A277015 -id:A277015 - 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

A276574 The infinite trunk of least squares beanstalk with reversed subsections.

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, Sep 07 2016

			This can be viewed as an irregular table, where after 0, each row has A260734(n) = 1, 2, 2, 4, 4, 5, 5, 7, ... terms:
0;
3;
8, 6;
15, 11;
24, 21, 18, 16;
35, 32, 30, 27;
48, 45, 43, 40, 38;
63, 59, 56, 53, 51;
80, 78, 75, 72, 70, 67, 64;
99, 96, 93, 90, 88, 85, 83;
120, 117, 115, 112, 108, 105, 102;
...
Each row begins with (n^2)-1 (see A005563), and each successive term is obtained by subtracting A002828(k) from the previous term k, until ((n-1)^2)-1 would be encountered, which is not listed second time (as it already occurs as the first term of the previous row), but instead, the current row is finished and the next row is started with the term ((n+1)^2)-1.

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

Cf. A005563 (left edge), A277023 (right edge).
Cf. A000196, A000290, A002828, A010052, A255131, A260734, A262689, A276572.
Used to construct A276573.
Cf. A277015 (tells which rows end with squares, listed in A277016).

(definec (A276574 n) (cond ((zero? n) n) ((= 1 n) 3) (else (let ((maybe_next (A255131 (A276574 (- 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, let k = A255131(a(n-1)). If k+1 is not a square, then a(n) = k, otherwise a(n) = A000290(2+A000196(k+1)) - 1.

Example section added and the formula rewritten to a simpler form (which is now correct) - Antti Karttunen, Oct 16 2016

A277016 Squares present in A276573 (the infinite trunk of least squares beanstalk).

Original entry on oeis.org

0, 16, 64, 144, 256, 361, 400, 576, 1296, 1600, 2025, 2304, 3136, 3600, 4624, 5184, 6400, 6889, 7056, 8464, 9216, 10816, 11881, 12544, 14400, 15376, 17424, 18496, 19600, 20736, 21609, 21904, 24336, 25600, 26896, 28224, 29929, 30976, 32400, 36864, 41616, 43264, 44521, 46656, 48400, 50176, 51984, 53824, 56169, 57600, 63504, 69696, 73984, 75625, 76176, 78400, 80656

Offset: 0

Antti Karttunen, Oct 03 2016

nonn

Indexing starts from zero because a(0)=0 is a special case in this sequence.

Cf. A276573, A277014, A277015 (square roots of these numbers), A277025.

(define (A277016 n) (A276573 (A277014 n)))

a(n) = A276573(A277014(n)).

A277023 a(n) = A276573(A260732(n)); For n >= 1, a(n) = the next larger term right after each (n^2)-1 in the infinite trunk of least squares beanstalk.

Original entry on oeis.org

0, 3, 6, 11, 16, 27, 38, 51, 64, 83, 102, 123, 144, 171, 198, 227, 256, 291, 326, 361, 400, 444, 486, 531, 576, 627, 678, 731, 786, 843, 902, 963, 1026, 1091, 1158, 1227, 1296, 1371, 1446, 1523, 1600, 1683, 1767, 1851, 1938, 2025, 2118, 2211, 2304, 2403, 2502, 2603, 2706, 2811, 2918, 3027, 3136, 3251, 3366, 3483, 3600, 3723, 3846

Offset: 0

Antti Karttunen, Oct 03 2016

nonn

Antti Karttunen, Table of n, a(n) for n = 0..750

Cf. A002828, A260732, A276573.

Cf. A277015 (the positions of squares in this sequence), A277024, A277025 A277026.

(define (A277023 n) (A276573 (A260732 n)))

a(n) = A276573(A260732(n)).

A277025 n for which A277023(4n) is square, thus A277024(4n) is zero.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 9, 10, 12, 14, 15, 17, 18, 20, 21, 23, 24, 26, 28, 30, 31, 33, 34, 35, 36, 37, 39, 40, 41, 42, 44, 45, 48, 51, 52, 54, 55, 56, 57, 58, 60, 63, 66, 68, 69, 70, 71, 72, 74, 78, 80, 81, 84, 88, 90, 91, 93, 95, 96, 98, 99, 102, 105, 107, 108, 110, 111, 112, 114, 117, 118, 120, 121, 123, 126, 129, 131

Offset: 0

Antti Karttunen, Oct 03 2016

nonn

Indexing starts from zero because a(0)=0 is a special case in this sequence.

Antti Karttunen, Table of n, a(n) for n = 0..120

Complement: A277026.

Cf. A002828, A277015, A277016, A277024.

A277024 a(n) = A277023(n) - n^2.

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, Oct 03 2016

nonn

Antti Karttunen, Table of n, a(n) for n = 0..750

Cf. A000290, A277023.

Cf. A277015 (positions of zeros), A277025 (positions where 4n is zero), A277026 (where 4n is nonzero).

(define (A277024 n) (- (A277023 n) (A000290 n)))

a(n) = A277023(n) - n^2.

A277014 Positions of squares in A276573.

Original entry on oeis.org

0, 6, 24, 53, 92, 132, 145, 208, 462, 568, 716, 815, 1106, 1270, 1625, 1822, 2245, 2415, 2474, 2960, 3222, 3777, 4148, 4377, 5022, 5359, 6064, 6436, 6815, 7204, 7508, 7611, 8452, 8888, 9339, 9797, 10383, 10744, 11237, 12775, 14413, 14982, 15409, 16146, 16741, 17352, 17978, 18612, 19423, 19916, 21944, 24067, 25533, 26099, 26289, 27053, 27831

Offset: 0

Antti Karttunen, Oct 03 2016

nonn

Indexing starts from zero, because a(0)=0 is a special case. After that the sequence gives the positions of ones in A276575.

Cf. A010052, A276573, A276575.

Cf. A277016 (the corresponding squares), A277015 (their square roots).

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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A276574 The infinite trunk of least squares beanstalk with reversed subsections.

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A277016 Squares present in A276573 (the infinite trunk of least squares beanstalk).

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A277023 a(n) = A276573(A260732(n)); For n >= 1, a(n) = the next larger term right after each (n^2)-1 in the infinite trunk of least squares beanstalk.

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A277025 n for which A277023(4n) is square, thus A277024(4n) is zero.

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A277024 a(n) = A277023(n) - n^2.

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A277014 Positions of squares in A276573.

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