A303953 -id:A303953 - cp's OEIS frontend

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

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).

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

Original entry on oeis.org

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).

cp's OEIS Frontend

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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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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Author

Keywords

Comments

Examples

Links

Crossrefs