A278518 -id:A278518 - cp's OEIS frontend

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).

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

A260731 a(n) = Number of steps to reach 0 starting from x=n and using the iterated process: x -> x - A002828(x), where A002828(x) = the least number of squares that add up to x.

Original entry on oeis.org

0, 1, 1, 1, 2, 2, 2, 2, 3, 4, 4, 4, 5, 5, 5, 5, 6, 6, 7, 7, 8, 8, 8, 8, 9, 10, 10, 10, 10, 11, 11, 11, 12, 12, 13, 13, 14, 14, 14, 14, 15, 15, 15, 16, 16, 17, 17, 17, 18, 19, 19, 19, 20, 20, 20, 20, 21, 21, 22, 22, 22, 23, 23, 23, 24, 24, 24, 25, 25, 25, 26, 26, 27, 27, 28, 28, 28, 29, 29, 29, 30, 31, 31, 31, 32, 32, 32, 32, 33, 33, 34, 34, 34, 35, 35, 35, 36, 36, 37, 37, 38

Offset: 0

Antti Karttunen, Aug 12 2015

nonn

Antti Karttunen, Table of n, a(n) for n = 0..10000

Cf. A002828, A255131, A260732, A260733, A260734.
Left inverse of A276573, A278517 and A278519. A278518(n) gives the number of times n occurs (run lengths).
Cf. also A261221.

A002828[n_] := Which[n == 0, 0, SquaresR[1, n] > 0, 1, SquaresR[2, n] > 0, 2, SquaresR[3, n] > 0, 3, True, 4]; a[0] = 0; a[n_] := a[n] = 1 + a[n - A002828[n]]; Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Nov 14 2016 *)

a(0) = 0; for >= 1, a(n) = 1 + A260731(A255131(n)).
From Antti Karttunen, Nov 28 2016: (Start)
For all n >= 0, a(A278517(n)) = a(A278519(n)) = a(A276573(n)) = n.
(End)

A278517 a(n) = smallest k for which A260731(k) = n.

Original entry on oeis.org

0, 1, 4, 8, 9, 12, 16, 18, 20, 24, 25, 29, 32, 34, 36, 40, 43, 45, 48, 49, 52, 56, 58, 61, 64, 67, 70, 72, 74, 77, 80, 81, 84, 88, 90, 93, 96, 98, 100, 104, 106, 109, 113, 116, 120, 121, 125, 128, 130, 133, 136, 139, 142, 144, 146, 148, 152, 155, 157, 160, 162, 164, 168, 169, 172, 176, 178, 180, 184, 187, 190, 193, 196, 200

Offset: 0

Antti Karttunen, Nov 28 2016

nonn

Antti Karttunen, Table of n, a(n) for n = 0..11363

Cf. A260731 (a left inverse), A278518 (first differences), A278519, A278521.

Cf. also A002828, A276573.

For all n >= 0, A260731(a(n)) = n.

A278519 a(n) = largest k for which A260731(k) = n.

Original entry on oeis.org

0, 3, 7, 8, 11, 15, 17, 19, 23, 24, 28, 31, 33, 35, 39, 42, 44, 47, 48, 51, 55, 57, 60, 63, 66, 69, 71, 73, 76, 79, 80, 83, 87, 89, 92, 95, 97, 99, 103, 105, 108, 112, 115, 119, 120, 124, 127, 129, 132, 135, 138, 141, 143, 145, 147, 151, 154, 156, 159, 161, 163, 167, 168, 171, 175, 177, 179, 183, 186, 189, 192, 195, 199, 201

Offset: 0

Antti Karttunen, Nov 28 2016

nonn

Antti Karttunen, Table of n, a(n) for n = 0..11363

Cf. A260731 (a left inverse), A278517, A278518 (first differences, after its initial 1), A278522.

(define (A278519 n) (if (zero? n) n (+ -1 (A278517 (+ 1 n)))))

a(0) = 0; for n > 0, a(n) = A278517(1+n)-1.

For all n >= 0, A260731(a(n)) = n.

A278521 a(n) = A276573(n) - A278517(n).

