A095269 -id:A095269 - cp's OEIS frontend

A095109 Sum of diving indices of all 4k+3 integers in range ]2^n,2^(n+1)].

0, 0, 0, 3, 13, 40, 194, 389, 1407, 3646, 13300, 39166, 131464, 389575, 1627565, 5556317, 19913811, 72249861, 272188855, 996509475

Offset: 1

Antti Karttunen, Jun 01 2004

Diving index is explained at A095269.

Antti Karttunen and J. Moyer, C-program for computing the initial terms of this sequence

Cf. A095110.

A095104 Diving index of the n-th 4k+3 prime (A002145(n)).

Original entry on oeis.org

0, 0, 0, 3, 0, 0, 3, 0, 0, 3, 0, 0, 0, 0, 7, 7, 0, 3, 0, 3, 0, 11, 0, 0, 3, 13, 61, 0, 0, 0, 0, 3, 3, 0, 3, 7, 0, 45, 3, 0, 0, 0, 0, 7, 7, 35, 0, 7, 35, 3, 0, 3, 3, 0, 3, 15, 0, 0, 3, 15, 3, 0, 0, 7, 3, 0, 45, 3, 0, 0, 3, 3, 7, 7, 0, 3, 0, 3, 0, 3, 0, 0, 7, 7, 0, 0, 0, 67, 0, 0, 3, 0

Offset: 1

Antti Karttunen, Jun 01 2004

nonn

Diving index of an odd number n is the first integer u > 1 where Sum_{i=1..u} J(i/n) results -1 and zero if never. Here J(i/n) is Jacobi symbol of i and n, which reduces to a Legendre symbol L(i/n) when n is a prime.

A. Karttunen and J. Moyer, C-program for computing the initial terms of this sequence

a(n)=A095105(n)+1 modulo A002145(n). Cf. A095106, A095108 (same sequence with zeros removed), A095269.

A095270 Length of maximal Motzkin path prefix in the Jacobi-vector of 4n+3.

Original entry on oeis.org

2, 6, 10, 14, 2, 22, 26, 30, 34, 38, 2, 46, 9, 54, 58, 62, 2, 70, 74, 78, 82, 86, 2, 94, 9, 102, 6, 110, 2, 118, 6, 6, 130, 134, 2, 142, 7, 150, 7, 158, 2, 166, 170, 174, 10, 182, 2, 190, 33, 198, 7, 18, 2, 214, 9, 12, 60, 230, 2, 238, 242, 6, 250, 254, 2, 262

Offset: 1

Antti Karttunen, Jun 01 2004

nonn

Antti Karttunen and J. Moyer, C-program for computing the initial terms of this sequence

Cf. A004767, A095269, A095110, A095105.

a(n) = A095269(n)-1 modulo A004767(n).

A095271 Diving index of A095101(n).

Original entry on oeis.org

3, 3, 10, 3, 3, 10, 7, 3, 7, 7, 3, 8, 8, 3, 11, 3, 34, 8, 19, 3, 10, 13, 61, 3, 7, 3, 7, 8, 3, 10, 3, 32, 7, 3, 58, 7, 3, 45, 3, 7, 3, 13, 31, 3, 8, 7, 3, 10, 7, 35, 3, 8, 7, 35, 3, 14, 8, 3, 13, 22, 3, 8, 3, 103, 7, 15, 3, 7, 40, 3, 7, 15, 55, 3, 113, 31, 3, 7, 13, 3, 118

Offset: 1

Antti Karttunen, Jun 01 2004

nonn

Antti Karttunen and J. Moyer, C-program for computing the initial terms of this sequence

Cf. A095101, A095108, A095269, A095275.

a(n) = A095269(A095275(n)). See comments at A095269.

A095361 Ratio A095110(n)/A000079(n-2) rounded down.

Original entry on oeis.org

2, 6, 12, 20, 37, 52, 106, 200, 369, 657, 1226, 2191, 4268, 8076, 15291, 28931, 55124, 105982, 202482, 391505

Offset: 1

Antti Karttunen, Jun 12 2004

nonn

This is the average length of maximum Motzkin path prefix (i.e. non-diving portion) found in the "Jacobi-vectors" of all 4k+3 integers in range ]2^n,2^(n+1)]. See A095269-A095270.

The ratios before rounding are: 2, 6, 12, 20, 37.875, 52.9375, 106.28125, 200.25, 369.179687, 657.445312, 1226.675781, 2191.126953, 4268.283691, 8076.054443, 15291.317139, 28931.598755, 55124.513184, 105982.564758, 202482.488968, 391505.689705.

A095362 gives the same ratios rounded to nearest integer. A095357 gives similar ratios computed only for 4k+3 primes.

a(1) = 2, a(n) = floor(A095110(n)/A000079(n-2)) for n > 1.

cp's OEIS Frontend

A095109 Sum of diving indices of all 4k+3 integers in range ]2^n,2^(n+1)].

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A095104 Diving index of the n-th 4k+3 prime (A002145(n)).

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A095270 Length of maximal Motzkin path prefix in the Jacobi-vector of 4n+3.

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A095271 Diving index of A095101(n).

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A095361 Ratio A095110(n)/A000079(n-2) rounded down.

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