A266117 -id:A266117 - cp's OEIS frontend

A266118 a(n) = index of n in A266117 or 0 if n is not present in that sequence.

1, 2, 3, 4, 5, 8, 9, 11, 12, 6, 7, 10, 14, 18, 23, 22, 19, 15, 16, 26, 21, 20, 27, 13, 31, 17, 37, 30, 28, 25, 59, 24, 39, 33, 70, 40, 41, 48, 47, 38, 35, 29, 74, 44, 64, 54, 53, 32, 45, 56, 49, 50, 34, 36, 51, 61, 62, 55, 52, 57, 42, 58, 66, 63, 43, 60, 75, 69, 68, 76, 77, 46, 73, 67, 85, 83, 94, 78, 71, 65, 89, 98, 136, 79, 119, 100

Offset: 1

Antti Karttunen, Dec 22 2015

It is conjectured that A266117 is not just injective, but also surjective on N, i.e., that it is a true permutation of natural numbers. In that case also this sequence is a true permutation, and no hypothetical zero-values are ever needed.

Antti Karttunen, Table of n, a(n) for n = 1..12634
Index entries for sequences that are permutations of the natural numbers

Inverse: A266117 (with provisions, see comment section).

A264990 a(n) = number of occurrences of a most frequent nonzero digit in factorial base representation (A007623) of n.

Original entry on oeis.org

0, 1, 1, 2, 1, 1, 1, 2, 2, 3, 1, 2, 1, 1, 1, 2, 2, 2, 1, 1, 1, 2, 1, 1, 1, 2, 2, 3, 1, 2, 2, 3, 3, 4, 2, 3, 1, 2, 2, 3, 2, 2, 1, 2, 2, 3, 1, 2, 1, 1, 1, 2, 2, 2, 1, 2, 2, 3, 2, 2, 2, 2, 2, 2, 3, 3, 1, 1, 1, 2, 2, 2, 1, 1, 1, 2, 1, 1, 1, 2, 2, 3, 1, 2, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 2, 1, 1, 1, 2, 2, 3, 1, 2, 1, 1, 1, 2, 2, 2, 1, 1, 1, 2, 1, 1, 1, 2

Offset: 0

Antti Karttunen, Dec 22 2015

			   n  A007623(n)   a(n) [highest number of times any nonzero digit occurs].
   0 =   0           0 (because no nonzero digits present)
   1 =   1           1
   2 =  10           1
   3 =  11           2
   4 =  20           1
   5 =  21           1
   6 = 100           1
   7 = 101           2
   8 = 110           2
   9 = 111           3
  10 = 120           1
  11 = 121           2
  12 = 200           1
  13 = 201           1
  14 = 210           1
  15 = 211           2
  16 = 220           2
  17 = 221           2
  18 = 300           1
and for n=63 we have:
  63 = 2211          2.

Antti Karttunen, Table of n, a(n) for n = 0..10080
Index entries for sequences related to factorial base representation.

Cf. A007623, A051903, A225901, A257511, A257684, A275735, A275811.
Cf. A265349 (positions of terms <= 1), A265350 (positions of term > 1).
Cf. also A266117, A266118.

a[n_] := Module[{k = n, m = 2, r, s = {}}, While[{k, r} = QuotientRemainder[k, m]; k != 0|| r != 0, AppendTo[s, r]; m++]; Max[Tally[Select[s, # > 0 &]][[;;,2]]]]; a[0] = 0; Array[a, 100, 0] (* Amiram Eldar, Jan 24 2024 *)

from sympy import prime, factorint
from operator import mul
import collections
def a007623(n, p=2): return n if nIndranil Ghosh, Jun 20 2017

a(0) = 0; for n >= 1, a(n) = max(A257511(n), a(A257684(n))).
Other identities. For all n >= 0:
From Antti Karttunen, Aug 15 2016: (Start)
a(n) = A275811(A225901(n)).
a(n) = A051903(A275735(n)).
(End)

Name changed by Antti Karttunen, Aug 15 2016

A265349 Numbers in whose factorial base representation (A007623) no digit > 0 occurs more than once.

Original entry on oeis.org

0, 1, 2, 4, 5, 6, 10, 12, 13, 14, 18, 19, 20, 22, 23, 24, 28, 36, 42, 46, 48, 49, 50, 54, 66, 67, 68, 72, 73, 74, 76, 77, 78, 82, 84, 85, 86, 96, 97, 98, 100, 101, 102, 106, 108, 109, 110, 114, 115, 116, 118, 119, 120, 124, 132, 138, 142, 168, 186, 192, 196, 204, 216, 220, 228, 234, 238, 240, 241, 242, 246, 258, 259, 260

Offset: 0

Antti Karttunen, Dec 22 2015

Zero is a special case in this sequence, thus a(0) = 0, and the indexing of natural numbers >= 1 present starts from a(1) = 1.

After a(0), positions of ones in A264990.

			23 (in factorial base "321") is present, because none of the digits (which all are nonzero) occurs twice.
48 (in factorial base "2000") is present, because the only nonzero digit, "2", occurs only once.
259 (in factorial base "20301") is present, because none of the nonzero digits occurs more than once.

