A277023 -id:A277023 - cp's OEIS frontend

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

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.

A277026 n for which A277023(4n) is not a square, thus A277024(4n) is nonzero.

Original entry on oeis.org

7, 8, 11, 13, 16, 19, 22, 25, 27, 29, 32, 38, 43, 46, 47, 49, 50, 53, 59, 61, 62, 64, 65, 67, 73, 75, 76, 77, 79, 82, 83, 85, 86, 87, 89, 92, 94, 97, 100, 101, 103, 104, 106, 109, 113, 115, 116, 119, 122, 124, 125, 127, 128, 130, 133, 134, 137, 140, 141, 143, 145, 146, 148, 151, 152, 154, 155, 163, 164, 167, 168, 169, 170, 173, 174, 176, 178, 179, 181, 184, 185

Offset: 1

Antti Karttunen, Oct 03 2016

nonn

Complement: A277025.

Cf. A277024.

A277015 Numbers whose squares are present in A276573 (the infinite trunk of least squares beanstalk).

Original entry on oeis.org

0, 4, 8, 12, 16, 19, 20, 24, 36, 40, 45, 48, 56, 60, 68, 72, 80, 83, 84, 92, 96, 104, 109, 112, 120, 124, 132, 136, 140, 144, 147, 148, 156, 160, 164, 168, 173, 176, 180, 192, 204, 208, 211, 216, 220, 224, 228, 232, 237, 240, 252, 264, 272, 275, 276, 280, 284

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..130

Cf. A000196, A002828, A276573, A277014, A277016 (the corresponding squares), A277025 (multiples of four present / 4).

Positions of squares in A277023, zeros in A277024.

(define (A277015 n) (A000196 (A277016 n)))

a(n) = A000196(A277016(n)) = A000196(A276573(A277014(n))).

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

cp's OEIS Frontend

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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A277026 n for which A277023(4n) is not a square, thus A277024(4n) is nonzero.

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A277015 Numbers whose squares are present in A276573 (the infinite trunk of least squares beanstalk).

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

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