A303845 -id:A303845 - cp's OEIS frontend

A304337 Lexicographically earliest fractal-like sequence such that the erasure of all pairs of contiguous terms of opposite parity leaves the sequence unchanged.

1, 2, 4, 3, 1, 5, 6, 2, 4, 8, 7, 3, 1, 5, 9, 10, 6, 2, 4, 8, 12, 11, 7, 3, 1, 5, 9, 13, 14, 10, 6, 2, 4, 8, 12, 16, 15, 11, 7, 3, 1, 5, 9, 13, 17, 18, 14, 10, 6, 2, 4, 8, 12, 16, 20, 19, 15, 11, 7, 3, 1, 5, 9, 13, 17, 21, 22, 18, 14, 10, 6, 2, 4, 8, 12, 16, 20, 24, 23, 19, 15, 11, 7, 3, 1

Offset: 1

Eric Angelini, May 11 2018

The sequence is fractal-like as it embeds an infinite number of copies of itself.
The sequence was built according to these rules (see, in the Example section, the parenthesization technique):
  1) no overlapping pairs of parentheses;
  2) always start the content inside a pair of parentheses with the smallest integer X not yet present inside another pair of parentheses;
  3) always end the content inside a pair of parentheses with the smallest integer Y not yet present inside another pair of parentheses such Y is not of the same parity as X;
  4) after a(1) = 1 and a(2) = 2, always try to extend the sequence with a duplicate of the oldest term of the sequence not yet duplicated; if this leads to a contradiction, open a new pair of parentheses.

			Parentheses are added around each pair of terms of opposite parity:
(1,2),(4,3),1,(5,6),2,4,(8,7),3,1,5,(9,10),6,2,4,8,(12,11),7,3,1,5,9,(13,14),10,6,2,4,8,12,(16,15),11,7,3,1,5,9,13,(17,18),14,10,6,
Erasing all the parenthesized contents yields
(...),(...),1,(...),2,4,(...),3,1,5,(....),6,2,4,8,(.....),7,3,1,5,9,(.....),10,6,2,4,8,12,(.....),11,7,3,1,5,9,13,(.....),14,10,6,
We see that the remaining terms slowly rebuild the starting sequence.

Carole Dubois, Table of n, a(n) for n = 1..5000

Cf. A303845 (same idea, but pairs of contiguous terms are erased if a prime by concatenation arises), A303948 (if pair has at least one digit in common), A303953 (if pair sums up to a square).

A316272 A fractal-like sequence: erasing all pairs of consecutive terms that include a prime and a composite number (in any order) leaves the sequence unchanged.

Original entry on oeis.org

1, 2, 3, 4, 1, 6, 5, 2, 3, 7, 8, 4, 1, 6, 9, 11, 5, 2, 3, 7, 13, 10, 8, 4, 1, 6, 9, 12, 17, 11, 5, 2, 3, 7, 13, 19, 14, 10, 8, 4, 1, 6, 9, 12, 15, 23, 17, 11, 5, 2, 3, 7, 13, 19, 29, 16, 14, 10, 8, 4, 1, 6, 9, 12, 15, 18, 31, 23, 17, 11, 5, 2, 3, 7, 13, 19, 29, 37, 20, 16, 14, 10, 8, 4, 1

Offset: 1

Eric Angelini and Jean-Marc Falcoz, Jun 28 2018

The sequence is fractal-like as it embeds an infinite number of copies of itself.
The sequence was built according to these rules (see, in the Example section, the parenthesization technique):
  1) no overlapping pairs of parentheses;
  2) always start the content inside a pair of parentheses either with the smallest prime P > 2 not yet present inside another pair of parentheses or with the smallest composite C > 1 not yet present inside another pair of parentheses ;
  3) always end the content inside a pair of parentheses either with the smallest composite C > 1 not yet present inside another pair of parentheses or with the smallest prime > 2 not yet present inside another pair of parentheses;
  4) after a(1) = 1 and a(2) = 2, always try to extend the sequence with a duplicate > 1 of the oldest term of the sequence not yet duplicated; if this leads to a contradiction, open a new pair of parentheses.

			Parentheses are added around each pair of terms made of a composite and a prime number (in any order):
(1,2),(3,4),1,(6,5),2,3,(7,8),4,1,6,(9,11),5,2,3,7,(13,10),8,4,1,6,9,(12,17),11,...
Erasing all the parenthesized contents yields
(...),(...),1,(...),2,3,(...),4,1,6,(....),5,2,3,7,(.....),8,4,1,6,9,(.....),11,...
We see that the remaining terms rebuild the starting sequence.

Eric Angelini, Table of n, a(n) for n = 1..20706

For other "erasing criteria", see A303845 (prime by concatenation), A274329 (pair summing up to a prime), A303936 (pair not summing up to a prime), A303948 (pair sharing a digit), A302389 (pair having no digit in common), A303950 (pair summing up to a Fibonacci), A303951 (pair not summing up to a Fibonacci), A303953 (pair summing up to a square), A303954 (pair not summing up to a square).

A342165 A fractal-like sequence: erase the terms that have a prime index, the non-erased terms rebuild the original sequence.

