%I A095090
%S A095090 1,1,2,3,6,9,17,33,60,108,202,360,703,1328,2519,4779,9103,17501,33473,
%T A095090 64761
%N A095090 Number of 4k+3 integers in range ]2^n,2^(n+1)] whose Jacobi-vector is a Motzkin-path (A095100).
%Y A095090 a(n) = (2^(n-2))-A095091(n) for n > 1. Cf. A095092.
%H A095090 A. Karttunen, J. Moyer: <a href="http://www.research.att.com/~njas/sequences/primestats.c.txt">C-program for computing the initial terms of this sequence</a>
%K A095090 nonn
%O A095090 1,3
%A A095090 Antti Karttunen (his-firstname.his-surname(AT)iki.fi), Jun 01 2004

%I A095100
%S A095100 3,7,11,15,23,27,31,35,39,47,55,59,63,71,75,79,83,87,95,103,111,119,
%T A095100 131,135,143,151,159,167,171,175,183,191,199,215,231,239,243,251,
%U A095100 255,263,271,279,287,295,299,303,311,319,327,335,343,351,359,363
%N A095100 Integers n of the form 4k+3 for which all sums Sum_{i=1..u} J(i/n) (with u ranging from 1 to (n-1)) are nonnegative, where J(i/n) is Jacobi symbol of i and n.
%C A095100 Integers whose Jacobi-vector forms a valid Motzkin-path.
%Y A095100 a(n) = 4*A095274(n)+3.Subset: A095102. Complement of A095101 in A004767. Cf. A095090.
%H A095100 A. Karttunen, J. Moyer: <a href="http://www.research.att.com/~njas/sequences/primestats.c.txt">C-program for computing the initial terms of this sequence</a>
%K A095100 nonn
%O A095100 1,1
%A A095100 Antti Karttunen (his-firstname.his-surname(AT)iki.fi), Jun 01 2004

%I A095091
%S A095091 0,0,0,1,2,7,15,31,68,148,310,664,1345,2768,5673,11605,23665,48035,
%T A095091 97599,197383
%N A095091 Number of 4k+3 integers in range ]2^n,2^(n+1)] whose Jacobi-vector is not a valid Motzkin-path (A095101).
%Y A095091 a(n) = (2^(n-2))-A095090(n) for n > 1. Cf. A095093.
%H A095091 A. Karttunen, J. Moyer: <a href="http://www.research.att.com/~njas/sequences/primestats.c.txt">C-program for computing the initial terms of this sequence</a>
%K A095091 nonn
%O A095091 1,5
%A A095091 Antti Karttunen (his-firstname.his-surname(AT)iki.fi), Jun 01 2004

%I A095101
%S A095101 19,43,51,67,91,99,107,115,123,127,139,147,155,163,179,187,195,203,
%T A095101 207,211,219,223,227,235,247,259,267,275,283,291,307,315,323,331,
%U A095101 339,347,355,367,379,387,403,411,423,427,435,443,451,459,463,467
%N A095101 Integers n of the form 4k+3 for which some of the sums Sum_{i=1..u} J(i/n) (with u ranging from 1 to (n-1)) is negative, where J(i/n) is Jacobi symbol of i and n.
%C A095100 Integers whose Jacobi-vector does not form a valid Motzkin-path.
%Y A095101 a(n) = 4*A095275(n)+3.Subset: A095103. Complement of A095100 in A004767. Cf. A095091.
%H A095101 A. Karttunen, J. Moyer: <a href="http://www.research.att.com/~njas/sequences/primestats.c.txt">C-program for computing the initial terms of this sequence</a>
%K A095101 nonn
%O A095101 1,1
%A A095101 Antti Karttunen (his-firstname.his-surname(AT)iki.fi), Jun 01 2004

%I A095109
%S A095109 0,0,0,3,13,40,194,389,1407,3646,13300,39166,131464,389575,1627565,
%T A095109 5556317,19913811,72249861,272188855,996509475
%N A095109 Sum of diving indices of all 4k+3 integers in range ]2^n,2^(n+1)].
%C A095109 Diving index is explained at A095269.
%Y A095109 Cf. A095110.
%H A095109 A. Karttunen, J. Moyer: <a href="http://www.research.att.com/~njas/sequences/primestats.c.txt">C-program for computing the initial terms of this sequence</a>
%K A095109 nonn
%O A095109 1,4
%A A095109 Antti Karttunen (his-firstname.his-surname(AT)iki.fi), Jun 01 2004

