A302656 -id:A302656 - cp's OEIS frontend

A383320 Lexicographically earliest sequence of distinct terms such that replacing each term k with Fibonacci(k) does not change the succession of digits.

0, 1, 5, 43, 3, 4, 9, 44, 37, 2, 33, 470, 140, 8, 7, 332, 41, 57, 81, 71, 35, 24, 578, 74, 93, 86, 58, 6, 61, 14, 242, 47, 46, 936, 9310, 13, 87, 148, 48, 19, 30, 12, 55, 77, 36, 270, 246, 51, 68, 97, 194, 4350, 50, 27, 72, 31, 359, 90, 22, 40, 278, 505, 23

Offset: 1

Dominic McCarty, Apr 23 2025

			Let b(n) = Fibonacci(a(n))
(a(n)): 0, 1, 5, 43, 3, 4, 9, 44, 37, 2, ...
(b(n)): 0, 1, 5, 433494437,           2, ...

Dominic McCarty, Table of n, a(n) for n = 1..10000

Cf. A383321 (Fibonacci(a(n))), A038546, A383318, A383322, A302656.

from sympy import fibonacci
from itertools import count
a, sa, sb = [0,1,5,43], "01543", "015433494437"
for _ in range(30):
    a.append(next(n for k in count(1) if not (n := int(sb[len(sa):len(sa)+k])) in a and not (len(sb) > len(sa) + k and sb[len(sa) + k] == "0")))
    sa += str(a[-1]); sb += str(fibonacci(a[-1]))
print(a)

A383322 Lexicographically earliest sequence of distinct terms such that replacing each term k with k! does not change the succession of digits.

Original entry on oeis.org

1, 2, 198, 15, 5, 24, 3, 0, 56, 4, 800, 260, 18, 181, 7, 120, 43, 26, 25, 78, 46, 6, 11, 45, 67, 2580, 8, 37, 34, 49, 61, 66, 465, 63, 9, 28, 62, 93, 960, 65, 410, 626, 13, 82, 98, 59, 32, 659, 453, 242, 255, 580, 939, 42, 70, 44, 932, 22, 55, 38, 389, 50

Offset: 1

Dominic McCarty, Apr 23 2025

Similarly to A302656, this sequence contains very large jumps. For example, a(131) = 4*10^47, a(702) = 496*10^199, a(2808) = 5712*10^643, etc.

			Let b(n) = a(n)!
(a(n)): 1, 2, 198, 15, 5, 24, 3, 0, 56, 4, 800, 260, 18, ...
(b(n)): 1, 2, 198155243056480026018[...] (350 digits omitted), ...

Dominic McCarty, Table of n, a(n) for n = 1..10000
Dominic McCarty, Table of n, a(n)! for n = 1..100

Cf. A033147, A383318, A383320, A302656.

from sympy import factorial
from itertools import count
a, sa, sb = [1, 2, 198], "12198", "12"+str(factorial(198))
for _ in range(20):
    a.append(next(n for k in count(1) if not (n := int(sb[len(sa):len(sa)+k])) in a and (0 not in a or not (len(sb) > len(sa) + k and sb[len(sa) + k] == "0"))))
    sa += str(a[-1]); sb += str(factorial(a[-1]))
print(a)

A376770 a(n) = digitsum of A376769(n).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 9, 1, 8, 1, 8, 1, 9, 8, 9, 1, 2, 7, 2, 6, 3, 6, 10, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 2, 7, 2, 6, 3, 2, 4, 10, 9, 4, 5, 5, 4, 6, 3, 7, 2, 8, 1, 9, 1, 8, 2, 7, 3, 6, 4, 5, 2, 9, 7, 9, 2, 9, 9, 6, 9, 3, 9, 1, 2, 3, 2, 8, 9, 5, 4, 4, 5, 5, 3, 4, 3, 3, 7, 1, 1, 3, 5, 1, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 2, 8, 4

Offset: 1

N. J. A. Sloane, Nov 04 2024

Michael S. Branicky, Table of n, a(n) for n = 1..1982

Cf. A007953, A302656, A376769.

Summary: the 16 sequences derived from A302656 are A376769-A376776, A377903-A377904, A377906-A377911.

A376773 Records in A376772.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 12, 21, 25, 44, 48, 52, 53, 58, 75, 77, 84, 96, 135, 146, 300, 317, 401, 452, 478, 1608, 1677, 1679, 1681, 1683, 1703, 1753, 1773, 13649, 13704, 124912, 124925, 125336, 128212, 128221, 128347, 128376, 128529

Offset: 1

N. J. A. Sloane, Nov 05 2024 (with thanks to Michael S. Branicky)

Numbers that are the slowest to appear in A302656.

