A091020 -id:A091020 - cp's OEIS frontend

A286417 Lexicographically earliest sequence of distinct positive terms such that the binary representation of the n-th prime contains the binary representation of a(n).

1, 3, 2, 7, 5, 6, 4, 9, 11, 13, 15, 18, 10, 21, 23, 26, 14, 29, 8, 17, 36, 19, 20, 12, 16, 25, 39, 43, 22, 24, 31, 32, 34, 69, 37, 75, 78, 35, 41, 45, 44, 53, 47, 48, 49, 71, 52, 27, 28, 50, 58, 55, 30, 59, 64, 65, 33, 67, 138, 70, 141, 73, 38, 77, 57, 61, 82

Offset: 1

Rémy Sigrist, May 08 2017

This sequence is a permutation of the natural numbers.
As for A160855, the scatterplot of this sequence shows (almost) straight lines:
- here those lines are related to the position of a(n) in the n-th prime (in binary representation),
- the scatterplot of the possible values of a(n) as a function of the n-th prime constitutes a network of lines in which the scatterplot of a(n) lies,
- see also the scatterplots in the Links section.
For any n>0, a(n) <= A000040(n); the first known values where equality occurs are a(2) = 3 and a(4) = 7.
The fixed points of the sequence belong to A091020.
The first fixed points are: 1, 5, 6, 31, 32, 50, 81, 1052, 3378.

			See illustration of the first terms in the Links section.

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, Illustration of the first terms
Rémy Sigrist, Colorized scatterplot of the first 100000 terms
Rémy Sigrist, Scatterplot of the numbers appearing in the binary representation of the first 10000 prime numbers (this scatterplot contains the scatterplot of a)
Rémy Sigrist, PARI program for A286417
Index entries for sequences that are permutations of the natural numbers

Cf. A000040, A091020, A160855.

A091021 Primes p=prime(k) such that in binary representation k is a substring of p.

Original entry on oeis.org

2, 11, 13, 47, 127, 131, 139, 167, 227, 229, 419, 421, 8419, 15391, 18503, 23053, 31333, 32119, 642049, 771769, 1240081, 2545453, 2951341, 9677999, 11390809, 22436899, 33056171, 63701437, 71298761, 465462769, 989883217, 1490754379, 1775394377, 1775394389

Offset: 1

Reinhard Zumkeller, Dec 14 2003

			229 = A000040(50): 50->110010, 229->11100101 = 1'110010'1, therefore 229 is a term.

Cf. A007088, A004676.

from sympy import primerange
[print(i, end=', ') for n, i in enumerate(primerange(1, 10**7)) if bin(n+1)[2:] in bin(i)[2:]]; # Nicholas Stefan Georgescu, Jan 03 2025

a(n) = A000040(A091020(n)).

A091020(n) = A049084(a(n)).

a(22)-a(32) from Donovan Johnson, May 03 2010

a(33)-a(34) from Donovan Johnson, May 08 2012

cp's OEIS Frontend

A286417 Lexicographically earliest sequence of distinct positive terms such that the binary representation of the n-th prime contains the binary representation of a(n).

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A091021 Primes p=prime(k) such that in binary representation k is a substring of p.

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