A095100 -id:A095100 - cp's OEIS frontend

A095274 a(n) = (A095100(n)-3)/4.

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, 37, 39, 41, 42, 43, 45, 47, 49, 53, 57, 59, 60, 62, 63, 65, 67, 69, 71, 73, 74, 75, 77, 79, 81, 83, 85, 87, 89, 90, 92, 93, 95, 97, 98, 99, 101, 103, 104, 107, 109, 111, 113

Offset: 0

Antti Karttunen, Jun 01 2004

nonn

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

Cf. A095100. Complement of A095275. Subset: A095272.

A095090 Number of 4k+3 integers in range ]2^n,2^(n+1)] whose Jacobi-vector is a Motzkin-path (A095100).

Original entry on oeis.org

1, 1, 2, 3, 6, 9, 17, 33, 60, 108, 202, 360, 703, 1328, 2519, 4779, 9103, 17501, 33473, 64761

Offset: 1

Antti Karttunen and Jun 01 2004

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

Cf. A095092, A095100.

is(m) = {my(s=0); for(i=1, m-1, if((s+=kronecker(i, m))<0, return(0))); 1; }
a(n) = {my(c=0); forstep(m=2^n+3-(n==1), 2^(n+1), 4, c+=is(m)); c; } \\ Jinyuan Wang, Jul 20 2020

a(n) = 2^(n-2) - A095091(n) for n > 1.

A095102 Odd primes p for which all sums Sum_{i=1..u} L(i/p) (with u ranging from 1 to (p-1)) are nonnegative, where L(i/p) is Legendre symbol of i and p, defined to be 1 if i is a quadratic residue (mod p) and -1 if i is a quadratic non-residue (mod p).

Original entry on oeis.org

3, 7, 11, 23, 31, 47, 59, 71, 79, 83, 103, 131, 151, 167, 191, 199, 239, 251, 263, 271, 311, 359, 383, 419, 431, 439, 479, 503, 563, 599, 607, 647, 659, 719, 743, 751, 839, 863, 887, 911, 919, 971, 983, 991, 1031, 1039, 1063, 1091, 1103, 1151

Offset: 1

Antti Karttunen, Jun 01 2004

nonn

All 4k+3 primes whose Legendre-vector (cf. A055094) forms a valid Dyck-path (cf. A014486).

T. D. Noe, Table of n, a(n) for n=1..1000
Peter Borwein, Stephen K.K. Choi and Michael Coons, Completely multiplicative functions taking values in {-1,1}, arXiv:0809.1691 [math.NT], 2008.
A. Karttunen and J. Moyer, C-program for computing the initial terms of this sequence

Intersection of A000040 and A095100. Subset of A080114 (see comments there). Complement of A095103 in A002145.

Cf. A095092.

isMotzkin[n_, k_] := Module[{s = 0, r = True}, Do[s += JacobiSymbol[i, n]; If[s < 0, r = False; Break[]], {i, 1, k}]; r]; A095102[max_] := Select[ Range[3, max, 4], PrimeQ[#] && isMotzkin[#, Quotient[#, 2]]&]; A095102[1151] (* Jean-François Alcover, Feb 16 2018, after Peter Luschny *)

isok(m) = {if(!isprime(m-(m<3)), return(0)); my(s=0); for(i=1, m-1, if((s+=kronecker(i, m))<0, return(0))); 1; } \\ Jinyuan Wang, Jul 20 2020

def A095102_list(n) :
    def is_Motzkin(n, k):
        s = 0
        for i in (1..k):
            s += jacobi_symbol(i, n)
            if s < 0: return False
        return True
    P = filter(is_prime, range(3, n+1, 4))
    return filter(lambda m: is_Motzkin(m, m//2), P)
A095102_list(1151) # Peter Luschny, Aug 09 2012

a(n) = 4*A095272(n) + 3.

A095101 Integers m of the form 4k+3 for which some of the sums Sum_{i=1..u} J(i/m) (with u ranging from 1 to (m-1)) is negative, where J(i/m) is Jacobi symbol of i and m.

Original entry on oeis.org

19, 43, 51, 67, 91, 99, 107, 115, 123, 127, 139, 147, 155, 163, 179, 187, 195, 203, 207, 211, 219, 223, 227, 235, 247, 259, 267, 275, 283, 291, 307, 315, 323, 331, 339, 347, 355, 367, 379, 387, 403, 411, 423, 427, 435, 443, 451, 459, 463, 467

Offset: 1

Antti Karttunen, Jun 01 2004

nonn

Integers whose Jacobi-vector does not form a valid Motzkin-path.

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

Subset of A095103. Complement of A095100 in A004767.

Cf. A095091.

isok(m) = {my(s=0); if(m%4==3, for(i=1, m-1, if((s+=kronecker(i, m))<0, return(1)))); 0; } \\ Jinyuan Wang, Jul 20 2020

def is_Motzkin(n, k):
    s = 0
    for i in (1..k) :
        s += jacobi_symbol(i, n)
        if s < 0 : return False
    return True
def A095101_list(n):
    return [m for m in range(3, n+1, 4) if not is_Motzkin(m, m//2)]
A095101_list(467) # Peter Luschny, Aug 08 2012

a(n) = 4*A095275(n) + 3.

A165469 Numbers of the form 4n+3 for which Sum_{i=1..u} J(i,4n+3) is never zero for any u in range [1,(2n+1)], where J(i,m) is the Jacobi symbol.

