A335365 -id:A335365 - cp's OEIS frontend

A335155 Start with 1; if n is in the sequence, so are n+5 and 3*n.

1, 3, 6, 8, 9, 11, 13, 14, 16, 18, 19, 21, 23, 24, 26, 27, 28, 29, 31, 32, 33, 34, 36, 37, 38, 39, 41, 42, 43, 44, 46, 47, 48, 49, 51, 52, 53, 54, 56, 57, 58, 59, 61, 62, 63, 64, 66, 67, 68, 69, 71, 72, 73, 74, 76, 77, 78, 79, 81, 82, 83, 84, 86, 87, 88, 89, 91, 92, 93, 94, 96, 97, 98, 99, 101, 102, 103, 104, 106

Offset: 1

N. J. A. Sloane, Jun 05 2020

This is the lexicographically earliest sequence of positive numbers with the property that if n is in the sequence, so are n+5 and 3*n.

Suggested by A335365 (which is the complement).

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).

Cf. A335365.

See also A005658, A005660, A005662, etc.

Vec(x*(1 + 2*x + 3*x^2 + 2*x^3 - x^6 - x^7 + x^8 - x^10 + x^11 - x^13 + x^14 - x^15 - x^16) / ((1 - x)^2*(1 + x)*(1 + x^2)) + O(x^60)) \\ Colin Barker, Jun 07 2020

From Colin Barker, Jun 07 2020: (Start)
G.f.: x*(1 + 2*x + 3*x^2 + 2*x^3 - x^6 - x^7 + x^8 - x^10 + x^11 - x^13 + x^14 - x^15 - x^16) / ((1 - x)^2*(1 + x)*(1 + x^2)).
a(n) = a(n-1) + a(n-4) - a(n-5) for n>17.
(End)

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 }.

A335393 a(1) = 1 and for any n >= 1, a(2n) = a(n) + 5, a(2n+1) = a(n) * 3.

Original entry on oeis.org

1, 6, 3, 11, 18, 8, 9, 16, 33, 23, 54, 13, 24, 14, 27, 21, 48, 38, 99, 28, 69, 59, 162, 18, 39, 29, 72, 19, 42, 32, 81, 26, 63, 53, 144, 43, 114, 104, 297, 33, 84, 74, 207, 64, 177, 167, 486, 23, 54, 44, 117, 34, 87, 77, 216, 24, 57, 47, 126, 37, 96, 86, 243

Offset: 1

Rémy Sigrist, Jun 05 2020

nonn

All terms belong to A335155.

For any k > 0, the value k appears A335392(k) times.

			a(12) = a(6) + 5 = a(3) + 5 + 5 = a(1) * 3 + 5 + 5 = 1 * 3 + 5 + 5 = 13.

Rémy Sigrist, Table of n, a(n) for n = 1..8192

Cf. A335155, A335365, A335392.

a(n) = { if (n==1, 1, n%2==0, a(n\2)+5, a(n\2)*3) }

a(2^k) = 1 + 5*k for any k >= 0.

a(2^k-1) = 2^(k-1) for any k > 0.

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

Original entry on oeis.org

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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See Links section.
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A335155(n) = A335393(a(n)).

cp's OEIS Frontend

A335155 Start with 1; if n is in the sequence, so are n+5 and 3*n.

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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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

A335393 a(1) = 1 and for any n >= 1, a(2*n) = a(n) + 5, a(2*n+1) = a(n) * 3.

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PARI

Formula

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

A335393 a(1) = 1 and for any n >= 1, a(2n) = a(n) + 5, a(2n+1) = a(n) * 3.