A335393 -id:A335393 - cp's OEIS frontend

A335409 a(n) is the least k such that A335155(n) = A335393(k).

1, 3, 2, 6, 7, 4, 12, 14, 8, 5, 28, 16, 10, 13, 32, 15, 20, 26, 64, 30, 9, 52, 128, 60, 18, 25, 256, 29, 36, 50, 512, 58, 17, 100, 1024, 116, 34, 11, 2048, 57, 68, 22, 4096, 114, 33, 44, 8192, 228, 66, 21, 16384, 27, 132, 42, 32768, 54, 65, 84, 31, 108, 130

Offset: 1

Rémy Sigrist, Jun 06 2020

For any n > 0, the binary representation of a(n) encodes a minimal way (in the sense of the number of operations) of obtaining A335155(n) by starting from 1 and then repeatedly adding 5 or multiplying by 3; the leading 1 corresponds to the starting value 1, and then the 0's correspond to adding 5 and the 1's correspond to multiplying by 3.

			The first terms, alongside their binary representation and A335155(n), are:
  n   a(n)  bin(a(n))  A335155(n)
  --  ----  ---------  ----------
   1     1          1   1 = 1
   2     3         11   3 = 1*3
   3     2         10   6 = 1+5
   4     6        110   8 = (1*3)+5
   5     7        111   9 = 1*3*3
   6     4        100  11 = 1+5+5
   7    12       1100  13 = (1*3)+5+5
   8    14       1110  14 = (1*3*3)+5
   9     8       1000  16 = 1+5+5+5
  10     5        101  18 = (1+5)*3
  11    28      11100  19 = (1*3*3)+5+5
  12    16      10000  21 = 1+5+5+5+5

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, PARI program for A335409

Cf. A335155, A335365, A335393.

PARI
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A335155(n) = A335393(a(n)).

A335392 a(n) is the number of ways to reach n by the process of starting from 1 and repeatedly adding 5 or multiplying by 3.

Original entry on oeis.org

1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 0, 2, 1, 0, 1, 0, 2, 2, 0, 1, 1, 2, 2, 0, 1, 1, 3, 2, 0, 1, 1, 3, 3, 0, 1, 2, 3, 3, 0, 1, 2, 4, 3, 0, 1, 2, 4, 5, 0, 1, 3, 4, 5, 0, 1, 3, 5, 5, 0, 1, 3, 5, 7, 0, 1, 5, 5, 7, 0, 1, 5, 6, 7, 0, 2, 5, 6, 9, 0, 2, 7

Offset: 1

Rémy Sigrist, Jun 05 2020

nonn

This sequence has connections with A018819, the number of ways to reach a number by the process of starting from 1 and repeatedly adding 1 or multiplying by 2.

			For n = 18:
- 18 can be expressed as (1+5)*3 and 1*3 + 5 + 5 + 5,
- so a(18) = 2.

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, Colored logarithmic scatterplot of (n, a(n)) for n = 1..100000 (where the color is function of n mod 5)

Cf. A018819, A335365, A335393.

for (n=1, #a=vector(87), print1 (a[n]=if (n==1, 1, if (n-5>0, a[n-5], 0)+if (n%3==0, a[n/3], 0))", "))

a(n) = 0 iff n belongs to A335365.

a(n) = #{ k > 0 such that A335393(k) = n }.

cp's OEIS Frontend

A335409 a(n) is the least k such that A335155(n) = A335393(k).

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