A274329 -id:A274329 - cp's OEIS frontend

A302389 A fractal-like sequence: erasing all pairs of contiguous terms that don't have a digit in common leaves the sequence unchanged.

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

Offset: 1

Eric Angelini and Lars Blomberg, May 03 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 > 1 not yet present inside another pair of parentheses;
  3) always end the content inside a pair of parentheses with the smallest integer Y > 1 not yet present inside another pair of parentheses such that X and Y have no digit in common;
  4) after a(1) = 1 and a(2) = 2, 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.

			Parentheses are added around each pair of terms that have no digit in common:
(1,2),(12,3),(13,4),(14,5),(15,6),(16,7),(17,8),(18,9),(19,20),(10,22),(21,30),(23,11),1,(31,24),2,12,(25,33),3,13,(32,40),4,14,
Erasing all the parenthesized contents yields
(...),(....),(....),(....),(....),(....),(....),(....),(.....),(.....),(.....),(.....),1,( .....),2,12,( .....),3,13,( .....),4,14,
We see that the remaining terms slowly rebuild the starting sequence.

Lars Blomberg, Table of n, a(n) for n = 1..998

For other erasing criteria, cf. A303845 (prime by concatenation), A303948 (pair sharing a digit), A274329 (pair summing up to a prime).

A303936 A fractal-like sequence: erasing all pairs of contiguous terms that do not sum up to a prime leaves the sequence unchanged.

Original entry on oeis.org

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

Offset: 1

Eric Angelini, May 03 2018

nonn

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 > 3 not yet present inside another pair of parentheses;
  3) always end the content inside a pair of parentheses with the smallest integer Y > 3 not yet present inside another pair of parentheses such that X and Y sum up to a composite number;
  4) after a(1) = 1, a(2) = 2 and a(3) = 3, 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 that don't sum up to a prime:
1, 2, 3, (4,5), (6,8), (9,7), 4, (13,11), (12,10), (19,14), 5, 6, (17,15), 8, 9, (20,16), 7, 4, 13,
Erasing all the parenthesized contents yields
1, 2, 3, (...), (...), (...), 4, (.....), (.....), (.....), 5, 6, (.....), 8, 9, (.....), 7, 4, 13,
We see that the remaining terms slowly rebuild the starting sequence.

Lars Blomberg, Table of n, a(n) for n = 1..1000

For other erasing criteria, cf. A303845 (prime by concatenation), A303948 (pair sharing a digit), A274329 (pair summing up to a prime), A302389 (pair having no digit in common).

A303950 A fractal-like sequence: erasing all pairs of contiguous terms that sum up to a Fibonacci number leaves the sequence unchanged.

Original entry on oeis.org

1, 2, 4, 9, 1, 3, 5, 2, 4, 6, 7, 9, 1, 3, 8, 13, 5, 2, 4, 6, 10, 11, 7, 9, 1, 3, 8, 12, 22, 13, 5, 2, 4, 6, 10, 14, 20, 11, 7, 9, 1, 3, 8, 12, 15, 19, 22, 13, 5, 2, 4, 6, 10, 14, 16, 18, 20, 11, 7, 9, 1, 3, 8, 12, 15, 17, 38, 19, 22, 13, 5, 2, 4, 6, 10, 14, 16

Offset: 1

Eric Angelini and Lars Blomberg, May 03 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 F not yet present inside another pair of parentheses;
  3) always end the content inside a pair of parentheses with the smallest integer I not yet present inside another pair of parentheses such that the sum F + I is a Fibonacci number;
  4) after 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 that sum up to a Fibonacci:
(1,2), (4,9), 1, (3,5), 2, 4, (6,7), 9, 1, 3, (8,13), 5, 2, 4, 6, (10,11), 7, 9, 1, 3, 8, (12,22), 13, 5, 2, 4, 6, 10, (14,20), 11, ...
Erasing all the parenthesized contents yields
(...), (...), 1, (...), 2, 4, (...), 9, 1, 3, (....), 5, 2, 4, 6, (.....), 7, 9, 1, 3, 8, (.....), 13, 5, 2, 4, 6, 10, (.....), 11, ...
We see that the remaining terms slowly rebuild the starting sequence.

Lars Blomberg, Table of n, a(n) for n = 1..998

Cf. A000045 (Fibonacci numbers).

For other "erasing criteria", cf. 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).

A303951 A fractal-like sequence: erasing all pairs of contiguous terms that don't sum up to a Fibonacci number leaves the sequence unchanged.

