A307081 Triangular array read by rows: row n consists of the numbers k from 1 to n sorted in ascending order according to the value of f(k) = A095112(k)/k.

1, 1, 2, 1, 3, 2, 1, 3, 2, 4, 1, 5, 3, 2, 4, 1, 5, 3, 2, 4, 6, 1, 7, 5, 3, 2, 4, 6, 1, 7, 5, 3, 2, 4, 6, 8, 1, 7, 5, 3, 9, 2, 4, 6, 8, 1, 7, 5, 3, 9, 2, 10, 4, 6, 8, 1, 11, 7, 5, 3, 9, 2, 10, 4, 6, 8, 1, 11, 7, 5, 3, 9, 2, 10, 4, 6, 8, 12, 1, 13, 11, 7, 5, 3, 9, 2, 10, 4, 6, 8, 12

Offset

1,3

Comments

f(n) = Sum_{p prime divisor of n} Sum{i=1..v_p(n)} 1/p^i, where v_p denotes the p-adic valuation. f is injective.

Links

Table of n, a(n) for n=1..91.

Example

Array begins:
  1
  1  2
  1  3  2
  1  3  2  4
  1  5  3  2  4
  1  5  3  2  4  6
  1  7  5  3  2  4  6
  1  7  5  3  2  4  6  8
  1  7  5  3  9  2  4  6  8
  1  7  5  3  9  2 10  4  6  8
  1 11  7  5  3  9  2 10  4  6  8
  1 11  7  5  3  9  2 10  4  6  8 12
  1 13 11  7  5  3  9  2 10  4  6  8 12
...

Mathematica

f[n_]:=If[n==1, 0, Plus@@(Sum[1/#[[1]]^i, {i, #[[2]]}]&)/@FactorInteger[n]]
row[n_]:=SortBy[Range[n], f]
For[n=1, n<=13, n++, r=row[n]; For[i=1, i<=n, i++, WriteString["stdout", r[[i]], ", "]]]

Prog

(PARI) f(n) = sumdiv(n, d, (1==omega(d))*(n/d)); \\ A095112
cmpf(na, nb) = f(na)/na-f(nb)/nb;
tabl(nn) = {for (n=1, nn, print(Vec(vecsort([1..n], cmpf, 1))); ); } \\ Michel Marcus, Mar 28 2019

Crossrefs

Cf. A095112, A307088, A307187.

Sequence in context: A085014 A082074 A132283 * A256440 A088370 A328719

Adjacent sequences: A307078 A307079 A307080 * A307082 A307083 A307084

Keyword

nonn,tabl

Author

Luc Rousseau, Mar 23 2019

Status

approved