A276583 -id:A276583 - cp's OEIS frontend

A276585 After a(0)=0, the first differences of A276583.

0, 3, 3, 2, 3, 4, 3, 3, 3, 3, 5, 3, 3, 5, 5, 3, 5, 3, 4, 3, 5, 3, 4, 2, 3, 5, 3, 4, 4, 3, 5, 3, 4, 3, 3, 3, 5, 3, 4, 3, 5, 3, 5, 3, 4, 3, 3, 4, 3, 5, 3, 4, 3, 6, 3, 3, 5, 3, 4, 3, 6, 5, 3, 5, 3, 4, 3, 6, 3, 4, 3, 5, 3, 4, 3, 6, 3, 6, 3, 5, 3, 4, 3, 6, 3, 5, 3, 3, 5, 3, 4, 3, 6, 3, 5, 3, 2, 3, 5, 3, 4, 3, 6, 3, 5, 3, 4, 3, 5, 3, 4, 3, 6, 3, 5, 3, 3, 3, 3, 5, 3

Offset: 0

Antti Karttunen, Sep 07 2016

nonn

Antti Karttunen, Table of n, a(n) for n = 0..10095

Cf. A053610, A276583.

a(n) = A053610(A276583(n)).

a(0) = 0; for n >= 1, a(n) = A276583(n) - A276583(n-1).

A179016 The infinite trunk of binary beanstalk: The only infinite sequence such that a(n-1) = a(n) - number of 1's in binary representation of a(n).

Original entry on oeis.org

0, 1, 3, 4, 7, 8, 11, 15, 16, 19, 23, 26, 31, 32, 35, 39, 42, 46, 49, 53, 57, 63, 64, 67, 71, 74, 78, 81, 85, 89, 94, 97, 101, 104, 109, 112, 116, 120, 127, 128, 131, 135, 138, 142, 145, 149, 153, 158, 161, 165, 168, 173, 176, 180, 184, 190, 193, 197, 200, 205, 209

Offset: 0

Carl R. White, Jun 24 2010

a(n) tells in what number we end in n steps, when we start climbing up the infinite trunk of the "binary beanstalk" from its root (zero). The name "beanstalk" is due to Antti Karttunen.

There are many finite sequences such as 0,1,2; 0,1,3,4,7,9; etc. obeying the same condition (see A218254) and as the length increases, so (necessarily) does the similarity to this infinite sequence.

Alois P. Heinz and Antti Karttunen, Table of n, a(n) for n = 0..16405 (first 1000 terms from Alois P. Heinz)
Paul Tek, Illustration of the first terms

Cf. A000120, A010062, A011371, A213710, A213711, A213717, A213730, A213731, A218600, A218616, A218789, A233271, A218602, A054429. First differences: A213712, complement: A213713.
A subsequence of A005187, i.e., a(n) = A005187(A213715(n)). For all n,
A071542(a(n)) = n, and furthermore A213708(n) <= a(n) <= A173601(n). (Cf. A218603, A218604).
Rows of A218254, when reversed, converge towards this sequence.
Cf. A276623, A219648, A219666, A255056, A276573, A276583, A276613 for analogous constructions, and also A259934.

TakeWhile[Reverse@ NestWhileList[# - DigitCount[#, 2, 1] &, 10^3, # > 0 &], # <= 209 &] (* Michael De Vlieger, Sep 12 2016 *)

a(0)=0, a(1)=1, and for n > 1, if n = A218600(A213711(n)) then a(n) = (2^A213711(n)) - 1, and in other cases, a(n) = a(n+1) - A213712(n+1). (This formula is based on Carl White's observation that this iterated/converging path must pass through each (2^n)-1. However, it would be very interesting to know whether the sequence admits more traditional recurrence(s), referring to previous, not to further terms in the sequence in their definition!) - Antti Karttunen, Oct 26 2012
a(n) = A218616(A218602(n)). - Antti Karttunen, Mar 04 2013
a(n) = A054429(A233271(A218602(n))). - Antti Karttunen, Dec 12 2013

Starting offset changed from 1 to 0 by Antti Karttunen, Nov 05 2012

A276573 The infinite trunk of least squares beanstalk: The only infinite sequence such that a(0) = 0 and a(n-1) = a(n) - least number of squares (A002828) that sum to a(n).

