A095105 -id:A095105 - cp's OEIS frontend

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

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.

A095107 Sum of maximal Dyck path prefix lengths of all 4k+3 primes in range ]2^n,2^(n+1)].

Original entry on oeis.org

2, 6, 10, 54, 106, 346, 1410, 3994, 12828, 46242, 158252, 542568, 1865796, 6822592, 25120772, 88926198, 327561126, 1212380122, 4358259840, 16111686770, 59936356888

Offset: 1

Antti Karttunen, Jun 01 2004

nonn

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

Cf. A095105, A095106, A095110.

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).

A095357 Ratio A095107(n)/A095008(n) rounded down.

Original entry on oeis.org

2, 6, 10, 18, 35, 49, 108, 181, 346, 651, 1236, 2348, 4240, 8454, 16537, 30963, 60986, 118814, 225337, 438305, 854049

Offset: 1

Antti Karttunen, Jun 12 2004

This is the average length of maximum Dyck path prefix (i.e. non-diving portion) found in the "Legendre-vectors" of all 4k+3 primes in range ]2^n,2^(n+1)]. See A095104-A095105.
The ratios before rounding are: 2, 6, 10, 18, 35.333333, 49.428571, 108.461538, 181.545455, 346.702703, 651.295775, 1236.34375, 2348.779221, 4240.445455, 8454.26518, 16537.703752, 30963.160864, 60986.990505, 118814.20247, 225337.874981, 438305.90522, 854049.74263.
Ratio (A095107(n)/A095008(n))/(A095110(n)/A000079(n-2)) starts as follows: 1, 1, 0.5, 0.75, 0.375, 0.4375, 0.40625, 0.34375, 0.289062, 0.277344, 0.25, 0.225586, 0.214844, 0.197021, 0.185425, 0.175293, 0.16391, 0.155701, 0.14756, 0.140224.

A095358 gives the same ratios rounded to nearest integer. A095361 gives similar ratios computed for all 4k+3 integers.

a(n) = floor(A095107(n)/A095008(n)).

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Links

Crossrefs

A095107 Sum of maximal Dyck path prefix lengths of all 4k+3 primes in range ]2^n,2^(n+1)].

Views

Author

Keywords

Links

Crossrefs

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

Views

Author

Keywords

Links

Crossrefs

Formula

A095357 Ratio A095107(n)/A095008(n) rounded down.

Views

Author

Keywords

Comments

Crossrefs

Formula