A165582 -id:A165582 - cp's OEIS frontend

A165578 Partial sums of A011627.

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

A165581 Legendre symbol (n,524287).

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

Offset: 0

Antti Karttunen, Sep 22 2009

524287 is the 7th Mersenne prime, A000668(7) = 2^19 - 1.

Periodic with period 524287. - Charles R Greathouse IV, Aug 06 2012

A. Karttunen, Table of n, a(n) for n = 0..524287

Partial sums: A165582.

a(n)=kronecker(n,524287) \\ Charles R Greathouse IV, Aug 06 2012

A226519 Irregular triangle read by rows: T(n,k) = Sum_{i=0..k} Legendre(i,prime(n)).

Original entry on oeis.org

1, 1, 0, 1, 0, -1, 0, 1, 2, 1, 2, 1, 0, 1, 0, 1, 2, 3, 2, 1, 0, 1, 0, 1, 0, 1, 2, 1, 0, -1, -2, -1, 0, -1, 0, 1, 2, 1, 2, 1, 0, -1, 0, 1, 0, -1, -2, -1, -2, -1, 0, 1, 0, -1, 0, 1, 2, 3, 2, 3, 2, 3, 2, 1, 0, -1, 0, 1, 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.

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

József Beck, 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.

G. C. Greubel, Rows n = 1..50 of the irregular triangle, flattened

A variant of A226518, which is the main entry for this triangle.

Cf. A165582, A226518.

A226519:= func< n,k | n eq 1 select k else  (&+[JacobiSymbol(j, NthPrime(n)): j in [0..k]]) >;
[A226519(n,k) : k in [1..NthPrime(n)-1], n in [1..15]]; // G. C. Greubel, Oct 05 2024

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

Table[P = Prime[n]; Table[JacobiSymbol[k,P], {k,P-1}]//Accumulate, {n,15}]// Flatten (* G. C. Greubel, Oct 05 2024 *)

def A226519(n,k): return k if n==1 else sum(jacobi_symbol(j, nth_prime(n)) for j in range(k+1))
flatten([[A226519(n,k) for k in range(1,nth_prime(n))] for n in range(1,16)]) # G. C. Greubel, Oct 05 2024

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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A165578 Partial sums of A011627.

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A165581 Legendre symbol (n,524287).

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A226519 Irregular triangle read by rows: T(n,k) = Sum_{i=0..k} Legendre(i,prime(n)).

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A165577 Partial sums of A011626.

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