A165477 -id:A165477 - cp's OEIS frontend

A165579 Partial sums of A011628.

0, 1, 2, 1, 2, 1, 0, 1, 2, 3, 2, 1, 0, 1, 2, 3, 4, 3, 4, 5, 4, 3, 2, 3, 2, 3, 4, 3, 4, 5, 6, 7, 8, 9, 8, 7, 8, 9, 10, 9, 8, 7, 6, 5, 4, 3, 4, 3, 2, 3, 4, 5, 6, 5, 4, 5, 6, 5, 6, 5, 6, 5, 6, 7, 8, 7, 8, 7, 6, 5, 4, 5, 6, 5, 6, 5, 6, 5, 4, 3, 2, 3, 2, 1, 0, 1, 0, -1, -2, -1, -2, -1, 0, -1, -2, -3, -4

Offset: 0

Antti Karttunen, Sep 22 2009

sign

Period 233.

A. Karttunen, Table of n, a(n) for n = 0..59881

Similar sequences: A165575-A165578, A165472, A165477, A165482, A165582, A165587, A165592, A165597.

Table[ JacobiSymbol[n, 233], {n, 0, 96}] // Accumulate (* Jean-François Alcover, Oct 08 2013 *)

A226518 Irregular triangle read by rows: T(n,k) = Sum_{i=0..k} Legendre(i,prime(n)).

Original entry on oeis.org

0, 1, 0, 1, 0, 0, 1, 0, -1, 0, 0, 1, 2, 1, 2, 1, 0, 0, 1, 0, 1, 2, 3, 2, 1, 0, 1, 0, 0, 1, 0, 1, 2, 1, 0, -1, -2, -1, 0, -1, 0, 0, 1, 2, 1, 2, 1, 0, -1, 0, 1, 0, -1, -2, -1, -2, -1, 0, 0, 1, 0, -1, 0, 1, 2, 3, 2, 3, 2, 3, 2, 1, 0, -1, 0, 1, 0, 0, 1, 2, 3, 4, 3, 4, 3, 4, 5, 4, 3, 4, 5, 4, 3, 4, 3, 4, 3, 2, 1, 0

Offset: 1

N. J. A. Sloane, Jun 19 2013

Strictly speaking, the symbol in the definition is the Legendre-Jacobi-Kronecker symbol, since the Legendre symbol is defined only for odd primes.
The classical Polya-Vinogradov theorem gives an upper bound.
There is a famous open problem concerning upper bounds on |T(n,k)| for small k.

			Triangle begins:
  0, 1;
  0, 1, 0;
  0, 1, 0, -1, 0;
  0, 1, 2,  1, 2, 1, 0;
  0, 1, 0,  1, 2, 3, 2,  1,  0,  1, 0;
  0, 1, 0,  1, 2, 1, 0, -1, -2, -1, 0, -1,  0;
  0, 1, 2,  1, 2, 1, 0, -1,  0,  1, 0, -1, -2, -1, -2, -1, 0;
  0, 1, 0, -1, 0, 1, 2,  3,  2,  3, 2,  3,  2,  1,  0, -1, 0, 1, 0;
  ...

R. Ayoub, An Introduction to the Analytic Theory of Numbers, Amer. Math. Soc., 1963; p. 320, Theorem 5.1.
Beck, József. Inevitable randomness in discrete mathematics. University Lecture Series, 49. American Mathematical Society, Providence, RI, 2009. xii+250 pp. ISBN: 978-0-8218-4756-5; MR2543141 (2010m:60026). See page 23.
Elliott, P. D. T. A. Probabilistic number theory. I. Mean-value theorems. Grundlehren der Mathematischen Wissenschaften, 239. Springer-Verlag, New York-Berlin, 1979. xxii+359+xxxiii pp. (2 plates). ISBN: 0-387-90437-9 MR0551361 (82h:10002a). See Vol. 1, p. 154.

Alois P. Heinz, Rows n = 1..70, flattened
D. A. Burgess, The distribution of quadratic residues and non-residues, Mathematika 4, 1957, 106--112. MR0093504 (20 #28)
Wikipedia, Legendre symbol.

Partial sums of rows of triangle in A226520.
See A226519 for another version.
Third and fourth columns give A226914, A226915.
See also A226523.
Cf. A165477 (131071st row), A165582.

a226518 n k = a226518_tabf !! (n-1) !! k
a226518_row n = a226518_tabf !! (n-1)
a226518_tabf = map (scanl1 (+)) a226520_tabf
-- Reinhard Zumkeller, Feb 02 2014

A226518:= func< n,k | n eq 1 select k else  (&+[JacobiSymbol(j, NthPrime(n)): j in [0..k]]) >;
[A226518(n,k) : k in [0..NthPrime(n)-1], n in [1..15]]; // G. C. Greubel, Oct 05 2024

with(numtheory);
T:=(n,k)->add(legendre(i,ithprime(n)),i=0..k);
f:=n->[seq(T(n,k),k=0..ithprime(n)-1)];
[seq(f(n),n=1..15)];

Table[p = Prime[n]; Table[JacobiSymbol[k, p], {k, 0, p-1}] // Accumulate, {n, 1, 15}] // Flatten (* Jean-François Alcover, Mar 07 2014 *)

print("# A226518 ");
cnt=1; for(j5=1,9,summ=0; for(i5=0,prime(j5)-1, summ=summ+kronecker(i5,prime(j5)); print(cnt,"  ",summ); cnt++)); \\ Bill McEachen, Aug 02 2013

def A226518(n,k): return k if n==1 else sum(jacobi_symbol(j, nth_prime(n)) for j in range(k+1))
flatten([[A226518(n,k) for k in range(nth_prime(n))] for n in range(1,16)]) # G. C. Greubel, Oct 05 2024

A165576 Partial sums of A165574.