Cf. A302656, A376772, A376774, A376775.

Summary: the 16 sequences derived from A302656 are A376769-A376776, A377903-A377904, A377906-A377911.

a(37)-a(46) from Dominic McCarty, Nov 08 2024

A376774 Indices of records in A376772.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 21, 25, 28, 30, 37, 40, 41, 50, 55, 57, 66, 67, 76, 85, 93, 94, 139, 148, 157, 165, 174, 175, 179, 188, 197, 269, 278, 279, 288, 297, 369, 378

Offset: 1

N. J. A. Sloane, Nov 05 2024 (with thanks to Michael S. Branicky)

Cf. A302656, A376772, A376773.

Summary: the 16 sequences derived from A302656 are A376769-A376776, A377903-A377904, A377906-A377911.

a(37)-a(46) from Dominic McCarty, Nov 08 2024

A383319 a(n) = prime(A383318(n)).

Original entry on oeis.org

64553, 5, 11, 2, 37, 157, 47, 17, 7, 353, 389, 149, 137, 19, 23, 43, 467, 2539, 479, 139, 2339, 359, 241, 491, 401, 41, 7219, 167, 2417, 3, 179, 227, 809, 811, 5449, 7159, 5479, 59, 127, 6073, 103, 409, 521, 31, 251, 3547, 937, 3943, 499, 7121, 5791, 367, 29

Offset: 1

Dominic McCarty, Apr 23 2025

Dominic McCarty, Table of n, a(n) for n = 1..10000

Cf. A383318, A067928, A383320, A383322, A302656.

from sympy import prime
from itertools import count
a, b, sa, sb = [6455], [64553], "6455", "64553"
for _ in range(30):
    a.append(next(n for k in count(1) if not (n := int(sb[len(sa):len(sa)+k])) in a and not (len(sb) > len(sa) + k and sb[len(sa) + k] == "0")))
    b.append(prime(a[-1]))
    sa += str(a[-1]); sb += str(b[-1])
print(b)

A383321 a(n) = Fibonacci(A383320(n)).

Original entry on oeis.org

0, 1, 5, 433494437, 2, 3, 34, 701408733, 24157817, 1, 3524578, 74938658661142424746936931013871484819301255773627024651689719443505027723135990224027850523592585, 81055900096023504197206408605, 21, 13

Offset: 1

Dominic McCarty, Apr 23 2025

Dominic McCarty, Table of n, a(n) for n = 1..34

Cf. A383320, A038546, A383318, A383322, A302656.

from sympy import fibonacci
from itertools import count
a, b, sa, sb = [0,1,5,43], [0,1,5,433494437], "01543", "015433494437"
for _ in range(10):
    a.append(next(n for k in count(1) if not (n := int(sb[len(sa):len(sa)+k])) in a and not (len(sb) > len(sa) + k and sb[len(sa) + k] == "0")))
    b.append(fibonacci(a[-1]))
    sa += str(a[-1]); sb += str(b[-1])
print(b)

A302799 Lexicographically earliest sequence of distinct terms such that adding 10 to each term produces a new sequence that has exactly the same succession of digits as the present one.

Original entry on oeis.org

1, 12, 2, 121, 3, 11, 32, 14, 22, 4, 321, 43, 31, 5, 34, 115, 44, 125, 54, 13, 56, 42, 36, 6, 52, 46, 16, 62, 562, 67, 25, 7, 27, 73, 51, 737, 8, 361, 74, 71, 83, 718, 48, 19, 37, 28, 58, 29, 47, 38, 68, 39, 57, 487, 84, 9, 674, 97, 94, 196, 8410, 710, 420, 684, 20, 720, 430, 69, 4307, 30

Offset: 1

Eric Angelini and Hans Havermann, Apr 13 2018

The sequence starts with a(1) = 1 and is always extended with the smallest integer not yet present that doesn't lead to a contradiction.

			    1 = a(1) is replaced by   1 + 10 =  11
   12 = a(2) is replaced by  12 + 10 =  22
    2 = a(3) is replaced by   2 + 10 =  12
  121 = a(4) is replaced by 121 + 10 = 131
    3 = a(5) is replaced by   3 + 10 =  13
   11 = a(6) is replaced by  11 + 10 =  21
   32 = a(7) is replaced by  32 + 10 =  42
   14 = a(8) is replaced by  14 + 10 =  24
etc.
We see that the first and the last column here (which are respectively the terms of the present sequence and the terms of the transformed one) share the same succession of digits (so far): 1,1,2,2,1,2,1,3,1,1,3,2,1,4,2,2,4,...

