cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-6 of 6 results.

A141095 Number of unordered pairs of coprime nonprime numbers that sum to 2n.

Original entry on oeis.org

1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0, 0, 2, 1, 0, 1, 0, 1, 2, 0, 1, 2, 0, 1, 3, 0, 1, 4, 1, 1, 1, 0, 3, 4, 1, 0, 4, 0, 4, 3, 0, 4, 5, 1, 1, 3, 1, 4, 6, 0, 0, 4, 1, 6, 6, 1, 5, 7, 1, 5, 3, 0, 8, 7, 0, 0, 10, 3, 9, 9, 0, 6, 5, 2, 8, 5, 2, 9, 10, 1, 5, 11, 2, 9, 12, 0, 3, 12, 4, 11, 5, 2, 12, 9, 1, 4, 14
Offset: 1

Views

Author

T. D. Noe, Jun 02 2008

Keywords

Comments

Nonprime numbers are 1 and the composite numbers. See A141097 for pairs of coprime composite numbers. It appears that a(n) > 0 except for the 26 values of 2n given in A141096.

Examples

			a(17)=2 because 34 = 1+33 = 9+25.

Links

Crossrefs

Cf. A141099, A141100.

Programs

  • Mathematica
    Table[cnt=0; Do[If[GCD[2n-i,i]==1 && !PrimeQ[i] && !PrimeQ[2n-i], cnt++ ], {i,1,n,2}]; cnt, {n,100}]

A141097 Number of unordered pairs of coprime composite numbers that sum to 2n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 2, 0, 1, 3, 0, 1, 0, 0, 3, 3, 0, 0, 3, 0, 3, 2, 0, 3, 4, 0, 1, 2, 1, 4, 5, 0, 0, 3, 1, 5, 5, 0, 4, 6, 0, 5, 2, 0, 7, 6, 0, 0, 9, 2, 8, 8, 0, 6, 4, 1, 8, 4, 1, 9, 9, 1, 4, 10, 2, 8, 11, 0, 3, 11, 3, 10, 4, 2, 12, 8, 1, 4, 13, 2
Offset: 1

Views

Author

T. D. Noe, Jun 02 2008

Keywords

Comments

See A141095 for pairs of coprime nonprime numbers. It appears that a(n) > 0 except for the 43 values of 2n given in A141098. Roberts says that A. M. Vaidya proved that a(n) > 0 for all sufficiently large n.

Examples

			a(17)=1 because 34 = 9+25.

References

  • Joe Roberts, "Lure of the Integers", The Mathematical Association of America, 1992, p. 190.

Links

Crossrefs

Cf. A141099, A141100.

Programs

  • Mathematica
    Table[cnt=0; Do[If[GCD[2n-i,i]==1 && !PrimeQ[i] && !PrimeQ[2n-i], cnt++ ], {i,3,n,2}]; cnt, {n,100}]

A141099 Number of unordered pairs of odd nonprime numbers that sum to 2n.

Original entry on oeis.org

1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 1, 1, 1, 1, 2, 0, 2, 3, 0, 2, 3, 1, 2, 3, 3, 2, 4, 2, 3, 5, 1, 4, 6, 1, 5, 5, 3, 4, 7, 3, 4, 7, 4, 4, 9, 4, 5, 8, 4, 7, 9, 5, 6, 8, 6, 7, 10, 6, 6, 13, 5, 7, 13, 5, 10, 11, 8, 8, 11, 9, 10, 14, 9, 9, 16, 7, 12, 15, 8, 12, 15, 9, 10, 17, 14, 11, 16, 12, 12, 19, 11, 13, 19
Offset: 1

Views

Author

T. D. Noe, Jun 02 2008, Jun 05 2008

Keywords

Comments

See A141100 for pairs of odd composite numbers. We have a(n) > 0 except for the 8 values of 2n given in A046458.

Examples

			a(18)=3 because 36 = 1+35 = 9+27 = 15+21.

Links

Crossrefs

Cf. A141095, A141097.

Programs

  • Mathematica
    Table[cnt=0; Do[If[ !PrimeQ[i] && !PrimeQ[2n-i], cnt++ ], {i,1,n,2}]; cnt, {n,100}]

A234537 Number of nontrivial non-Goldbach partitions of 2n into two odd parts (with smaller part greater than 1).

