A088934 -id:A088934 - cp's OEIS frontend

A088910 Conjectured minimal required number k of terms in a representation n=sum_(i=1..k)e_i*(p_i)^2 by distinct primes p_i, where e_i is 1 or -1.

4, 3, 4, 4, 1, 2, 5, 5, 4, 1, 4, 4, 3, 2, 4, 3, 2, 5, 5, 4, 3, 2, 4, 3, 2, 1, 4, 4, 3, 2, 3, 5, 4, 3, 2, 4, 3, 4, 3, 3, 2, 5, 5, 4, 3, 2, 4, 3, 2, 1, 4, 4, 3, 2, 3, 5, 4, 3, 2, 4, 5, 4, 3, 3, 4, 3, 5, 4, 3, 4, 3, 3, 2, 3, 2, 4, 3, 4, 3, 4, 4, 3, 4, 3, 5, 4, 4, 3, 4, 5, 5, 4, 3, 4, 4, 3, 2, 3, 4, 4, 3, 4, 5, 5, 4

Offset: 0

Hugo Pfoertner, Oct 24 2003

nonn

It is conjectured that all sequence terms are <=5. The terms with a(n)=5 were provided by W. Edwin Clark.

			The following are representation with the minimal number of terms:
  a(0)=4: 0=7^2-11^2-17^2+19^2;
  a(1)=3: 1=7^2+11^2-13^2;
  a(4)=1: 4=2^2;
  a(5)=2: 5=3^2-2^2;
  a(6)=5: 6=-(2^2)+3^2+7^2+11^2-13^2.

Robert E. Dressler, Louis Pigno, and Robert Young, Sums of squares of primes, Nordisk Mat. Tidskr. 24 (1976), 39-40.
Hugo Pfoertner, Conjectured minimal representations of n by squaresof distinct primes (Table for n<=400).

Cf. A088934 (maximum required prime in representation), A048261, A088908, A088909.

A089294 Smallest prime p_k such that n can be written as a "plus-minus" sum n=sum_(i=1..k)e_i*(p_i)^2 with distinct primes p_i<=p_k, where e_i is 1 or -1.

Original entry on oeis.org

19, 13, 19, 13, 2, 3, 13, 19, 13, 3, 13, 7, 5, 3, 13, 7, 5, 13, 13, 7, 5, 5, 13, 13, 7, 5, 13, 13, 7, 5, 5, 13, 13, 7, 5, 13, 7, 7, 5, 13, 7, 17, 11, 11, 7, 7, 17, 11, 13, 7, 17, 11, 11, 7, 7, 17, 11, 13, 7, 11, 11, 7, 7, 11, 13, 7, 17, 11, 11, 7, 7, 17, 11, 13, 7, 17, 11, 11, 7, 7, 17, 11

Offset: 0

W. Edwin Clark Nov 06 2003

nonn

The first terms where this sequence differs from A088934 are n=22,31,52,87,89,98,118,127,...

			a(22)=13 because 22 can be represented as -13^2+11^2+7^2+5^2-2^2.
The corresponding representation with the minimum number of terms is
22=19^2-13^2-11^2-7^2 (A088934(22)=19, A088910(22)=4 terms)

The positions of the records in this sequence are given in A089295. Cf. A088910, A088934.

A089297 Maximum prime required in the representation of the square of the n-th prime A000040(n) by a signed sum of squares of distinct other primes minimizing the number of terms in the sum.

Original entry on oeis.org

29, 37, 19, 19, 19, 31, 19, 17, 19, 23, 29, 31, 31, 37, 37, 47, 43, 47, 53, 59, 59, 59, 71, 67, 71, 73, 79, 79, 83, 83, 97, 97, 101, 101, 109, 109, 127, 127, 127, 137, 139, 131, 149, 139, 151, 149, 163, 167, 191, 173, 167, 179, 179, 191, 191, 193, 193, 193, 199, 211

Offset: 1

Hugo Pfoertner, Nov 18 2003

nonn

a(1) requires 6 squares, a(2) requires 8, a(3) requires 5 and a(4) through a(70) require 3. - David Wasserman, Sep 01 2005

			The first representations different from those in A089296 are
a(6)=31: 13^2 = 169 = 31^2 - 29^2 + 7^2 = -31^2 + 29^2 + 17^2
a(10)=23: 29^2 = 841 = 23^2 + 19^2 - 7^2
a(11)=29: 31^2 = 961 = 29^2 + 13^2 - 7^2

T. Lassila, H. Pfoertner et al., Sum of unique prime squares? Thread in NG sci.math, Oct 21 2003.

Cf. A088934 representation of n by distinct squares of primes, A089296 representation of (n-th prime)^2 with maximum term minimized.

More terms from David Wasserman, Sep 01 2005

cp's OEIS Frontend

A088910 Conjectured minimal required number k of terms in a representation n=sum_(i=1..k)e_i*(p_i)^2 by distinct primes p_i, where e_i is 1 or -1.

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A089294 Smallest prime p_k such that n can be written as a "plus-minus" sum n=sum_(i=1..k)e_i*(p_i)^2 with distinct primes p_i<=p_k, where e_i is 1 or -1.

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A089297 Maximum prime required in the representation of the square of the n-th prime A000040(n) by a signed sum of squares of distinct other primes minimizing the number of terms in the sum.

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