cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A205727 Number of odd semiprimes <= n^2.

Original entry on oeis.org

0, 0, 1, 2, 4, 6, 8, 11, 14, 19, 23, 28, 33, 37, 46, 51, 56, 66, 73, 80, 88, 96, 108, 118, 126, 134, 148, 159, 172, 183, 197, 207, 220, 234, 249, 263, 280, 297, 309, 323, 338, 356, 376, 393, 412, 427, 449, 465, 482, 505, 527, 544, 561, 582, 606, 634, 658
Offset: 1

Views

Author

Keith Backman, Jan 30 2012

Keywords

Comments

The Goldbach conjecture being true would imply that for every integer j, there exists at least one integer k such that (j^2)-(k^2) is an odd semiprime; i.e., for 2j=p+q, j=(p+q)/2 and k=(p-q)/2 results in (j^2)-(k^2)=pq. [Note that in many cases, 2j can be expressed as the sum of more than one set of two primes.] See A205728 for related series where p must be distinct from q.

Crossrefs

Cf. A046315, A205726, A205728.

Programs

  • Mathematica
    SemiPrimeQ[n_Integer] := If[Abs[n] < 2, False, (2 == Plus @@ Transpose[FactorInteger[Abs[n]]][[2]])];nn = 100;  t = Select[Range[1, nn^2, 2], SemiPrimeQ]; Table[Length[Select[t, # <= n^2 &]], {n, nn}] (* T. D. Noe, Jan 30 2012 *)
With[{osp=Table[{n,PrimeOmega[n]},{n,1,10001,2}]},Table[ Count[ Select[ osp,#[[1]]<=k^2&],?(#[[2]]==2&)],{k,60}]] (* _Harvey P. Dale, Dec 29 2017 *)
  • PARI
    a(n) = sum(k=1, n^2, (k%2) && (bigomega(k) == 2)); \\ Michel Marcus, Feb 24 2018

A205728 Number of odd, nonsquare semiprimes <= n^2.

Original entry on oeis.org

0, 0, 0, 1, 2, 4, 5, 8, 11, 16, 19, 24, 28, 32, 41, 46, 50, 60, 66, 73, 81, 89, 100, 110, 118, 126, 140, 151, 163, 174, 187, 197, 210, 224, 239, 253, 269, 286, 298, 312, 326, 344, 363, 380, 399, 414, 435, 451, 468, 491, 513, 530, 546, 567, 591, 619, 643, 664
Offset: 1

Views

Author

Keith Backman, Jan 30 2012

Keywords

Comments

Like A205727 (see comments thereto), this looks at odd semiprimes, but excludes squares. This then relates to the Goldbach conjecture 2j=p+q with the additional restriction that j, p, and q are not equal.

Links

Crossrefs

Cf. A205726, A205727.

Programs

  • Mathematica
    SemiPrimeQ[n_Integer] := If[Abs[n] < 2, False, (2 == Plus @@ Transpose[FactorInteger[Abs[n]]][[2]])]; nn = 100;  t = Select[Range[1, nn^2, 2], SemiPrimeQ[#] && ! IntegerQ[Sqrt[#]] &]; Table[Length[Select[t, # <= n^2 &]], {n, nn}] (* T. D. Noe, Jan 30 2012 *)

A038108 Number of prime pairs {p,q}, such that pq < n^2.

Original entry on oeis.org

0, 2, 6, 8, 13, 16, 22, 26, 34, 39, 48, 55, 62, 75, 82, 89, 103, 113, 126, 135, 149, 163, 179, 190, 202, 220, 236, 252, 270, 288, 304, 320, 340, 360, 381, 403, 425, 443, 462, 483, 508, 532, 556, 581, 604, 633, 655, 678, 709, 738, 761, 782
Offset: 2

Views

Author

Joe K. Crump (joecr(AT)carolina.rr.com)

Keywords

Comments

Number of semiprimes (A001358) < n^2. [Michel Marcus, Sep 02 2013]

Examples

			a(3)=2 because only the prime pairs (2,2) and (2,3) form products < 9.

Crossrefs

Cf. A205726.

Programs

  • PARI
    a(n) = {sqn = n^2; idp = primepi(sqn\2); nbp = 0; for (i = 1, idp, p = prime(i); for (j = 1, i, if (p * prime(j) < sqn, nbp++););); nbp;} \\ Michel Marcus, Sep 02 2013
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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