%I A095110
%S A095110 2,6,24,80,303,847,3401,12816,47255,168306,628058,2243714,8741445,
%T A095110 33079519,125266470,474015314,1806320048,6945673364,26539784794,102630867522
%N A095110 Sum of max. Motzkin path prefix-lengths of all 4k+3 integers in range ]2^n,2^(n+1)].
%Y A095110 Cf. A095270, A095109.
%H A095110 A. Karttunen, J. Moyer: <a href="http://www.research.att.com/~njas/sequences/primestats.c.txt">C-program for computing the initial terms of this sequence</a>
%K A095110 nonn
%O A095110 1,1
%A A095110 Antti Karttunen (his-firstname.his-surname(AT)iki.fi), Jun 01 2004

%I A095269
%S A095269 0,0,0,0,3,0,0,0,0,0,3,0,10,0,0,0,3,0,0,0,0,0,3,0,10,0,7,0,3,0,7,
%T A095269 7,0,0,3,0,8,0,8,0,3,0,0,0,11,0,3,0,34,0,8,19,3,0,10,13,61,0,3,0,
%U A095269 0,7,0,0,3,0,7,0,8,0,3,0,10,0,0,0,3,0,32,0,7,0,3,0,58,0,7,0,3,0,0
%N A095269 Diving index of 4n+3.
%C A095269 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.
%Y A095269 a(n)=A095270(n)+1 modulo A004767(n). Cf. A095109, A095271 (same sequence with zeros removed).
%H A095269 A. Karttunen, J. Moyer: <a href="http://www.research.att.com/~njas/sequences/primestats.c.txt">C-program for computing the initial terms of this sequence</a>
%K A095269 nonn
%O A095269 0,5
%A A095269 Antti Karttunen (his-firstname.his-surname(AT)iki.fi), Jun 01 2004

%I A095270
%S A095270 2,6,10,14,2,22,26,30,34,38,2,46,9,54,58,62,2,70,74,78,82,86,2,94,
%T A095270 9,102,6,110,2,118,6,6,130,134,2,142,7,150,7,158,2,166,170,174,10,
%U A095270 182,2,190,33,198,7,18,2,214,9,12,60,230,2,238,242,6,250,254,2,262
%N A095270 Length of max. Motzkin path prefix in the Jacobi-vector of 4n+3.
%Y A095270 a(n)=A095269(n)-1 modulo A004767(n). Cf. A095110, A095105.
%H A095270 A. Karttunen, J. Moyer: <a href="http://www.research.att.com/~njas/sequences/primestats.c.txt">C-program for computing the initial terms of this sequence</a>
%K A095270 nonn
%O A095270 1,1
%A A095270 Antti Karttunen (his-firstname.his-surname(AT)iki.fi), Jun 01 2004

%I A095271
%S A095271 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,
%T A095271 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,
%U A095271 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
%N A095271 Diving index of A095101(n).
%F A095271 a(n) = A095269(A095275(n)). See comments at A095269.
%Y A095271 Cf. A095108.
%H A095271 A. Karttunen, J. Moyer: <a href="http://www.research.att.com/~njas/sequences/primestats.c.txt">C-program for computing the initial terms of this sequence</a>
%K A095271 nonn
%O A095271 1,1
%A A095271 Antti Karttunen (his-firstname.his-surname(AT)iki.fi), Jun 01 2004

%I A095274
%S A095274 0,1,2,3,5,6,7,8,9,11,13,14,15,17,18,19,20,21,23,25,27,29,32,33,35,
%T A095274 37,39,41,42,43,45,47,49,53,57,59,60,62,63,65,67,69,71,73,74,75,77,
%U A095274 79,81,83,85,87,89,90,92,93,95,97,98,99,101,103,104,107,109,111,113
%N A095274 a(n) = (A095100(n)-3)/4.
%Y A095274 Complement of A095275. Subset: A095272.
%H A095274 A. Karttunen, J. Moyer: <a href="http://www.research.att.com/~njas/sequences/primestats.c.txt">C-program for computing the initial terms of this sequence</a>
%K A095274 nonn
%O A095274 0,3
%A A095274 Antti Karttunen (his-firstname.his-surname(AT)iki.fi), Jun 01 2004

%I A095275
%S A095275 4,10,12,16,22,24,26,28,30,31,34,36,38,40,44,46,48,50,51,52,54,55,
%T A095275 56,58,61,64,66,68,70,72,76,78,80,82,84,86,88,91,94,96,100,102,105,
%U A095275 106,108,110,112,114,115,116,118,120,121,122,124,126,128,130,132
%N A095275 a(n) = (A095101(n)-3)/4.
%Y A095275 Complement of A095274. Subset: A095273.
%H A095275 A. Karttunen, J. Moyer: <a href="http://www.research.att.com/~njas/sequences/primestats.c.txt">C-program for computing the initial terms of this sequence</a>
%K A095275 nonn
%O A095275 0,1
%A A095275 Antti Karttunen (his-firstname.his-surname(AT)iki.fi), Jun 01 2004