Hans Havermann, Table of n, a(n) for n = 1..1000
Hans Havermann, Graph of the first 100000 terms

Cf. A302656 for another transformation in the same spirit that preserves the succession of digits in the sequence.

A303002 Replacing each term of this sequence S with the product of its digits produces a new sequence S' such that S' and S share the same succession of digits.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 26, 16, 28, 12, 18, 34, 29, 13, 14, 21, 19, 111, 31, 27, 37, 1111, 33, 11111, 111111, 1111111, 113, 43, 17, 131, 71, 11111111, 111111111, 1111111111, 11111111111, 311, 1113, 111111111111, 1111111111111, 11111111111111, 111111111111111, 1111111111111111, 11111111111111111, 111111111111111111, 1111111111111111111

Offset: 1

Eric Angelini and Jean-Marc Falcoz, Apr 17 2018

The sequence starts with a(1) = 1 and is always extended with the smallest integer not yet present that doesn't lead to a contradiction.

Huge repunits appear quickly and leave almost no space for non-repunits in the sequence; a(112) = A002275(82), a(113) = 3111, a(114) = A002275(83) and nothing but repunits will show from there until at least a(303) = A002275(350).

			The first nine terms are replaced by themselves;
11 = a(10) is replaced by the product 1 * 1 =  1;
26 = a(11) is replaced by the product 2 * 6 = 12;
16 = a(12) is replaced by the product 1 * 6 =  6;
28 = a(13) is replaced by the product 2 * 8 = 16;
12 = a(14) is replaced by the product 1 * 2 =  2;
18 = a(15) is replaced by the product 1 * 8 =  8;
34 = a(16) is replaced by the product 3 * 4 = 12;
29 = a(17) is replaced by the product 2 * 9 = 18;
13 = a(18) is replaced by the product 1 * 3 =  3;
14 = a(19) is replaced by the product 1 * 4 =  4;
etc.
We see that the first and the last column here (the terms of S and S') share the same succession of digits: 1,1,2,6,1,6,2,8,1,2,1,8,3,4,...

Jean-Marc Falcoz, Table of n, a(n) for n = 1..302 (shortened by _N. J. A. Sloane_, Jan 18 2019)

Cf. A302656 where the word "product" is replaced by "sum".

Cf. A002275 (repunits).

A340348 Add the two terms of the first pair of integers to produce b(1); add the two terms of the second pair of integers to produce b(2); add the two terms of the third pair of integers to produce b(3), etc. This sequence and the b(n) sequence share the same succession of digits.

Original entry on oeis.org

1, 109, 2, 7, 3, 24, 4, 28, 5, 39, 6, 22, 8, 45, 9, 87, 10, 12, 11, 73, 13, 46, 14, 8696, 15, 106, 16, 157, 17, 117, 18, 43, 19, 29, 20, 49, 21, 40, 23, 487, 25, 36, 26, 35, 27, 30, 31, 140, 32, 139, 33, 51, 34, 285, 37, 2883, 38, 454, 41, 99, 42, 192, 44, 828, 47, 489, 48, 215, 50, 477, 52, 251, 53, 1087, 54, 267

Offset: 1

Eric Angelini and Carole Dubois, Jan 05 2021

This is the lexicographically earliest sequence of distinct integers > 0 with this property. The sequence is conjectured to be a permutation of the integers > 0.

			1 + 109 = 110
2 + 7 = 9
3 + 24 = 27
4 + 28 = 32
5 + 39 = 44
6 + 22 = 28
8 + 45 = 53; etc.
The 3rd column and the starting sequence share the same succession of digits.

Carole Dubois, Table of n, a(n) for n = 1..10000

Cf. A302656.

cp's OEIS Frontend

A383320 Lexicographically earliest sequence of distinct terms such that replacing each term k with Fibonacci(k) does not change the succession of digits.

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A383322 Lexicographically earliest sequence of distinct terms such that replacing each term k with k! does not change the succession of digits.

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A376770 a(n) = digitsum of A376769(n).

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A376773 Records in A376772.

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A376774 Indices of records in A376772.

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A383319 a(n) = prime(A383318(n)).

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A383321 a(n) = Fibonacci(A383320(n)).

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A302799 Lexicographically earliest sequence of distinct terms such that adding 10 to each term produces a new sequence that has exactly the same succession of digits as the present one.

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A303002 Replacing each term of this sequence S with the product of its digits produces a new sequence S' such that S' and S share the same succession of digits.

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A340348 Add the two terms of the first pair of integers to produce b(1); add the two terms of the second pair of integers to produce b(2); add the two terms of the third pair of integers to produce b(3), etc. This sequence and the b(n) sequence share the same succession of digits.

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