A241922 -id:A241922 - cp's OEIS frontend

A100570 Numbers that are not the sum of a square and a semiprime.

1, 2, 3, 12, 17, 28, 32, 72, 108, 117, 297, 657

Offset: 1

Jonathan Vos Post, Nov 29 2004

No others up to 300000. Computed in collaboration with Ray Chandler. It appears that this sequence is finite, that is, that almost every positive integer is the sum of a semiprime and a square number. There are probably no further exceptions after a(12)=657.
The statement about the finiteness of this sequence (namely, a(n)<=657) is much stronger than the Goldbach binary conjecture. Indeed, a much weaker conjecture, that this sequence contains no perfect squares >1, already implies the Goldbach conjecture. Cf. comment in A241922. - Vladimir Shevelev, May 01 2014
From Daniel Mikhail, Nov 23 2020: (Start)
There are no new terms in this sequence between 658 and 2^28.
Notably, A014090 (numbers that are not the sum of a square and one prime) is a known infinite sequence. (End)

			From _Daniel Mikhail_, Nov 23 2020: (Start)
An integer m is in this set if, for any primes, p and q, there does not exist a natural k, such that m-k^2 = p*q.
Consider m=12 and all k such that k^2 < 12: k is either 0,1,4, or 9.
  12 - 0 = 12 = 2*2*2*3 => not semiprime;
  12 - 1 = 11 => not semiprime;
  12 - 4 = 8 = 2*2*2 => not semiprime;
  12 - 9 = 3 => not semiprime.
Therefore, 12 is a term. (End)

Daniel Mikhail, Brief glance of the search for integer solutions to a(n)-k^2 = semiprime

Cf. A000290, A001358, A046903, A241922, A014090.

lim = 657; Complement[Range[lim],Select[Flatten[Outer[Plus,Select[Range[lim], PrimeOmega[#] == 2 &],Table[i^2, {i, 0, Sqrt[lim]}]]], # <= lim &]] (* Robert Price, Apr 10 2019 *)

An integer is not an element for any integers i, j of the pairwise sum of {A001358(i)} and {A000290(j)}.

A241927 Smallest k^2>=1 such that n-k^2 is semiprime p*q in Fermi-Dirac arithmetic (A176525) with additional requirement that, if n is a square, then p and q are of the same parity; or a(n)=2 if there is no such k^2.

Original entry on oeis.org

2, 2, 2, 2, 2, 2, 1, 2, 1, 4, 1, 4, 1, 4, 1, 1, 9, 4, 1, 2, 1, 1, 1, 4, 4, 4, 1, 1, 1, 4, 4, 4, 1, 1, 1, 1, 1, 4, 1, 1, 9, 4, 4, 9, 1, 1, 1, 4, 4, 4, 1, 1, 1, 4, 4, 1, 9, 1, 1, 9, 4, 4, 1, 1, 1, 1, 4, 4, 1, 1, 9, 4, 4, 9, 1, 1, 1, 1, 4, 4, 4, 25, 1, 4, 9, 1, 1, 1, 4, 4, 4, 1, 1, 1, 1, 1, 4, 4, 1, 1, 1, 4, 4, 4, 25

Offset: 1

Vladimir Shevelev, May 02 2014

nonn

A semiprime in Fermi-Dirac arithmetic is a product of two distinct terms of A050376, or, equivalently, an infinitary semiprime. The conjecture that every even number>=4 is a sum of two A050376 terms is a weaker form of the Goldbach conjecture; as such, it is natural to refer to it as a Goldbach conjecture in Fermi-Dirac arithmetic (FDGC).
Let us prove that the condition {a(m^2) differs from 2} is equivalent to the FDGC.
Indeed, from the FDGC for a perfect square n>=4, we have 2*sqrt(n)=p+q (pA050376 terms of the same parity). Thus n=((p+q)/2)^2 and n-((p-q)/2)^2=p*q is Fermi-Dirac semiprime. Hence, a(n)>=1 is a square not exceeding ((p-q)/2)^2. Thus the condition {a(m^2) differs from 2} is necessary for the truth of the FDGC.
Let us prove that the condition {a(m^2) differs from 2} is also sufficient. Indeed, a(m^2)-k^2 = p*q, where, say, pA050376, and p,q are of the same parity. If p,q are primes, then the proof repeats one in A241922. Let, e.g., p=s^2A050376). Consider two principal cases: 1) m-k = s, m+k = s*q; 2) m-k = s^2, m+k = q. In 1) k=m-s, in 2) k=m-s^2. In view of the minimality of k, we should accept 2) and thus m-k=p, m+k=q. So, 2*m=p+q as the FDGC requires.
The sequence of numbers n for which a(n)=2 begins 1, 2, 3, 4, 5, 6, 8, 20, ... (A241947).

