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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)
OFFSET

1,2

COMMENTS

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

LINKS

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

EXAMPLE

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.

MATHEMATICA

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]]}]]]

PROG

(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

CROSSREFS

Sequence in context: A162963 A297860 A002094 * A264867 A115725 A305577

Adjacent sequences:  A212706 A212707 A212708 * A212710 A212711 A212712

KEYWORD

nonn

AUTHOR

Alonso del Arte, May 24 2012

STATUS

approved

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