Original entry on oeis.org

1, 2, 3, 4, 9, 5, 3, 10, 6, 7, 8, 13, 11, 23, 12, 22, 14, 20, 15, 19, 16, 18, 17, 4, 9, 25, 21, 34, 24, 31, 26, 29, 27, 28, 30, 59, 32, 57, 33, 56, 35, 54, 36, 53, 37, 52, 38, 51, 39, 50, 40, 49, 41, 48, 42, 47, 43, 46, 44, 45, 55, 89, 58, 86, 60, 84, 61, 83

Offset: 1

Lars Blomberg and Eric Angelini, May 03 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 C > 2 not yet present inside another pair of parentheses;
  3) always end the content inside a pair of parentheses with the smallest integer I > 2 not yet present inside another pair of parentheses such that the sum C + I is not a Fibonacci number;
  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 that don't sum up to a Fibonacci:
1, 2, (3,4), (9,5), 3, (10,6), (7,8), (13,11), (23,12), (22,14), (20,15), (19,16), (18,17), 4, 9, (25,21), ...
Erasing all the parenthesized contents yields
1, 2, (...), (...), 3, (....), (...), (.....), (.....), (.....), (.....), (.....), (.....), 4, 9, (.....), ...
We see that the remaining terms slowly rebuild the starting sequence.

Lars Blomberg, Table of n, a(n) for n = 1..999

Cf. A000045 (Fibonacci numbers).

For other "erasing criteria", cf. 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).

A303953 A fractal-like sequence: erasing all pairs of contiguous terms that sum up to a square leaves the sequence unchanged.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 10, 4, 7, 9, 5, 6, 8, 17, 10, 4, 7, 11, 14, 9, 5, 6, 8, 12, 13, 17, 10, 4, 7, 11, 15, 21, 14, 9, 5, 6, 8, 12, 16, 20, 13, 17, 10, 4, 7, 11, 15, 18, 31, 21, 14, 9, 5, 6, 8, 12, 16, 19, 30, 20, 13, 17, 10, 4, 7, 11, 15, 18, 22, 27, 31, 21, 14

Offset: 1

Lars Blomberg and Eric Angelini, May 03 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 R > 3 not yet present inside another pair of parentheses;
  3) always end the content inside a pair of parentheses with the smallest integer E > 3 not yet present inside another pair of parentheses such that the sum R + E is a square number;
  4) after a(1) = 1, a(2) = 2, a(3) = 3, a(4) = 4 and a(5) = 5, always try to extend the sequence with a duplicate of the oldest term > 5 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 that sum up to a square:
1, 2, 3, (4,5), (6,10), 4, (7,9), 5, 6, (8,17), 10, 4, 7, (11,14), 9, 5, 6, 8, (12,13), 17, 10,
Erasing all the parenthesized contents yields
1, 2, 3, (...), (....), 4, (...), 5, 6, (....), 10, 4, 7, (.....), 9, 5, 6, 8, (.....), 17, 10,
We see that the remaining terms slowly rebuild the starting sequence.

Lars Blomberg, Table of n, a(n) for n = 1..998

Cf. A000290 (Square numbers).

For other "erasing criteria", cf. 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).

A303954 A fractal-like sequence: erasing all pairs of contiguous terms that don't sum up to a square leaves the sequence unchanged.

Original entry on oeis.org

1, 2, 7, 3, 1, 8, 4, 5, 6, 10, 9, 16, 11, 14, 12, 13, 15, 21, 17, 19, 18, 31, 20, 29, 22, 27, 23, 2, 7, 42, 24, 25, 26, 38, 28, 36, 30, 34, 32, 49, 33, 3, 1, 8, 41, 35, 46, 37, 44, 39, 61, 40, 60, 43, 57, 45, 4, 5, 59, 47, 53, 48, 52, 50, 71, 51, 70, 54, 67

Offset: 1

Lars Blomberg and Eric Angelini, May 03 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 N not yet present inside another pair of parentheses;
  3) always end the content inside a pair of parentheses with the smallest integer E not yet present inside another pair of parentheses such that the sum N + E is not a square number;
  4) after a(1) = 1 and a(2) = 2, always try to extend the sequence with a duplicate of the oldest term > 5 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 that don't sum up to a square:
(1,2), (7,3), 1, (8,4), (5,6), (10,9), (16,11), (14,12), (13,15), (21,17), (19,18), (31,20), (29,22), (27,23), 2, 7, (42,24),
Erasing all the parenthesized contents yields
(...), (...), 1, (...), (...), (....), (.....), (.....), (.....), (.....), (.....), (.....), (.....), (.....), 2, 7, (.....),
We see that the remaining terms slowly rebuild the starting sequence.