Original entry on oeis.org

0, 3, 6, 8, 11, 15, 16, 18, 21, 24, 27, 30, 32, 35, 38, 40, 43, 45, 48, 51, 53, 56, 59, 63, 64, 67, 70, 72, 75, 78, 80, 83, 85, 88, 90, 93, 96, 99, 102, 105, 108, 112, 115, 117, 120, 123, 126, 128, 131, 134, 136, 139, 143, 144, 147, 149, 152, 155, 158, 160, 162, 165, 168, 171, 173, 176, 179, 183, 186, 189, 192, 195

Offset: 0

Antti Karttunen, Sep 07 2016

nonn

Antti Karttunen, Table of n, a(n) for n = 0..10028

Cf. A002828, A005563, A255131, A260731, A260733, A262689, A276572, A276574, A276575 (first differences), A277016 (squares present), A277015 (their square roots), A277888 (primes), A278486 (numbers one more than a prime), A278265, A278487, A278488, A278491 (another subsequence), A278497, A278498, A278499, A278513, A278516, A278517, A278518, A278519, A278521, A278522.
Cf. A277890 & A277891 (number of even and odd terms in each range. The latter seem to be slightly more numerous), A277889.
Positions of nonzero terms in A278515.
Subsequence of A278489, no common terms with A278490.
Cf. also A179016, A259934, A276583, A276613, A276623 for similar constructions.

(define (A276573 n) (A276574 (A276572 n)))

a(n) = A276574(A276572(n)).
Other identities and observations. For all n >= 0:
A260731(a(n)) = n.
a(A260733(n+1)) = A005563(n).
A278517(n) <= a(n) <= A278519(n).
A010873(a(n)) = A278499(n). [Terms reduced modulo 4.]
A010877(a(n)) = A278488(n). [modulo 8.]
A046523(a(n)) = A278497(n). [Least number with the same prime signature.]
A008683(a(n)) = A278513(n).
A065338(a(n)) = A278498(n).
A278509(a(n)) = A278265(n).
A278216(a(n)) = A278516(n). [Number of children the n-th node of the trunk has.]

Definition clarified and more identities added to the Formula section by Antti Karttunen, Nov 28 2016

A276623 The infinite trunk of ternary beanstalk: The only infinite sequence such that a(n-1) = a(n) - A053735(a(n)), where A053735(n) = base-3 digit sum of n.

Original entry on oeis.org

0, 2, 4, 8, 10, 12, 16, 20, 26, 28, 30, 34, 38, 42, 46, 52, 56, 62, 68, 72, 80, 82, 84, 88, 92, 96, 100, 106, 110, 116, 122, 126, 134, 140, 144, 152, 160, 164, 170, 176, 180, 188, 194, 198, 204, 212, 216, 224, 232, 242, 244, 246, 250, 254, 258, 262, 268, 272, 278, 284, 288, 296, 302, 306, 314, 322, 326, 332, 338, 342, 350, 356, 360

Offset: 0

Antti Karttunen, Sep 11 2016

Antti Karttunen, Table of n, a(n) for n = 0..6250

Cf. A004128, A024023, A053735, A054861, A261231 (left inverse), A261233, A276622, A276624, A276603 (terms divided by 2), A276604 (first differences).
Cf. A179016, A219648, A219666, A255056, A259934, A276573, A276583, A276613 for similar constructions.
Cf. also A263273.

(define (A276623 n) (A276624 (A276622 n)))

a(n) = A276624(A276622(n)).
Other identities. For all n >= 0:
A261231(a(n)) = n.
a(A261233(n)) = A024023(n) = 3^n - 1.

A276613 The infinite trunk of greedy cubes beanstalk: The only infinite sequence such that a(n-1) = a(n) - number of cubes that sum to a(n) with greedy algorithm (A055401).

Original entry on oeis.org

0, 7, 14, 21, 26, 33, 40, 47, 52, 59, 63, 70, 77, 84, 89, 96, 103, 110, 115, 124, 131, 138, 145, 150, 157, 164, 171, 176, 183, 187, 194, 201, 208, 215, 222, 229, 236, 241, 248, 255, 262, 267, 274, 278, 285, 292, 299, 304, 311, 318, 330, 339, 342, 349, 356, 363, 368, 375, 382, 389, 394, 401, 405, 412, 419, 426, 431, 438, 445

Offset: 0

Antti Karttunen, Sep 07 2016

nonn

Antti Karttunen, Table of n, a(n) for n = 0..10236

Cf. A055401, A261225, A276612, A276614, A276615 (first differences).

Cf. also A179016, A259934, A276573, A276583, A276623 for similar constructions.

(define (A276613 n) (A276614 (A276612 n)))

a(n) = A276614(A276612(n)).

A276582 Simple self-inverse permutation of natural numbers: after a(0)=0, list each block of A261224(n) numbers in reverse order, from A261222(n) to A261223(1+n).

