

A241927


Smallest k^2>=1 such that nk^2 is semiprime p*q in FermiDirac 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.


4



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
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OFFSET

1,1


COMMENTS

A semiprime in FermiDirac 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 FermiDirac 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 (p<q are both A050376 terms of the same parity). Thus n=((p+q)/2)^2 and n((pq)/2)^2=p*q is FermiDirac semiprime. Hence, a(n)>=1 is a square not exceeding ((pq)/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, p<q are both in A050376, 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^2<q (other cases are considered analogously), such that m^2  k^2 = s^2*q (it is clear that s is also in A050376). Consider two principal cases: 1) mk = s, m+k = s*q; 2) mk = s^2, m+k = q. In 1) k=ms, in 2) k=ms^2. In view of the minimality of k, we should accept 2) and thus mk=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).


REFERENCES

V. S. Shevelev, Multiplicative functions in the FermiDirac arithmetic, Izvestia Vuzov of the NorthCaucasus region, Nature sciences 4 (1996), 2843 (in Russian; MR 2000f: 11097, pp. 39123913).


LINKS

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


EXAMPLE

a(17)=9, since 9 is the smallest square such that 179 = 8 = 2*4 is a FermiDirac semiprime.


CROSSREFS

Cf. A000290, A001358, A050376, A176525, A100570, A152522, A152451, A156537, A241922.
Sequence in context: A273429 A273915 A270969 * A297033 A194318 A306459
Adjacent sequences: A241924 A241925 A241926 * A241928 A241929 A241930


KEYWORD

nonn


AUTHOR

Vladimir Shevelev, May 02 2014


STATUS

approved