Original entry on oeis.org

0, 2, 2, 0, 2, 3, 0, 0, 1, 0, 2, 1, 0, 1, 2, 0, 0, 0, 0, 2, 1, 0, 1, 2, 0, 0, 0, 0, 1, 1, 0, 2, 1, 0, 0, 0, 0, 1, 2, 1, 2, 3, 2, 1, 0, 2, 1, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 1, 0, 2, 1, 0, 1, 3, 2, 2, 2, 2, 2, 0, 1, 2, 2, 2, 2, 2, 0, 0, 2, 3, 1, 2, 3, 2, 1, 0, 1, 2, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 2, 2, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 2, 0, 0

Offset: 0

Antti Karttunen, Nov 28 2016

nonn

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

Cf. A276573, A278517, A278518, A278522.

(define (A278521 n) (- (A276573 n) (A278517 n)))

a(n) = A276573(n) - A278517(n).

A278522 a(n) = A278519(n) - A276573(n).

Original entry on oeis.org

0, 0, 1, 0, 0, 0, 1, 1, 2, 0, 1, 1, 1, 0, 1, 2, 1, 2, 0, 0, 2, 1, 1, 0, 2, 2, 1, 1, 1, 1, 0, 0, 2, 1, 2, 2, 1, 0, 1, 0, 0, 0, 0, 2, 0, 1, 1, 1, 1, 1, 2, 2, 0, 1, 0, 2, 2, 1, 1, 1, 1, 2, 0, 0, 2, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 1, 2, 0, 0, 0, 0, 0, 0, 0, 2, 1, 1, 0, 2, 1, 2, 2, 1, 1, 1, 1, 2, 2, 2, 1, 0, 0, 1, 1, 2, 2, 1, 2, 2, 1, 1, 1, 2, 0, 1, 2, 1

Offset: 0

Antti Karttunen, Nov 28 2016

nonn

First 3 occurs as a(133).

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

Cf. A276573, A278518, A278519, A278521.

(define (A278522 n) (- (A278519 n) (A276573 n)))

a(n) = A278519(n) - A276573(n).

A278516 Number of children the n-th node (counted from the root 0) has in the infinite trunk of least squares beanstalk tree: a(n) = A278216(A276573(n)).

Original entry on oeis.org

4, 4, 1, 3, 3, 2, 2, 2, 1, 4, 3, 2, 2, 4, 1, 1, 3, 1, 3, 3, 1, 3, 3, 3, 1, 2, 1, 2, 2, 1, 3, 3, 1, 3, 1, 1, 2, 4, 2, 1, 1, 1, 3, 1, 4, 3, 2, 2, 2, 1, 1, 2, 2, 2, 3, 1, 2, 3, 2, 2, 2, 1, 3, 3, 1, 2, 3, 2, 1, 1, 2, 4, 2, 2, 3, 1, 2, 1, 3, 3, 1, 3, 3, 1, 1, 1, 2, 3, 1, 3, 2, 3, 1, 3, 1, 1, 3, 2, 2, 3, 1, 2, 2, 1, 3, 3, 2, 2, 1, 1, 2, 1, 1, 3, 3, 2, 1, 4, 2, 1, 3

Offset: 0

Antti Karttunen, Nov 28 2016

nonn

Naturally, all terms must be > 0 and <= 4.

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

Cf. A002828, A276573, A278216, A278518.

(define (A278516 n) (A278216 (A276573 n)))

a(n) = A278216(A276573(n)).

cp's OEIS Frontend

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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A260731 a(n) = Number of steps to reach 0 starting from x=n and using the iterated process: x -> x - A002828(x), where A002828(x) = the least number of squares that add up to x.

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A278517 a(n) = smallest k for which A260731(k) = n.

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A278519 a(n) = largest k for which A260731(k) = n.

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A278521 a(n) = A276573(n) - A278517(n).

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A278522 a(n) = A278519(n) - A276573(n).

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A278516 Number of children the n-th node (counted from the root 0) has in the infinite trunk of least squares beanstalk tree: a(n) = A278216(A276573(n)).

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