Original entry on oeis.org

1, 2, 3, 2, 4, 3, 5, 2, 4, 3, 6, 5, 7, 2, 4, 3, 8, 6, 9, 5, 7, 2, 10, 4, 3, 8, 6, 9, 11, 5, 12, 7, 2, 10, 4, 3, 13, 8, 6, 9, 14, 11, 15, 5, 12, 7, 16, 2, 10, 4, 3, 13, 17, 8, 6, 9, 14, 11, 18, 15, 19, 5, 12, 7, 16, 2, 20, 10, 4, 3, 21, 13, 22, 17, 8, 6, 9, 14

Offset: 1

Jean-Marc Falcoz and Eric Angelini, Mar 03 2021

To build the sequence, we start with a(1) = 1 and always extend it with the smallest integer not yet used, except in the case where the number is imposed by the constraint (i.e. if the index is nonprime). This fractal-like sequence takes arbitrarily large values.

			Original sequence: 1,2,3,2,4,3,5,2,4,3,6,5,7,2,4,3,8,6,9,5,7,2,10,4,3
Erasing: 1,(2,3,)2,(4,)3,(5,)2,4,3,(6,)5,(7,)2,4,3,(8,)6,(9,)5,7,2,(10,)4,3
Non-erased: 1,( )2,(  )3,(  )2,4,3,(  )5,(  )2,4,3,(  )6,(  )5,7,2,(   )4,3
The non-erased terms rebuild the original sequence.

Michael S. Branicky, Table of n, a(n) for n = 1..10000

Cf. A303845.

a(n) = while(n>1 && !isprime(n), n-=primepi(n)); primepi(n)+1; \\ Kevin Ryde, Mar 03 2021

from sympy import isprime
def aupton(terms):
  alst, idx = [1], 1
  for n in range(2, terms+1):
    if isprime(n): an = max(alst) + 1
    else: an, idx = alst[idx], idx + 1
    alst.append(an)
  return alst
print(aupton(78)) # Michael S. Branicky, Mar 03 2021

a(prime(i)) = i + 1. - Michael S. Branicky, Mar 04 2021

a(26) and beyond from Michael S. Branicky, Mar 03 2021

A351329 A fractal-like sequence: erase all triples of adjacent terms that have an even sum; the remaining terms rebuild the starting sequence.

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 1, 2, 10, 5, 7, 3, 9, 11, 12, 4, 13, 14, 15, 6, 8, 1, 2, 10, 5, 16, 18, 20, 7, 22, 24, 26, 3, 28, 30, 32, 9, 34, 36, 38, 11, 12, 4, 13, 14, 40, 17, 19, 15, 21, 23, 42, 6, 25, 44, 27, 8, 46, 29, 31, 1, 33, 35, 48, 2, 37, 50, 39, 10, 52, 41, 43, 5, 45, 47, 54, 16, 49, 56, 51, 18, 20, 7, 22

Offset: 1

Eric Angelini and Carole Dubois, Feb 07 2022

This is the lexicographically earliest such sequence starting with a(1) = 1 and showing no duplicate term in any triple to be erased.
The sequence is fractal-like as it embeds an infinite number of copies of itself.
The sequence was built according to these rules (see, in the Example section, the parenthesization technique):
  1) no overlapping triple of parentheses; a triple is made of integers X, Y and Z;
  2) always start the content inside a pair of parentheses with the smallest integer X > 1 not yet present inside another pair of parentheses and not leading to a contradiction;
  3) always follow X with the smallest integer Y > 1 not yet present inside another pair of parentheses and not leading to a contradiction;
  4) always end the content inside a pair of parentheses with the smallest integer Z > 1 not yet present inside another pair of parentheses and not leading to a contradiction such that X + Y + Z is even;
  5) after a(1) = 1, a(2) = 2 and a(3) = 3, always try to extend the sequence with a duplicate > 2 of the oldest term of the sequence not yet duplicated; if this leads to a contradiction, open a new pair of parentheses.

Eric Angelini, Fabriquons une suite fractale, January 22nd 2022, personal blog (in French).

For other erasing criteria, cf. A303845 (prime by concatenation), A303948 (pair sharing a digit), A274329 (pair summing up to a prime), A351330 (triples having an odd sum).

Parentheses are added around each triple of terms that have an even sum:
(1, 2, 3), (4, 6, 8), 1, 2, (10, 5, 7), 3, (9, 11, 12), 4, (13, 14, 15), 6, 8, 1, 2, 10, 5, (16, 18, 20), 7, (22, 24, 26), 3, (28, 30, 32), 9, (34, 36, 38), 11, 12, 4, 13, 14, (40, 17, 19), 15, (21, 23, 42), 6, (25, 44, 27), 8, (46, 29, 31), 1, ...
Erasing all the parenthesized contents yields
(...), (...), 1, 2, (...), 3, (...), 4, (...), 6, 8, 1, 2, 10, 5, (...), 7, (...), 3, (...), 9, (...), 11, 12, 4, 13, 14, (...), 15, (...), 6, (...), 8, (...), 1, ...
We see that the remaining terms slowly rebuild the starting sequence.

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A304337 Lexicographically earliest fractal-like sequence such that the erasure of all pairs of contiguous terms of opposite parity leaves the sequence unchanged.

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A316272 A fractal-like sequence: erasing all pairs of consecutive terms that include a prime and a composite number (in any order) leaves the sequence unchanged.

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A342165 A fractal-like sequence: erase the terms that have a prime index, the non-erased terms rebuild the original sequence.

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A351329 A fractal-like sequence: erase all triples of adjacent terms that have an even sum; the remaining terms rebuild the starting sequence.

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