Lars Blomberg, Table of n, a(n) for n = 1..1000

Cf. A000290 (Square numbers).

For other "erasing criteria", cf. 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).

A351330 A fractal-like sequence: erase all triples of contiguous terms that have an odd sum; the remaining terms rebuild the starting sequence.

Original entry on oeis.org

1, 2, 4, 6, 8, 3, 1, 2, 5, 7, 9, 4, 11, 13, 15, 6, 17, 19, 21, 8, 3, 1, 2, 5, 7, 10, 23, 12, 9, 25, 14, 16, 4, 18, 20, 27, 11, 22, 29, 24, 13, 15, 6, 17, 19, 26, 31, 28, 21, 33, 30, 32, 8, 34, 36, 35, 3, 38, 37, 40, 1, 39, 42, 44, 2, 46, 48, 41, 5, 50, 43, 52, 7, 45, 54, 56, 10, 58, 60, 47, 23, 12, 9, 25, 14

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 odd;
  5) after a(1) = 1, a(2) = 2 and a(3) = 4, 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.

			Parentheses are added around each triple of terms that have an odd sum:
(1, 2, 4), (6, 8, 3), 1, 2, (5, 7, 9), 4, (11, 13, 15), 6, (17, 19, 21), 8, 3, 1, 2, 5, 7, (10, 23, 12), 9, (25, 14, 16), 4, (18, 20, 27), 11, (22, 29, 24), 13, 15, 6, 17, 19, (26, 31, 28), 21, (33, 30, 32), 8, (34, 36, 35), 3, (38, 37, 40), 1, (39, 42, 44), 2,...
Erasing all the parenthesized contents yields
(...), (...), 1, 2, (...), 4, (...), 6, (...), 8, 3, 1, 2, 5, 7, (...), 9, (...), 4, (...), 11, (...), 13, 15, 6, 17, 19, (...), 21, (...), 8, (...), 3, (...), 1, (...), 2,...
We see that the remaining terms slowly rebuild the starting sequence.

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), A351329 (triples having an even sum).

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).

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.

A338265 If a(n) has at least one common digit with a(n-1), underline a(n). The non-underlined terms reproduce the sequence itself.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 21, 20, 11, 31, 23, 22, 24, 25, 26, 27, 28, 29, 32, 30, 12, 41, 34, 33, 35, 36, 37, 38, 39, 43, 40, 13, 51, 45, 42, 44, 46, 47, 48, 49, 54, 50, 14, 61, 56, 52, 53, 55, 57, 58, 59, 65, 60, 15, 71, 67, 62, 63, 64, 66, 68, 69, 76, 70, 16, 81, 78, 72

Offset: 1

Eric Angelini, Oct 19 2020

This is a kind of fractal sequence as it contains an infinite number of copies of itself. The underlined terms of the sequence are the lexicographically earliest permutation of the natural numbers > 10.

			If we erase the parenthesized terms, we get the sequence itself: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, (11, 12, 13, 14, 15, 16, 17, 18, 19, 21, 20), 11, (31, 23, 22, 24, 25, 26, 27, 28, 29, 32, 30), 12, (41, 34, 33, 35, 36, 37, 38, 39, 43, 40), 13, (51, 45, 42, 44, 46, 47, 48, 49, 54, 50), 14, (61, 56, 52, 53, 55, 57, 58, 59, 65, 60), 15, (71, 67, 62, 63, 64, 66, 68, 69, 76, 70), 16, (81, 78, 72, ...) etc.

Cf. A274329.

cp's OEIS Frontend

A302389 A fractal-like sequence: erasing all pairs of contiguous terms that don't have a digit in common leaves the sequence unchanged.

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A303936 A fractal-like sequence: erasing all pairs of contiguous terms that do not sum up to a prime leaves the sequence unchanged.

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A303950 A fractal-like sequence: erasing all pairs of contiguous terms that sum up to a Fibonacci number leaves the sequence unchanged.

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A303951 A fractal-like sequence: erasing all pairs of contiguous terms that don't sum up to a Fibonacci number leaves the sequence unchanged.

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A303953 A fractal-like sequence: erasing all pairs of contiguous terms that sum up to a square leaves the sequence unchanged.

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A303954 A fractal-like sequence: erasing all pairs of contiguous terms that don't sum up to a square leaves the sequence unchanged.

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A351330 A fractal-like sequence: erase all triples of contiguous terms that have an odd sum; the remaining terms rebuild the starting sequence.

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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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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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Formula

A338265 If a(n) has at least one common digit with a(n-1), underline a(n). The non-underlined terms reproduce the sequence itself.

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