Original entry on oeis.org

0, 1, 2, 3, 4, 3, 4, 3, 4, 5, 4, 5, 6, 7, 6, 5, 6, 7, 8, 7, 6, 5, 6, 7, 8, 9, 10, 11, 10, 9, 8, 9, 10, 11, 12, 13, 14, 15, 14, 15, 14, 13, 12, 13, 14, 13, 14, 13, 14, 15, 16, 17, 18, 17, 18, 17, 16, 15, 14, 13, 12, 13, 14, 13, 14, 13, 14, 13, 14, 15, 16, 15, 16, 15, 16, 17, 16, 15

Offset: 0

Antti Karttunen, Sep 22 2009

nonn

Period 263. There are no negative values as 263 is one of the primes in A095102.

A. Karttunen, Table of n, a(n) for n = 0..59701

Similar sequences: A165575-A165579, A165472, A165477, A165482, A165582, A165587, A165592, A165597.

Accumulate[JacobiSymbol[Range[0,90],263]] (* Harvey P. Dale, Sep 01 2021 *)

A165476 Legendre symbol (n,131071).

Original entry on oeis.org

0, 1, 1, -1, 1, 1, -1, 1, 1, 1, 1, 1, -1, -1, 1, -1, 1, 1, 1, -1, 1, -1, 1, 1, -1, 1, -1, -1, 1, 1, -1, 1, 1, -1, 1, 1, 1, -1, -1, 1, 1, -1, -1, 1, 1, 1, 1, 1, -1, 1, 1, -1, -1, -1, -1, 1, 1, 1, 1, 1, -1, -1, 1, 1, 1, -1, -1, -1, 1, -1, 1, -1, 1, 1, -1, -1, -1, 1, 1, -1, 1, 1, -1, 1, -1, 1

Offset: 0

Antti Karttunen, Sep 21 2009

131071 is the 6th Mersenne prime, A000668(6).

See row 12251 in triangle A226520, A000040(12251) = 131071. - Reinhard Zumkeller, Feb 02 2014

A. Karttunen, Table of n, a(n) for n = 0..131071
Wikipedia, Legendre symbol.

Partial sums: A165477.

Cf. A097343.

a165476 = flip legendreSymbol 131071
-- Where the function legendreSymbol is defined in A097343.
-- Reinhard Zumkeller, Feb 02 2014

a[n_] := JacobiSymbol[n, 2^17 - 1];
Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Jan 13 2023 *)

A165578 Partial sums of A011627.

Original entry on oeis.org

0, 1, 0, 1, 2, 3, 2, 1, 0, 1, 0, 1, 2, 1, 2, 3, 4, 5, 4, 5, 6, 5, 4, 3, 2, 3, 4, 5, 4, 3, 2, 1, 0, 1, 0, -1, 0, 1, 0, -1, -2, -3, -2, -1, 0, 1, 2, 1, 2, 3, 2, 3, 2, 3, 2, 3, 4, 5, 6, 5, 6, 7, 8, 7, 8, 7, 6, 5, 6, 5, 6, 7, 6, 5, 4, 5, 6, 5, 6, 5, 6, 7, 8, 9, 8, 9, 8, 7, 6, 5, 4, 5, 4, 3, 4, 5, 4, 5, 4

Offset: 0

Antti Karttunen, Sep 22 2009

Period 229.

A. Karttunen, Table of n, a(n) for n = 0..229

Similar sequences: A165575-A165579, A165472, A165477, A165482, A165582, A165587, A165592, A165597.

Accumulate[JacobiSymbol[Range[0,120],229]] (* Harvey P. Dale, Feb 12 2022 *)

a(n)=sum(k=1,n%229,kronecker(k,229)) \\ Charles R Greathouse IV, Feb 11 2013

a(n) = -(a(n-1) + a(n-2) + ... + a(n-228)) = a(n-229). - Charles R Greathouse IV, Feb 11 2013

A165577 Partial sums of A011626.

Original entry on oeis.org

0, 1, 0, 1, 2, 1, 0, 1, 0, 1, 2, 3, 4, 3, 2, 1, 2, 1, 0, 1, 0, 1, 0, 1, 0, 1, 2, 3, 4, 5, 6, 5, 4, 5, 6, 5, 6, 5, 4, 3, 4, 3, 2, 3, 4, 3, 2, 3, 4, 5, 4, 3, 2, 3, 2, 1, 0, 1, 0, 1, 0, -1, 0, 1, 2, 3, 2, 1, 0, 1, 2, 3, 2, 3, 4, 5, 6, 7, 8, 9, 8, 9, 10, 9, 10, 11, 10, 11, 10, 11, 12, 11, 12, 11, 10, 9

Offset: 0

Antti Karttunen, Sep 22 2009

sign

Period 227.

A. Karttunen, Table of n, a(n) for n = 0..59701

Similar sequences: A165575-A165579, A165472, A165477, A165482, A165582, A165587, A165592, A165597.

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A165579 Partial sums of A011628.

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A226518 Irregular triangle read by rows: T(n,k) = Sum_{i=0..k} Legendre(i,prime(n)).

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A165576 Partial sums of A165574.

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A165476 Legendre symbol (n,131071).

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A165578 Partial sums of A011627.

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A165577 Partial sums of A011626.

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