A361376 Rewrite A129912(n), a product of distinct primorials P(i) = A002110(i) instead as a sum of powers 2^(i-1).

0, 1, 2, 3, 4, 5, 6, 8, 7, 9, 10, 16, 11, 17, 12, 13, 18, 19, 32, 14, 33, 20, 15, 21, 34, 35, 22, 24, 64, 23, 36, 25, 65, 37, 26, 66, 38, 27, 67, 40, 128, 39, 41, 28, 68, 129, 29, 69, 42, 130, 48, 43, 30, 70, 72, 131, 49, 31, 71, 44, 73, 256, 132, 45, 50, 257, 133, 74, 51, 46, 80, 75, 258, 134, 136

Offset

1,3

Comments

Permutation of nonnegative numbers.

Links

Michael De Vlieger, Table of n, a(n) for n = 1..15303 (a(15303) = 2^29.)

Michael De Vlieger, Log log scatterplot of a(n), n = 1..10^6.

Michael De Vlieger, Plot terms S(n) = A272011(a(n)) at (x,y) = (n,S(n,k)) for n = 1..2^11.

Formula

Let S(n) be the set of indices of primorials P(i), reverse sorted, such that A129912(n) = Product_{k=1..m} S(n,k), where m = | S(n) |. Then a(n) = Sum_{k=1..m} 2^(S(n,k)-1).

Example

a(1) = 0 by convention.
a(8) = 8 comes before a(9) = 7, since we interpret 8 = 2^3 instead as P(4) = 210, while for a(9), 7 = 2^2 + 2^1 + 2^0 becomes P(3)*P(2)*P(1) = 30*6*2 = 360. Because 210 < 360, 8 appears before 7 in this sequence.
Table relating a(n), n=1..19 with the set S(n) of indices of distinct primorial factors of A129912(n):
   n A129912(n)  S(n)   a(n)  A272011(a(n))
  -----------------------------------------
   1         1            0
   2         2   1        1   0
   3         6   2        2   1
   4        12   2,1      3   1,0
   5        30   3        4   2
   6        60   3,1      5   2,0
   7       180   3,2      6   2,1
   8       210   4        8   3
   9       360   3,2,1    7   2,1,0
  10       420   4,1      9   3,0
  11      1260   4,2     10   3,1
  12      2310   5       16   4
  13      2520   4,2,1   11   3,1,0
  14      4620   5,1     17   4,0
  15      6300   4,3     12   3,2
  16     12600   4,3,1   13   3,2,0
  17     13860   5,2     18   4,1
  18     27720   5,2,1   19   4,1,0
  19     30030   6       32   5
  ...

Mathematica

a6939[n_] := Product[Prime[n + 1 - i]^i, {i, n}];
g[m_] := Block[{f, j = 1},
  f[n_, i_, e_] :=
   If[n < m, Block[{p = Prime[i + 1]}, If[e == 1, Sow@ n];
     f[n p^e, i + 1, e];
     If[e > 1, f[n p^(e - 1), i + 1, e - 1]]]];
  Sort@ Reap[While[a6939[j] < m, f[2^j, 1, j]; j++]][[-1, 1]] ];
Map[Total@
     Map[2^(# - 1) &,
      Table[LengthWhile[#1, # >= j &], {j, #2}] & @@ {#, Max[#]} ] &[
FactorInteger[#][[All, -1]]] &, g[2^31]] (* Michael De Vlieger, Jun 08 2023, after Giovanni Resta at A129929 *)

Crossrefs

Cf. A002110, A129912, A272011, A283477.

Sequence in context: A215501 A117333 A368577 * A358143 A078840 A358124

Adjacent sequences: A361373 A361374 A361375 * A361377 A361378 A361379

Keyword

nonn

Author

Michael De Vlieger, Jun 08 2023

Status

approved