A297409 Lexicographically earliest sequence of distinct positive terms such that for any n > 1 and any prime number p, if p divides a(n) then the p-adic valuation of a(n) is one plus the p-adic valuation of a(n-1).

1, 2, 3, 5, 6, 4, 7, 10, 11, 13, 14, 12, 8, 15, 9, 17, 19, 21, 18, 20, 23, 22, 28, 24, 16, 29, 26, 31, 30, 25, 33, 34, 35, 37, 38, 39, 41, 42, 36, 27, 43, 46, 44, 40, 47, 51, 45, 50, 52, 53, 55, 57, 58, 59, 61, 62, 60, 56, 48, 32, 64, 65, 66, 63, 49, 67, 69

Offset

1,2

Comments

For any prime p and any run of k contiguous terms divisible by p, say a(m+1), a(m+2), ..., a(m+k), we have v_p(a(m+i)) = i for i = 1..k and v_p(a(m)) = 0 and v_p(a(m+k+1)) = 0 (where v_p denotes the p-adic valuation of a number).

For any n > 0, A003557(a(n+1)) is a unitary divisor of a(n).

This sequence is likely a permutation of the natural numbers.

Links

Rémy Sigrist, Table of n, a(n) for n = 1..10000

Rémy Sigrist, PARI program for A297409

Example

The run of multiples of 2 that contains 2^16, alongside its bounding terms, is:
  n       a(n)    Prime factorization of a(n)
  -----   -----   ---------------------------
  67528   67561   13 * 5197
  67529   67562   2 * 11 * 37 * 83
  67530   67556   2^2 * 16889
  67531   67560   2^3 * 3 * 5 * 563
  67532   67536   2^4 * 3^2 * 7 * 67
  67533   67424   2^5       * 7^2 * 43
  67534   65856   2^6 * 3   * 7^3
  67535   67456   2^7 * 17 * 31
  67536   67328   2^8 * 263
  67537   67072   2^9 * 131
  67538   66560   2^10 * 5 * 13
  67539   63488   2^11 * 31
  67540   61440   2^12 * 3 * 5
  67541   57344   2^13 * 7
  67542   49152   2^14 * 3
  67543   32768   2^15
  67544   65536   2^16
  67545   67565   5 * 13513

Maple

N:= 200: # to get terms before the first term > N
Avail:= [$2..N]:
A[1]:= 1:
found:= true;
for n from 2 while found do
  found:= false;
  for j from 1 to nops(Avail) do
    m:= Avail[j];
    Fm:= ifactors(m)[2];
    if andmap(t -> A[n-1] mod t[1]^(t[2]-1)=0 and A[n-1] mod t[1]^t[2] <> 0, Fm) then
       found:= true;
       A[n]:= m;
       Avail:= subsop(j=NULL, Avail);
       break
    fi
  od;
od:
seq(A[i], i=1..n-1); # Robert Israel, Jan 07 2018

Prog

(PARI) See Links section.

Crossrefs

Cf. A003557.

Sequence in context: A294269 A330647 A209260 * A335858 A137760 A368428

Adjacent sequences: A297406 A297407 A297408 * A297410 A297411 A297412

Keyword

nonn,look

Author

Rémy Sigrist, Dec 29 2017

Status

approved