Original entry on oeis.org

3, 7, 15, 23, 31, 39, 47, 55, 71, 79, 87, 95, 111, 119, 143, 151, 159, 167, 183, 191, 199, 215, 231, 239, 255, 263, 271, 287, 303, 311, 327, 335, 359, 383, 391, 399, 407, 431, 439, 447, 455, 471, 479, 503, 511, 519, 527, 535, 551, 559, 591, 599, 615, 623

Offset: 1

Antti Karttunen, Oct 06 2009

nonn

a(n) = A004767(A165468(n)). Setwise difference of A095100 and A165608. A165580 gives the primes in this sequence.

A165608 Numbers of the form 4n+3 for which Sum_{i=1..u} J(i,4n+3) is never negative for any u in range [1,2n+1], but obtains at least one value zero, where J(i,m) is the Jacobi symbol.

Original entry on oeis.org

11, 27, 35, 59, 63, 75, 83, 103, 131, 135, 171, 175, 243, 251, 279, 295, 299, 319, 343, 351, 363, 371, 375, 395, 415, 419, 495, 539, 563, 567, 575, 607, 635, 659, 675, 703, 711, 731, 735, 755, 783, 867, 875, 895, 899, 903, 927, 943, 971, 999, 1063, 1067

Offset: 1

Antti Karttunen, Oct 06 2009

nonn

Setwise difference of A095100 and A165469.

a(n) = A004767(A165607(n)). Subset of A095100. A165977 gives the primes only. Cf. A165603.

Definition edited by Jonathan Sondow, Sep 27 2011

A166409 Odd numbers corresponding to the positions of zeros in A166406.

Original entry on oeis.org

5, 13, 17, 21, 29, 33, 37, 41, 45, 53, 57, 61, 65, 69, 73, 77, 85, 89, 93, 97, 99, 101, 105, 109, 113, 117, 125, 129, 133, 137, 141, 145, 147, 149, 153, 157, 161, 165, 173, 177, 181, 185, 189, 193, 197, 201, 205, 207, 209, 213, 217, 221, 229, 233, 237, 241

Offset: 1

Antti Karttunen, Oct 21 2009, Oct 22 2009

nonn

Those odd numbers 2n+1 for which the sum of i in [1,2n+1] with J(i,2n+1)=-1 is equal to the sum of i in [1,2n+1] with J(i,2n+1)=+1. Here J(i,k) is the Jacobi symbol.

Probably a union of A077425 & A165603: It is clear that A077425 is a subsequence of this sequence. For the remaining terms to be equal to A165603, it is at least required that the intersection of A165603 and A095100 be empty.

Cf. A077425, A095100, A165603.

from sympy import jacobi_symbol as J
def a(n):
    l=0
    m=0
    for i in range(1, 2*n + 2):
        if J(i, 2*n + 1)==-1: l+=i
        elif J(i, 2*n + 1)==1: m+=i
    return l - m
print([2*n + 1 for n in range(201) if a(n)==0]) # Indranil Ghosh, Jun 12 2017

A227194 Natural numbers for which all sums Sum_{i=1..u} K(i/n) (with u ranging from 1 to n) are nonnegative, where K(i/n) is Kronecker symbol of i and n.

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 9, 10, 11, 12, 14, 15, 16, 23, 24, 25, 26, 27, 30, 31, 34, 35, 36, 38, 39, 42, 44, 47, 48, 49, 54, 55, 56, 58, 59, 62, 63, 64, 66, 71, 74, 75, 76, 79, 81, 82, 83, 86, 87, 94, 95, 96, 100, 103, 106, 108, 110, 111, 114, 119, 120, 121, 130, 131

Offset: 1

Antti Karttunen, Jul 06 2013

nonn

Subsets: A095100, A095102 (gives the prime terms). Cf. A227196-A227198.

firstdiving(n) = {s=0;for(k=1,n,s=s+kronecker(k,n);if(s<0,return(k)));return(0)}
for(n=1,500,if((0==firstdiving(n)),print1(n,", ")))

cp's OEIS Frontend

A095274 a(n) = (A095100(n)-3)/4.

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A095090 Number of 4k+3 integers in range ]2^n,2^(n+1)] whose Jacobi-vector is a Motzkin-path (A095100).

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A095102 Odd primes p for which all sums Sum_{i=1..u} L(i/p) (with u ranging from 1 to (p-1)) are nonnegative, where L(i/p) is Legendre symbol of i and p, defined to be 1 if i is a quadratic residue (mod p) and -1 if i is a quadratic non-residue (mod p).

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A095101 Integers m of the form 4k+3 for which some of the sums Sum_{i=1..u} J(i/m) (with u ranging from 1 to (m-1)) is negative, where J(i/m) is Jacobi symbol of i and m.

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Sage

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A165469 Numbers of the form 4n+3 for which Sum_{i=1..u} J(i,4n+3) is never zero for any u in range [1,(2n+1)], where J(i,m) is the Jacobi symbol.

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A165608 Numbers of the form 4n+3 for which Sum_{i=1..u} J(i,4n+3) is never negative for any u in range [1,2n+1], but obtains at least one value zero, where J(i,m) is the Jacobi symbol.

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A166409 Odd numbers corresponding to the positions of zeros in A166406.

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A227194 Natural numbers for which all sums Sum_{i=1..u} K(i/n) (with u ranging from 1 to n) are nonnegative, where K(i/n) is Kronecker symbol of i and n.

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