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 1, 1, 2, 2, 2, 2, 3, 4, 4, 5, 4, 4, 7, 6, 6, 7, 7, 6, 8, 9, 8, 10, 10, 8, 12, 10, 10, 14, 12, 11, 13, 13, 12, 15, 15, 12, 16, 17, 13, 18, 18, 16, 21, 18, 17, 20, 20, 18, 21, 20, 18, 22, 23, 17, 26, 25, 21, 28, 25, 23, 27, 28, 26, 27, 27, 24
Offset: 1

Views

Author

Wesley Ivan Hurt, Dec 27 2013

Keywords

Comments

Number of partitions of 2n into two odd parts with at least 1 composite part less than 2n-1.

Examples

			a(15) = 4; there are exactly 4 partitions of 2*15 = 30 into two odd parts with at least one composite part less than 2*15 - 1 = 29: (27,3), (25,5), (21,9), (15,15).

Links

Crossrefs

Cf. A002375, A141100.

Programs

  • Mathematica
    Table[Ceiling[n/2] - 1 - Sum[(PrimePi[i] - PrimePi[i - 1])*(PrimePi[2 n - i] - PrimePi[2 n - i - 1]), {i, 3, n}], {n, 100}]
  • PARI
    a(n)=my(s); forstep(k=3,n,2, if(!isprime(k) || !isprime(2*n-k), s++)); s \\ Charles R Greathouse IV, Jul 30 2016
  • Python
    from sympy import isprime
def a(n): return sum(1 for k in range(3, n + 1, 2) if not isprime(k) or not isprime(2*n - k))
print([a(n) for n in range(1, 101)]) # Indranil Ghosh, Jul 11 2017

Formula

a(n) = ceiling(n/2) - 1 - Sum_{i=3..n} A010051(i) * A010051(2n-i).

A234716 Number of odd composite integers k, such that n-1 < k < 2n-2.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 1, 1, 2, 1, 1, 2, 2, 3, 4, 3, 3, 4, 5, 5, 6, 5, 5, 6, 6, 6, 7, 6, 7, 8, 8, 8, 9, 9, 9, 9, 9, 9, 10, 10, 10, 11, 11, 12, 13, 12, 13, 14, 15, 14, 15, 14, 14, 15, 15, 14, 15, 14, 15, 16, 17, 18, 19, 19, 19, 19, 19, 20, 21, 20, 20, 21, 22, 23
Offset: 1

Views

Author

Wesley Ivan Hurt, Dec 29 2013

Keywords

Comments

Number of partitions of 2n into two odd parts such that the largest part is an odd composite less than 2n-2.

Examples

			a(9) = 2; There are two partitions of 2(9) = 18 into two odd parts such that the largest part is an odd composite less than 2(9)-2 = 16: (15,3) and (9,9).

Links

Crossrefs

Cf. A002375, A141100.

Programs

  • Maple
    with(numtheory); A234716:=n->floor((n-1)/2) - pi(2*n-3) + pi(n-1); seq(A234716(n), n=1..100);
  • Mathematica
    Table[Floor[(n - 1)/2] - PrimePi[2 n - 3] + PrimePi[n - 1], {n, 100}]

Formula

a(n) = floor((n-1)/2) - pi(2n-3) - pi(n-1).

A226982 a(n) = ceiling(n/2) - primepi(n).

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 4, 4, 5, 5, 5, 5, 5, 5, 6, 6, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 10, 10, 11, 11, 11, 11, 12, 12, 13, 13, 13, 13, 13, 13, 14, 14, 15, 15, 15
Offset: 1

Views

Author

Wesley Ivan Hurt, Jun 25 2013

Keywords

Comments

The number of partitions of 2n into exactly two parts such that the smaller part is an odd composite integer, n > 1.
Sequence decreases by 1 when n is an even prime and increases by 1 when n is an odd composite. - Wesley Ivan Hurt, Dec 27 2013

Examples

			a(18) =2. 2*18=38 has two partitions into exactly two odd parts with smallest part composite: (27,9) and (21,15). - _Wesley Ivan Hurt_, Dec 27 2013

Links

Crossrefs

Cf. A000720, A004526, A002375, A141100.

Programs

  • Maple
    seq(ceil(n/2)-numtheory[pi](n),n=1..100);

Formula

a(n) = floor((n+1)/2) - pi(n) = A004526(n+1) - A000720(n).
a(n) = n - A004526(n) - A000720(n). - Wesley Ivan Hurt, Dec 27 2013
Showing 1-6 of 6 results.

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