			a(17)=9, since 9 is the smallest square such that 17-9 = 8 = 2*4 is a Fermi-Dirac semiprime.

V. S. Shevelev, Multiplicative functions in the Fermi-Dirac arithmetic, Izvestia Vuzov of the North-Caucasus region, Nature sciences 4 (1996), 28-43 (in Russian; MR 2000f: 11097, pp. 3912-3913).

Peter J. C. Moses, Table of n, a(n) for n = 1..10000

Cf. A000290, A001358, A050376, A176525, A100570, A152522, A152451, A156537, A241922.

A241947 Numbers n for which A241927(n) = 2.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 20

Offset: 1

Vladimir Shevelev, May 03 2014

If the sequence contains no perfect squares>4, then the Goldbach conjecture in Fermi-Dirac arithmetic (FDGC) is true (see comment in A241927).
Essentially, the sequence is the Fermi-Dirac analog of A100570. Since A100570 is conjecturally finite, it is natural to suppose that this sequence is also finite.
There is not another term up to 10^6. - Peter J. C. Moses, May 05 2014
Thus, if 20 is the last term of the sequence, then the FDGC is true. - Vladimir Shevelev, May 05 2014

Cf. A241927, A241922, A100570.

Name corrected by Michel Marcus, Dec 14 2018

A242165 Smallest k>=0, such that n+/-k are both Fermi-Dirac primes (A050376).

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 3, 2, 0, 0, 1, 0, 3, 2, 3, 0, 1, 0, 3, 2, 3, 0, 1, 0, 9, 4, 3, 6, 5, 0, 9, 2, 3, 0, 1, 0, 3, 2, 3, 0, 1, 0, 3, 2, 9, 0, 5, 6, 3, 4, 9, 0, 1, 0, 9, 4, 3, 6, 5, 0, 15, 2, 3, 0, 1, 0, 7, 4, 3, 4, 5, 0, 1, 0, 1, 0, 5, 4, 3, 14, 9, 0, 7, 10, 9, 4, 13, 6, 7, 0

Offset: 2

Vladimir Shevelev, May 05 2014

nonn

The existence of a(n)>=0 for all n >= 2 is equivalent to the Goldbach conjecture in Fermi-Dirac arithmetic (cf. comment in A241927) that every even number >= 4 is a sum of two terms of A050376 (it is slightly weaker than Goldbach conjecture for primes).

V. S. Shevelev, Multiplicative functions in the Fermi-Dirac arithmetic, Izvestia Vuzov of the North-Caucasus region, Nature sciences 4 (1996), 28-43 (in Russian; MR 2000f: 11097, pp. 3912-3913).

Peter J. C. Moses, Table of n, a(n) for n = 2..10001

Cf. A082467, A241922, A241927, A241947.

a(A050376(n)) = 0.

cp's OEIS Frontend

A100570 Numbers that are not the sum of a square and a semiprime.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A241927 Smallest k^2>=1 such that n-k^2 is semiprime p*q in Fermi-Dirac arithmetic (A176525) with additional requirement that, if n is a square, then p and q are of the same parity; or a(n)=2 if there is no such k^2.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

A241947 Numbers n for which A241927(n) = 2.

Views

Author

Keywords

Comments

Crossrefs

Extensions

A242165 Smallest k>=0, such that n+/-k are both Fermi-Dirac primes (A050376).

Views

Author

Keywords

Comments

References

Links

Crossrefs

Formula