Original entry on oeis.org

0, 1, 3, 2, 5, 4, 8, 7, 6, 11, 10, 9, 14, 13, 12, 18, 17, 16, 15, 23, 22, 21, 20, 19, 28, 27, 26, 25, 24, 34, 33, 32, 31, 30, 29, 40, 39, 38, 37, 36, 35, 47, 46, 45, 44, 43, 42, 41, 54, 53, 52, 51, 50, 49, 48, 61, 60, 59, 58, 57, 56, 55, 69, 68, 67, 66, 65, 64, 63, 62, 77, 76, 75, 74, 73, 72, 71, 70, 86, 85, 84, 83, 82, 81, 80, 79, 78

Offset: 0

Antti Karttunen, Sep 07 2016

nonn

Maps between A276583 and A276584.

Antti Karttunen, Table of n, a(n) for n = 0..9996
Index entries for sequences that are permutations of the natural numbers

Cf. A261222, A261223, A261224, A276581, A276583, A276584.

(define (A276582 n) (if (zero? n) n (+ (- (A261223 (+ 1 (A276581 n))) n) (A261222 (A276581 n)))))

a(0)=0; for n >= 1, a(n) = (A261223(1+A276581(n))- n)+A261222(A276581(n)).

A276581 After a(0)=0, each n occurs A261224(n) times.

Original entry on oeis.org

0, 1, 2, 2, 3, 3, 4, 4, 4, 5, 5, 5, 6, 6, 6, 7, 7, 7, 7, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 11, 12, 12, 12, 12, 12, 12, 12, 13, 13, 13, 13, 13, 13, 13, 14, 14, 14, 14, 14, 14, 14, 15, 15, 15, 15, 15, 15, 15, 15, 16, 16, 16, 16, 16, 16, 16, 16, 17, 17, 17, 17, 17, 17, 17, 17, 17, 18

Offset: 0

Antti Karttunen, Sep 07 2016

nonn

Auxiliary function for computing A276582 & A276583.

Antti Karttunen, Table of n, a(n) for n = 0..9996

Cf. A261223, A261224, A276582, A276583.

Table[Table[n, {-1 + Length@ NestWhileList[# - Block[{m = #, c = 1}, While[a = (# - Floor[Sqrt@ #]^2) &@ m; a != 0, c++; m = a]; c] &, ((n + 1)^2) - 1, # != n^2 - 1 &]}], {n, 0, 18}] // Flatten (* Michael De Vlieger, Sep 08 2016 *)

(define (A276581 n) (let loop ((k 0)) (if (>= (A261223 (+ 1 k)) n) k (loop (+ 1 k)))))

A276584 The infinite trunk of greedy squares beanstalk with reversed subsections.

Original entry on oeis.org

0, 3, 8, 6, 15, 11, 24, 21, 18, 35, 32, 27, 48, 43, 38, 63, 59, 56, 51, 80, 78, 74, 71, 66, 99, 95, 91, 88, 83, 120, 117, 114, 110, 107, 102, 143, 138, 135, 131, 128, 123, 168, 164, 161, 158, 154, 151, 146, 195, 192, 186, 183, 179, 176, 171, 224, 219, 213, 210, 206, 203, 198, 255, 251, 248, 242, 239, 235, 232, 227

Offset: 0

Antti Karttunen, Sep 07 2016 and Sep 09 2016

nonn

Antti Karttunen, Table of n, a(n) for n = 0..9996

Cf. A000196, A000290, A010052, A053610, A260740, A276582, A276583.

(definec (A276584 n) (cond ((zero? n) n) ((= 1 n) 3) (else (let ((maybe_next (A260740 (A276584 (- n 1))))) (if (zero? (A010052 (+ 1 maybe_next))) maybe_next (+ -1 (A000290 (+ 2 (A000196 (+ 1 maybe_next))))))))))

a(0) = 0; a(1) = 3; for n > 1, if A260740(a(n-1))+1 is not a square, then a(n) = A260740(a(n-1)), otherwise a(n) = A000290(2+A000196(A260740(a(n-1)))) - 1.

cp's OEIS Frontend

A276585 After a(0)=0, the first differences of A276583.

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A179016 The infinite trunk of binary beanstalk: The only infinite sequence such that a(n-1) = a(n) - number of 1's in binary representation of a(n).

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A276573 The infinite trunk of least squares beanstalk: The only infinite sequence such that a(0) = 0 and a(n-1) = a(n) - least number of squares (A002828) that sum to a(n).

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A276623 The infinite trunk of ternary beanstalk: The only infinite sequence such that a(n-1) = a(n) - A053735(a(n)), where A053735(n) = base-3 digit sum of n.

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A276613 The infinite trunk of greedy cubes beanstalk: The only infinite sequence such that a(n-1) = a(n) - number of cubes that sum to a(n) with greedy algorithm (A055401).

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A276582 Simple self-inverse permutation of natural numbers: after a(0)=0, list each block of A261224(n) numbers in reverse order, from A261222(n) to A261223(1+n).

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A276581 After a(0)=0, each n occurs A261224(n) times.

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A276584 The infinite trunk of greedy squares beanstalk with reversed subsections.

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