A080114 -id:A080114 - cp's OEIS frontend

A080112 Positions of A080114 in A000040.

2, 3, 4, 5, 6, 9, 11, 12, 15, 17, 20, 22, 23, 27, 32, 36, 39, 43, 46, 52, 54, 56, 58, 64, 72, 76, 81, 83, 85, 92, 96, 103, 109, 111, 118, 120, 128, 132, 133, 146, 150, 154, 156, 157, 164, 166, 167, 173, 175, 179, 182, 185, 190, 200, 202, 207, 215, 222, 225, 228, 229

Offset: 1

Antti Karttunen, Feb 11 2003

nonn

Complement of A080113. Characteristic function: A080110.

A080112 := proc(n) option remember; local i; if(1 = n) then RETURN(2); fi; i := A080112(n-1)+1; while(i > 0) do if(A080110(i) > 0) then RETURN(i); fi; i := i+1; od; end;

s[p_, u_] := Sum[JacobiSymbol[j, p], {j, 1, u}]; Select[Range[2, 300], (p = Prime[#]; AllTrue[Range[(p - 1)/2], s[p, #] >= 0 &]) &] (* Jean-François Alcover, Mar 07 2016 *)

@CachedFunction
def A080112(n) :
     if 1 == n : return 2
     i = A080112(n-1) + 1
     while i > 0 :
         if A080110(i) > 0 : return i
         i += 1
[A080112(n) for n in (1..61)] # Peter Luschny, Aug 09 2012

A080115 Primes not in A080114.

Original entry on oeis.org

2, 17, 19, 29, 41, 43, 53, 61, 67, 73, 89, 97, 101, 107, 109, 113, 127, 137, 139, 149, 157, 163, 173, 179, 181, 193, 197, 211, 223, 227, 229, 233, 241, 257, 269, 277, 281, 283, 293, 307, 313, 317, 331, 337, 347, 349, 353, 367, 373, 379, 389, 397, 401, 409, 421

Offset: 1

Antti Karttunen, Feb 11 2003

nonn

Cf. A080113.

Maple
```
A080115 := n -> ithprime(A080113(n));
```

A080120 Dyck path encodings of Legendre's candelabras formed for primes in A080114. (I.e., symmetric rooted plane trees constructed from their quadratic residue sets.)

Original entry on oeis.org

10, 1010, 110100, 1011100010, 101100110010, 1111010110011001010000, 110110111100010101110000100100, 101100101111000100110111000010110010, 1111011110010101110010011011000101011000010000

Offset: 1

Antti Karttunen, Feb 11 2003

nonn

For the 2nd, 5th and 8th term of the sequence, the quadratic residue set of the corresponding prime (5,13,37, of the form 4k+1) has been converted from symmetric to complementarily symmetric as 1001->1010, 101100001101->101100110010, 101100101111000100001000111101001101->101100101111000100110111000010110010, for the others (of the form 4k+3), it is the quadratic residue set encoded as in A055094 (with +1 mapped to 1 and -1 to 0).

Antti Karttunen, Illustration of initial terms

Same sequence in decimal: A080118. Cf. A080114.

A080120 := n -> convert(A080118(n),binary);

a(n) = A063171(A080119(n)).

A095094 Number of A080114-primes in range ]2^n,2^(n+1)].

Original entry on oeis.org

1, 2, 2, 2, 3, 4, 7, 10, 16, 30, 51, 88, 153, 277, 509, 905, 1660, 3079, 5535, 10234, 19053

Offset: 1

Antti Karttunen, Jun 01 2004

A. Karttunen and J. Moyer, C-program for computing the initial terms of this sequence
Index entries for sequences related to occurrences of various subsets of primes in range ]2^n,2^(n+1)]

Cf. A080114.

is(p) = {if(!isprime(p), return(0)); my(s=0); for(i=1, (p-1)/2, if((s+=kronecker(i, p))<0, return(0))); 1; }
a(n) = sum(p=2^n+1, 2^(n+1), is(p)); \\ Jinyuan Wang, Jul 20 2020

a(n) = A036378(n) - A095095(n).

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.

A166051 Nonsquare integers of the form 4n+1 for which Sum_{i=1..u} J(i,4n+1) is never negative for any u in range [1,(2n)], where J(i,k) is the Jacobi symbol.

Original entry on oeis.org

5, 13, 21, 37, 85, 93, 165

Offset: 1

Antti Karttunen, Oct 08 2009

nonn

Conjecture: There are no more terms after 165. (Checked up to A016813(290511) = 1162045.) If this is true, then also 5, 13 and 37 are only 4k+1 primes in A080114.

Setwise difference of A016754 and A166049.

def is_what(n, k):
    s = 0
    for i in (1..k):
        s += jacobi_symbol(i, n)
        if s < 0: return False
    return not is_square(n)
def A166051_list(n):
    return [m for m in range(1, n + 1, 4) if is_what(m, m // 2)]
A166051_list(1000) # Peter Luschny, Aug 08 2012

cp's OEIS Frontend

A080112 Positions of A080114 in A000040.

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A080115 Primes not in A080114.

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A080120 Dyck path encodings of Legendre's candelabras formed for primes in A080114. (I.e., symmetric rooted plane trees constructed from their quadratic residue sets.)

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A095094 Number of A080114-primes in range ]2^n,2^(n+1)].

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PARI

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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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Author

Keywords

Comments

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Crossrefs

Programs

Mathematica

PARI

Sage

Formula

A166051 Nonsquare integers of the form 4n+1 for which Sum_{i=1..u} J(i,4n+1) is never negative for any u in range [1,(2n)], where J(i,k) is the Jacobi symbol.

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Sage