The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.



Please make a donation to keep the OEIS running. We are now in our 56th year. In the past year we added 10000 new sequences and reached almost 9000 citations (which often say "discovered thanks to the OEIS").
Other ways to donate

(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A212709 Positive integers not of the form p * c^2 + b^2, with p prime and c and b nonzero integers. 1
1, 2, 5, 10, 25, 58, 130 (list; graph; refs; listen; history; text; internal format)



Many numbers can be ruled out from membership in this sequence with the case c = 1, which then corresponds to p + b^2 (see A064233).

If a positive integer is of the form p * c^2 + b^2, then it may potentially have two different factorizations in Z[sqrt(-p)] (assuming that is not a unique factorization domain, of course): the familiar factorization in Z, and (c + b sqrt(-p))(c - b sqrt(-p)).

There are no more terms <= 2*10^9. - Donovan Johnson, May 30 2012


Table of n, a(n) for n=1..7.


Since 24 can be expressed as 5 * 2^2 + 2^2, it is not in the sequence.

No such expression exists for 25, hence it is in the sequence.

Since 26 can be expressed as 17 * 1^2 + 3^2, it is not in the sequence.


max = 10^5; Complement[Range[max], Flatten[Table[Prime[p]a^2 + b^2, {p, PrimePi[max]}, {a, Ceiling[Sqrt[max/2]]}, {b, Ceiling[Sqrt[max]]}]]]


(PARI) v=vectorsmall(10^5, i, 1); forprime(p=2, #v, for(a=1, sqrtint(#v\p), b=0; while((t=p*a^2+b++^2)<=#v, v[t]=0))); for(i=1, #v, if(v[i], print1(i", "))) \\ Charles R Greathouse IV, May 29 2012


Sequence in context: A162963 A297860 A002094 * A264867 A115725 A305577

Adjacent sequences:  A212706 A212707 A212708 * A212710 A212711 A212712




Alonso del Arte, May 24 2012



Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified December 4 20:44 EST 2020. Contains 338938 sequences. (Running on oeis4.)