A256547 Smallest k>=1 such that n^4 + (n+1)^4 + ... + (n+k)^4 is prime or a(n)=0 if there is no such k.

1, 1, 1, 1, 5, 1, 4, 1, 1, 2, 0, 1, 1, 1, 29, 1, 0, 0, 29, 2, 29, 0, 29, 29, 1, 1, 1, 2, 0, 2, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 0, 29, 1, 0, 5, 0, 9, 1, 0, 0, 0, 2, 0, 1, 0, 29, 0, 2, 0, 0, 0, 14, 1, 0, 9, 0, 1, 1, 0, 0, 29, 1, 0, 1, 0, 0, 0, 1, 0, 14, 0, 1, 9, 2

Offset

1,5

Comments

Every term is 0, 1, 2, 4, 5, 9, 14, or 29.

a(n)=0 if and only if n is in A256546.

From Vladimir Shevelev, Apr 09 2015: (Start)

Indeed, denote by S_k(n) = n^4 + (n+1)^4 + ... + (n+k)^4. If n=1, k=m-1, then, as is known,

s(m) = S_(m-1)(1) = 1^4 + 2^4 + ... + m^4 = (6*m^5 + 15*m^4 + 10*m^3 - m)/30 (1)

such that

S_k(n) = s(n+k) - s(n-1). (2)

Since S_(-1)(n) = 0, then S_k(n) as a polynomial is divisible by k+1. Put

S*_k(n) = S_k(n)/(k+1). So we have

S_k(n) = S*_k(n)*(k+1) = T_k(n)/30*(k+1), (3)

where T_k(n) = 30*S_k(n) is (by (1)) a polynomial with integer coefficients.

For k>=1, it is clear that (3) could be prime for some n only if k+1 is a divisor of 30, i.e., k = 1,2,4,5,9,14 or 29. The smallest n when all these values of a(n) appeared is n=62. If for some n all numbers n^4 + (n+1)^4 + ... + (n+k)^4 are composite for k = 1,2,4,5,9,14 and 29, then a(n)=0. (End)

Links

Peter J. C. Moses, Table of n, a(n) for n = 1..1000

Formula

1) If P_1(n) is prime, then a(n)=1;

2) if P_1(n) is composite, but P_2(n) is prime, then a(n)=2;

3) if P_1(n) and P_2(n) are composite, but P_3(n) is prime, then a(n)=4;

4) if P_1(n), P_2(n), and P_3(n) are composite, but P_4(n) is prime, then a(n)=5;

5) if P_1(n), P_2(n), P_3(n), and P_4(n) are composite, but P_5(n) is prime, then a(n)=9;

6) if P_1(n), P_2(n), P_3(n), P_4(n), and P_5(n) are composite, but P_6(n) is prime, then a(n)=14;

7) if P_1(n), P_2(n), P_3(n), P_4(n), P_5(n), and P_6(n) are composite, but P_7(n) is prime, then a(n)=29;

8) otherwise a(n)=0.

Here P_i(n), i=1,...,7, are defined in comment in A256546.

Crossrefs

Cf. A000583, A256385, A256503, A256546.

Sequence in context: A096615 A295882 A127551 * A023138 A194704 A108170

Adjacent sequences: A256544 A256545 A256546 * A256548 A256549 A256550

Keyword

nonn

Author

Vladimir Shevelev and Peter J. C. Moses, Apr 01 2015

Status

approved