A329428 -id:A329428 - 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.

A330356 Starting from n: as long as the decimal representation contains a prime number, replace the largest and leftmost such substring with the index of the corresponding prime number; a(n) corresponds to the final value.

Original entry on oeis.org

0, 1, 1, 1, 4, 1, 6, 4, 8, 9, 10, 1, 1, 6, 14, 6, 16, 4, 18, 8, 10, 1, 1, 9, 14, 9, 16, 14, 18, 10, 10, 1, 1, 9, 14, 9, 16, 1, 18, 10, 40, 6, 6, 14, 44, 14, 46, 6, 48, 49, 10, 1, 1, 16, 14, 9, 16, 14, 18, 4, 60, 18, 18, 18, 64, 18, 66, 8, 68, 69, 40, 10, 6, 1

Offset: 0

Rémy Sigrist, Dec 11 2019

This sequence is a variant of A329428.

			For n = 8601:
- let pi = A000720,
- 8601 gives 8 followed by pi(601) = 8110,
- 8110 gives pi(811) followed by 0 = 1410,
- 1410 gives 1 followed by pi(41) followed by 0 = 1130,
- 1130 gives pi(113) followed by 0 = 300,
- 300 gives pi(3) followed by 00 = 200,
- 200 gives pi(2) followed by 00 = 100,
- no prime number appears in 100,
- hence a(8601) = 100.

Rémy Sigrist, Table of n, a(n) for n = 0..10000
Rémy Sigrist, PARI program for A330356

Cf. A000720, A007097, A062115, A327539, A329428.

PARI
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See Links section.
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a(n) <= n with equality iff n belongs to A062115.
a(prime(k)) = a(k) for any k > 0 where prime(k) denotes the k-th prime number.
a(A007097(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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