, Y, i: X := convert(P, multiset): n := X[-1][1]: Y := map(proc (t) options operator, arrow: t[1] end proc, X): for i to n do if member(i, Y) = false then X := [op(X), [i, 0]] end if end do: X := sort(X, proc (s, t) options operator, arrow: evalb(s[1] < t[1]) end proc): X := map(ff, X); X := cat(op(X)): n := parse(X): n := convert(n, decimal, binary): (1/2)*n end proc: add(partovi(partition(n)[j]), j = 1 .. numbpart(n)) end proc: seq(a(n), n = 1 .. 27); # the subprogram partovi (due to W. Edwin Clark) yields the viabin number of a given partition. # Emeric Deutsch, Sep 06 2017 # second Maple program: b:= proc(n, i, l, r) option remember; `if`(n=0, r, `if`(i>n, 0, b(n, i+1, l, r)+b(n-i, i$2, ((x-> 2*x+1)@@(i-l))(2*r)))) end: a:= n-> b(n, 1, 0$2): seq(a(n), n=1..30); # Alois P. Heinz, Mar 01 2021
b[n_, i_, l_, r_] := b[n, i, l, r] = If[n == 0, r, If[i>n, 0,
b[n, i+1, l, r] + b[n-i, i, i, Nest[2#+1&, 2r, i-l]]]];
a[n_] := b[n, 1, 0, 0];
Array[a, 30] (* Jean-François Alcover, Mar 02 2021, after Alois P. Heinz *)
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