A329424 -id:A329424 - cp's OEIS frontend

A327539 Starting from n: as long as the decimal representation starts with a positive even number, divide the largest such prefix by 2; a(n) corresponds to the final value.

0, 1, 1, 3, 1, 5, 3, 7, 1, 9, 5, 11, 3, 13, 7, 15, 1, 17, 9, 19, 5, 11, 11, 13, 3, 15, 13, 17, 7, 19, 15, 31, 1, 33, 17, 35, 9, 37, 19, 39, 5, 11, 11, 13, 11, 15, 13, 17, 3, 19, 15, 51, 13, 53, 17, 55, 7, 57, 19, 59, 15, 31, 31, 33, 1, 35, 33, 37, 17, 39, 35, 71

Offset: 0

Rémy Sigrist, Nov 29 2019

For n > 0, as long as we have a number whose decimal representation is the concatenation of a positive even number, say u, and a possibly empty string of odd digits, say v, we replace this number with the concatenation of u/2 and v; eventually only odd digits remain.

			For n = 10000:
- 10000 gives 10000/2 = 5000,
- 5000 gives 5000/2 = 2500,
- 2500 gives 2500/2 = 1250,
- 1250 gives 125/2 = 625,
- 625 gives 62/2 followed by 5 = 315,
- 315 has only odd digits, so a(10000) = 315.

Rémy Sigrist, Table of n, a(n) for n = 0..10000
Rémy Sigrist, Logarithmic scatterplot of the first 1000000 terms
Rémy Sigrist, Scatterplot of the ordinal transform of the first 1000000 terms

See A329249, A329424 and A329428 for similar sequences.

Cf. A007283, A014261, A020714, A005009, A005010, A215145.

Array[FixedPoint[If[AllTrue[#, OddQ], FromDigits@ #, FromDigits@ Flatten@ Join[IntegerDigitsFromDigits[First[#]]/2, Last[#]] &@ TakeDrop[#, Position[#, ?EvenQ][[-1, -1]] ] ] &@ IntegerDigits[#] &, #] &, 71] (* _Michael De Vlieger, Dec 01 2019 *)

a(n) = if (n==0, 0, n%2==0, a(n/2), 10*a(n\10)+(n%10))

a(n) <= n with equality iff n = 0 or n belongs to A014261.
a(2*n) = a(n).
a(10*k + v) = 10*a(k) + v for any k >= 0 and v in {1, 3, 5, 7, 9}.
a(n) = 1 iff n is a power of 2.
a(n) = 3 iff n belongs to A007283.
a(n) = 5 iff n belongs to A020714.
a(n) = 7 iff n belongs to A005009.
a(n) = 9 iff n belongs to A005010.
a(n) = a(n+1) iff n belongs to A215145.

A330355 Starting from n: as long as the decimal representation contains a positive multiple of 3, divide the largest and leftmost such substring by 3; a(n) corresponds to the final value.

Original entry on oeis.org

0, 1, 2, 1, 4, 5, 2, 7, 8, 1, 10, 11, 4, 11, 14, 5, 4, 17, 2, 11, 20, 7, 22, 7, 8, 25, 22, 1, 28, 7, 10, 11, 4, 11, 14, 5, 4, 17, 2, 11, 40, 41, 14, 41, 44, 5, 14, 47, 4, 41, 50, 17, 52, 17, 2, 55, 52, 11, 58, 17, 20, 7, 22, 7, 8, 25, 22, 1, 28, 7, 70, 71, 8

Offset: 0

Rémy Sigrist, Dec 11 2019

This sequence is a variant of A329424.

			For n = 193:
- 193 gives 1 followed by 93/3 = 131,
- 131 gives 1 followed by 3/3 followed by 1 = 111,
- 111 gives 111/3 = 37,
- 37 gives 3/3 followed by 7 = 17,
- neither 1, 7 nor 17 are divisible by 3, so a(193) = 17.

Robert Israel, Table of n, a(n) for n = 0..10000
Rémy Sigrist, PARI program for A330355

Cf. A325112, A327539, A329424.

f:= proc(n) option remember; local L,m,i,d,np1,j,s;
  L:= convert(n,base,10);
  m:= nops(L);
  for d from m to 1 by -1 do
    for i from 1 to m-d+1 do
      s:= convert(L[i..i+d-1],`+`);
      if s > 0 and s mod 3 = 0 then
        np1:= add(L[j]*10^(j-1),j=1..i-1)+1/3*add(L[j]*10^(j-1),j=i..i+d-1);
        return procname(np1 + 10^(2+ilog10(np1)-(i+d))*add(L[j]*10^(j-1),j=i+d..m));
      fi
    od
  od;
  n
end proc:
map(f, [$0..100]); # Robert Israel, Dec 25 2019

PARI
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See Links section.
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a(n) <= n with equality iff n = 0 or n belongs to A325112.

a(3^k) = 1 for any k >= 0.

cp's OEIS Frontend

A327539 Starting from n: as long as the decimal representation starts with a positive even number, divide the largest such prefix by 2; a(n) corresponds to the final value.

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