Antti Karttunen, Table of n, a(n) for n = 0..10080
Indranil Ghosh, Python program for computing this sequence.
Index entries for sequences related to factorial base representation.

Cf. A007623, A264990.
Cf. A265350 (complement).
Cf. A000142 (a subsequence).
Cf. also A266117.

q[n_] := Module[{k = n, m = 2, r, s = {}}, While[{k, r} = QuotientRemainder[k, m]; k != 0|| r != 0, AppendTo[s, r]; m++]; Max[Tally[Select[s, # > 0 &]][[;;,2]]] == 1]; q[0] = True; Select[Range[0, 260], q] (* Amiram Eldar, Jan 24 2024 *)

A266121 Lexicographically first injection of natural numbers beginning with a(1)=1 such that 1+(a(n)*a(n+1)) is a fibbinary number (A003714), i.e., has no adjacent 1's in its base-2 representation.

Original entry on oeis.org

1, 3, 5, 4, 2, 8, 9, 7, 12, 6, 14, 24, 11, 13, 20, 16, 10, 26, 40, 17, 15, 39, 28, 19, 27, 25, 23, 48, 22, 30, 44, 31, 33, 32, 18, 36, 29, 47, 45, 52, 21, 55, 49, 84, 61, 43, 51, 53, 80, 34, 64, 37, 35, 59, 75, 117, 93, 91, 57, 41, 100, 82, 50, 104, 42, 98, 106, 90, 114, 72, 58, 144, 65, 63, 151, 56, 38, 54, 76, 71, 60

Offset: 1

Antti Karttunen, Dec 23 2015

It is conjectured that this sequence is not only injective, but also surjective on N, i.e., that it is a true permutation of natural numbers.
A similar sequence, but with condition that "(a(n)*a(n+1)) must be a member of A003714" yields a sequence: 1, 2, 4, 5, 8, 9, 16, 10, 13, 20, ... (A269361) which certainly is not a bijection, because it contains only terms of A091072.
Also, with above condition and the initial value a(1) = 3 the algorithm generates A269363 which contains only terms of A091067. See also A266191.

			After the initial a(1) = 1, for obtaining the value of a(2) we try the first unused number, which is 2, but (1*2)+1 = 3, which in binary is "11", thus 2 is not qualified at this point of time. So next we try 3, and (1*3)+1 = 4, which in binary is "100", and that satisfies our criterion (no adjacent 1-bits), thus we set a(2) = 3.
For a(3), we test with the least unused numbers 2, 4, 5, etc., yielding products (3*2)+1 = 7 = "111", (3*4)+1 = 13 = "1101" and (3*5)+1 = 16 = "10000" in binary, and only 5 satisfies the criterion, thus we set a(3) = 5.

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences that are permutations of the natural numbers

Left inverse: A266122 (also the right inverse if this sequence is a permutation of natural numbers).
Cf. A003714, A003987, A091067, A091072, A269361, A269363, A269365, A269366, A269367.
Cf. also A266191 and A266117 for similar permutations.

Minor typo in the description corrected by Antti Karttunen, Feb 25 2016

A266191 Sequence beginning with a(1)=1, a(2)=2, a(3)=3 and then after each new term a(n) is selected as the least unused number for which a(n)a(n-1)a(n-2) is a fibbinary number (A003714), i.e., has no adjacent 1's in its base-2 representation.

Original entry on oeis.org

1, 2, 3, 6, 4, 7, 12, 8, 11, 15, 16, 22, 24, 5, 39, 14, 10, 59, 28, 20, 67, 31, 9, 63, 19, 18, 27, 38, 17, 54, 47, 34, 44, 51, 33, 88, 30, 32, 43, 48, 21, 86, 23, 42, 35, 46, 84, 70, 91, 168, 71, 55, 36, 75, 99, 40, 135, 110, 41, 60, 111, 80, 120, 62, 160, 134, 56, 141, 76, 112, 64, 78, 119, 113, 103, 183, 37, 167, 366, 73

Offset: 1

Antti Karttunen, Dec 23 2015

It is conjectured that this sequence is not only injective, but also surjective on N, i.e., that it is a true permutation of natural numbers.

cp's OEIS Frontend

A266118 a(n) = index of n in A266117 or 0 if n is not present in that sequence.

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A264990 a(n) = number of occurrences of a most frequent nonzero digit in factorial base representation (A007623) of n.

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A265349 Numbers in whose factorial base representation (A007623) no digit > 0 occurs more than once.

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Mathematica

A266121 Lexicographically first injection of natural numbers beginning with a(1)=1 such that 1+(a(n)*a(n+1)) is a fibbinary number (A003714), i.e., has no adjacent 1's in its base-2 representation.

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Extensions

A266191 Sequence beginning with a(1)=1, a(2)=2, a(3)=3 and then after each new term a(n) is selected as the least unused number for which a(n)*a(n-1)*a(n-2) is a fibbinary number (A003714), i.e., has no adjacent 1's in its base-2 representation.

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Crossrefs

A266191 Sequence beginning with a(1)=1, a(2)=2, a(3)=3 and then after each new term a(n) is selected as the least unused number for which a(n)a(n-1)a(n-2) is a fibbinary number (A003714), i.e., has no adjacent 1's in its base-2 representation.