A214517 -id:A214517 - cp's OEIS frontend

A001912 Numbers k such that 4*k^2 + 1 is prime.

1, 2, 3, 5, 7, 8, 10, 12, 13, 18, 20, 27, 28, 33, 37, 42, 45, 47, 55, 58, 60, 62, 63, 65, 67, 73, 75, 78, 80, 85, 88, 90, 92, 102, 103, 105, 112, 115, 118, 120, 125, 128, 130, 132, 135, 140, 142, 150, 153, 157, 163, 170, 175, 192, 193, 198, 200

Offset: 1

N. J. A. Sloane

Complement of A094550. - Hermann Stamm-Wilbrandt, Sep 16 2014
Positive integers whose square is the sum of two triangular numbers in exactly one way (A000217(k) + A000217(k+1) = k*(k+1)/2 + (k+1)*(k+2)/2 = (k+1)^2). In other words, positive integers k such that A052343(k^2) = 1. - Altug Alkan, Jul 06 2016
4*a(n)^2 + 1 = A002496(n+1). - Hal M. Switkay, Apr 03 2022

E. Kogbetliantz and A. Krikorian, Handbook of First Complex Prime Numbers, Gordon and Breach, NY, 1971, p. 1.
M. Kraitchik, Recherches sur la Théorie des Nombres. Gauthiers-Villars, Paris, Vol. 1, 1924, Vol. 2, 1929, see Vol. 1, p. 11.
C. S. Ogilvy, Tomorrow's Math. 2nd ed., Oxford Univ. Press, 1972, p. 116.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 1..10000
A. J. C. Cunningham, Binomial Factorisations, Vols. 1-9, Hodgson, London, 1923-1929. [Annotated scans of a few pages from Volumes 1 and 2]
E. Kogbetliantz and A. Krikorian Handbook of First Complex Prime Numbers, Gordon and Breach, NY, 1971 [Annotated scans of a few pages]
Marek Wolf, Search for primes of the form m^2+1, arXiv:0803.1456 [math.NT], 2008-2010.

Cf. A002496, A005574, A062325, A090693, A094550, A214517 (first differences).

[n: n in [1..100] | IsPrime(4*n^2+1)] // Vincenzo Librandi, Nov 21 2010

A001912 := proc(n)
    option remember;
    if n = 1 then
        1;
    else
        for a from procname(n-1)+1 do
            if isprime(4*a^2+1) then
                return a;
            end if;
        end do:
    end if;
end proc: # R. J. Mathar, Aug 09 2012

Select[Range[200], PrimeQ[4#^2 + 1] &] (* Alonso del Arte, Dec 20 2013 *)

is(n)=isprime(4*n^2 + 1) \\ Charles R Greathouse IV, Apr 28 2015

a(n) = A005574(n+1)/2.

A214516 Differences between the numbers n such that n^2 + 1 is prime.

Original entry on oeis.org

1, 2, 2, 4, 4, 2, 4, 4, 2, 10, 4, 14, 2, 10, 8, 10, 6, 4, 16, 6, 4, 4, 2, 4, 4, 12, 4, 6, 4, 10, 6, 4, 4, 20, 2, 4, 14, 6, 6, 4, 10, 6, 4, 4, 6, 10, 4, 16, 6, 8, 12, 14, 10, 34, 2, 10, 4, 6, 14, 10, 6, 4, 4, 20, 2, 4, 4, 16, 6, 40, 8, 12, 14, 6, 8, 10, 40, 2

Offset: 1

T. D. Noe, Aug 06 2012

nonn

Sequence A005574 has the values of n. This sequence is the first differences of A005574. Note that a(1) is the only odd value.

T. D. Noe, Table of n, a(n) for n = 1..10000

Cf. A002496 (primes of the form 1+n^2), A005574 (values of n).

Differences[Select[Range[100], PrimeQ[1 + #^2] &, 101]]

a(n) = 2*A214517(n) for n > 1.

A214518 Record differences between the numbers n such that 4*n^2 + 1 is prime.

Original entry on oeis.org

1, 2, 5, 7, 8, 10, 17, 20, 23, 44, 50, 56, 65, 76, 106, 144, 165, 173

Offset: 1

T. D. Noe, Aug 06 2012

			a(1) = 1 because 4*1^2 + 1 = 5 and 4*2^2 + 1 = 17 are primes.
a(2) = 2 because 4*3^2 + 1 = 37 is prime, 4*4^2 + 1 = 65 is composite, and 4*5^2 + 1 = 101 is prime.
a(3) = 5 because 4*13^2 + 1 is prime, 4*n^2 + 1 is composite for n = 14..17, and 4*18^2 + 1 is prime.

Cf. A121326 (primes of the form 1+4*n^2), A001912 (values of n).

Cf. A214517 (differences), A214519 (where record differences occur).

n = 1; last = 1; t = {1}; While[Length[t] < 15, n++; p = 1 + 4*n^2; If[PrimeQ[p], If[n - last > t[[-1]], AppendTo[t, n - last]]; last = n]]; t

A214519 Least number m such that 4m^2 + 1 is prime and the next prime of this form is 4(m + A214518(n))^2 + 1.

Original entry on oeis.org

1, 3, 13, 20, 47, 92, 175, 248, 1695, 1768, 22685, 41367, 49532, 178582, 420452, 1940278, 13957468, 20258760

Offset: 1

T. D. Noe, Aug 06 2012

Cf. A121326 (primes of the form 1+4*n^2), A214517, A214518 (record differences).

n = 1; last = 1; t = {1}; t2 = {1}; While[Length[t] < 10, n++; p = 1 + 4 n^2; If[PrimeQ[p], If[n - last > t[[-1]], AppendTo[t, n - last]; AppendTo[t2, last]]; last = n]]; t2

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A001912 Numbers k such that 4*k^2 + 1 is prime.

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A214516 Differences between the numbers n such that n^2 + 1 is prime.

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A214518 Record differences between the numbers n such that 4*n^2 + 1 is prime.

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A214519 Least number m such that 4m^2 + 1 is prime and the next prime of this form is 4(m + A214518(n))^2 + 1.

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cp's OEIS Frontend

A001912 Numbers k such that 4*k^2 + 1 is prime.

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Author

Keywords

Comments

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Crossrefs

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Magma

Maple

Mathematica

PARI

Formula

A214516 Differences between the numbers n such that n^2 + 1 is prime.

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Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

A214518 Record differences between the numbers n such that 4*n^2 + 1 is prime.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

A214519 Least number m such that 4*m^2 + 1 is prime and the next prime of this form is 4*(m + A214518(n))^2 + 1.

Views

Author

Keywords

Crossrefs

Programs

Mathematica

A214519 Least number m such that 4m^2 + 1 is prime and the next prime of this form is 4(m + A214518(n))^2 + 1.