A256385 -id:A256385 - cp's OEIS frontend

A256581 Number of conditions on m under which m^n + (m+1)^n + ... + (m+k)^n is composite for every k>=1 (see comment).

2, 3, 2, 7, 5, 7, 7, 11, 5, 7, 7, 31, 23, 11, 9, 15, 17, 31, 31, 47, 23, 15, 29, 47, 23, 15, 7, 15, 11, 31, 47, 95, 47, 15, 11, 127, 95, 47, 39, 63, 71, 63, 63, 95, 47, 31, 71, 95, 71, 47, 31, 31, 47, 63, 39, 47, 23, 15, 23, 255, 191, 127, 111, 95, 71, 127

Offset: 1

Vladimir Shevelev, Apr 02 2015

nonn

We consider a(n), n>=2, conditions of the form: all numbers P_i(m) are composite, i = 1, ..., a(n), where P_i(m) is a polynomial of power n+1. It could be proved that S_k(m)= m^n + (m+1)^n + ... + (m+k)^n, as a polynomial in m of degree n+1, is divisible by k+1. Let S*_k(m) = S_k(m)/(k+1). So we have
S_k(m)=S*_k(m)*(k+1)=(T_k(m)/b(n))*(k+1), (1)
where b(n)=A064538(n) and, by the definition of A064538, T_k(m) = b(n)*S*_k(m) is a polynomial with integer coefficients.
It is clear that (1) could be prime only if k+1>=2 is a divisor of b(n). In this case we should require that (1) be a composite number. We have exactly A000005(b(n))-1 such requirements. In case of n=1, a(n)=2 (see A089306, A077654).
Remark. Sometimes some considered conditions satisfy trivially. For example, both a(3)=2 conditions for every  m>=2 evidently hold, such that every number of the form m^3 + (m+1)^3 + ... +(m+k)^3 is composite.
Note that essentially this method is useful only in case of even n. Indeed, according to our comment in A001017, in case of odd n>=3 the number m^n + (m+1)^n + ... + (m+k)^n is composite for every k>=1. - Vladimir Shevelev, Apr 06 2015

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

Cf. A000005, A064538, A089306 (a(1)=2), A256385 (a(2)=3), A256546 (a(4)=7).

For n>=2, a(n) = A000005(A064538(n))-1.

More terms from Peter J. C. Moses, Apr 02 2015

A256546 Numbers n such that n^4 + (n+1)^4 + ... + (n+k)^4 is composite for every k>=0.

Original entry on oeis.org

11, 17, 18, 22, 29, 32, 35, 39, 41, 44, 46, 49, 50, 51, 53, 55, 57, 59, 60, 61, 64, 66, 69, 70, 73, 75, 76, 77, 79, 81, 86, 92, 95, 96, 101, 102, 103, 107, 112, 113, 114, 116, 117, 118, 120, 125, 131, 133, 135, 137, 138, 141, 143, 144, 147, 148, 149, 150, 151

Offset: 1

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

nonn

Number n is in the sequence if and only if the following seven numbers are all composite:
P_1(n) = 2n^4 + 4n^3 + 6n^2 + 4n + 1,
P_2(n) = 3n^4 + 12n^3 + 30n^2 + 36n + 17,
P_3(n) = 5n^4 + 40n^3 + 180n^2 + 400n + 354,
P_4(n) = 6n^4 + 60n^3 + 330n^2 + 900n + 979,
P_5(n) = 10n^4 + 180n^3 + 1710n^2 + 8100n + 15333,
P_6(n) = 15n^4 + 420n^3 + 6090n^2 + 44100n + 127687,
P_7(n) = 30n^4 + 1740n^3 + 51330n^2 + 756900n + 4463999.
For a generalization, see comment in A256581.

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

Cf. A000583, A256385, A256503, A089306, A077654, A256385, A256581.

[n: n in [0..2*10^2] | not IsPrime(2*n^4+4*n^3+6*n^2 +4*n+1) and not IsPrime(3*n^4+12*n^3+30*n^2+36*n+17) and not IsPrime(5*n^4+40*n^3+180*n^2+400*n+354) and not IsPrime(6*n^4+60*n^3+330*n^2+900*n+979) and not IsPrime(10*n^4+ 180*n^3+1710*n^2+8100*n+15333) and not IsPrime(15*n^4+ 420*n^3+6090*n^2+44100*n+127687) and not IsPrime(30*n^4+ 1740*n^3+51330*n^2+756900*n+4463999)]; // Vincenzo Librandi, Apr 03 2015

A256503 Smallest k>=1 such that n^2 + (n+1)^2 + ... + (n+k)^2 is prime or a(n)=0 if there is no such k.

Original entry on oeis.org

1, 1, 5, 1, 1, 2, 1, 0, 1, 0, 0, 1, 5, 1, 0, 0, 1, 5, 1, 0, 5, 1, 5, 1, 1, 0, 0, 0, 1, 1, 0, 1, 0, 1, 1, 0, 0, 5, 1, 0, 0, 1, 0, 0, 0, 0, 1, 5, 0, 1, 5, 2, 0, 0, 0, 2, 0, 0, 0, 1, 0, 2, 5, 0, 1, 2, 0, 0, 1, 1, 0, 1, 0, 0, 0, 2, 0, 0, 1, 0, 5, 1, 0, 1, 1, 2, 1

Offset: 1

Vladimir Shevelev, Mar 31 2015

nonn

Every term is either 0 or 1 or 2 or 5.

a(n)=0 if and only if n is in A256385.

Cf. A000290, A256385, A089306.

1) if 2n^2+2n+1 is prime, then a(n)=1;
2) if 2n^2+2n+1 is not prime, but 3n^2+6n+5 is prime, then a(n)=2;
3) if 2n^2+2n+1 and 3n^2+6n+5 are both composite numbers, but 6n^2+30n+55 is prime, then a(n)=5;
4) otherwise, a(n)=0.

More terms from Peter J. C. Moses, Mar 31 2015

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.

Original entry on oeis.org

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

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

nonn

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)

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

Cf. A000583, A256385, A256503, A256546.

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.

A256812 Smallest k>=1 such that n^6 + (n+1)^6 + ... + (n+k)^6 is prime or a(n)=0 if there is no such k.

Original entry on oeis.org

0, 2, 0, 2, 0, 0, 0, 0, 5, 0, 0, 6, 20, 0, 0, 5, 20, 0, 20, 0, 5, 13, 5, 5, 0, 0, 0, 0, 5, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 6, 20, 0, 41, 2, 0, 5, 13, 0, 0, 0, 0, 6, 0, 0, 0, 5, 0, 0, 0, 0, 0, 0, 0, 0, 5, 0, 20, 0, 20, 41, 0, 0, 0, 5, 0, 0, 0, 41, 0, 13, 20

Offset: 1

Vladimir Shevelev and Peter J. C. Moses, Apr 10 2015

nonn

Using similar arguments as in comment in A256547, we obtain that every term is 0,2,5,6,13,20,41.

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

Cf. A001014, A089306, A256385, A256503, A256546, A256547, A256581.

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A256581 Number of conditions on m under which m^n + (m+1)^n + ... + (m+k)^n is composite for every k>=1 (see comment).

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A256546 Numbers n such that n^4 + (n+1)^4 + ... + (n+k)^4 is composite for every k>=0.

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A256503 Smallest k>=1 such that n^2 + (n+1)^2 + ... + (n+k)^2 is prime or a(n)=0 if there is no such k.

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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.

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A256812 Smallest k>=1 such that n^6 + (n+1)^6 + ... + (n+k)^6 is prime or a(n)=0 if there is no such k.

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