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.

A113693 Semiprimes in A054556.

Original entry on oeis.org

4, 15, 34, 249, 391, 565, 771, 886, 1915, 3814, 5149, 5739, 6046, 7354, 9169, 10765, 11611, 15814, 16321, 18429, 20665, 22426, 24259, 28141, 29499, 32311, 36769, 39106, 43161, 48291, 52786, 53709, 57481, 60394, 63379, 65409, 67471, 69565
Offset: 1

Views

Author

Jonathan Vos Post, Nov 05 2005

Keywords

Comments

This sequence contains semiprimes from the center straight up the y-axis in the semiprime spiral of A113688-A113689. Semiprimes from the center straight down the y-axis in the semiprime spiral are A113691. Semiprimes from the center straight right along the x-axis in the semiprime spiral are A113690. Semiprimes from the center straight left along the x-axis in the semiprime spiral are A113692.

Examples

			a(27) = 4*97^2 - 9*97 + 6 = 36769 = 83 * 443.
a(28) = 4*100^2 - 9*100 + 6 = 39106 = 2 * 19553.
a(27) and a(28) are horizontally adjacent in the prime spiral, hence part of a clump and not isolated semiprimes as in A113688.
a(45) = 4*157^2 - 9*157 + 6 = 97189 = 17 * 5717 is the greatest member under 10^5.
		

Crossrefs

Programs

  • Magma
    IsSemiprime:= func; [s: n in [2..150] | IsSemiprime(s) where s is 4*n^2 - 9*n + 6]; // Vincenzo Librandi, Sep 22 2012
  • Mathematica
    Select[Table[4 n^2 - 9 n + 6, {n, 140}], PrimeOmega[#] == 2 &] (* Vincenzo Librandi, Sep 22 2012 *)

Formula

{a(n)} = Intersection of A001358 and A054556. Semiprimes of the form 4*k^2 - 9*k + 6.

A113690 Semiprimes in A054552.

Original entry on oeis.org

86, 298, 371, 1243, 1541, 2426, 2627, 3053, 4258, 5366, 5663, 6281, 6602, 6931, 7613, 8327, 9073, 9458, 10661, 13283, 14702, 15191, 16706, 18293, 18838, 23486, 25361, 26002, 26651, 27973, 28646, 34318, 35063, 36577, 38123, 41311, 43786, 44627
Offset: 1

Views

Author

Jonathan Vos Post, Nov 05 2005

Keywords

Comments

This sequence, A113690, contains semiprimes from the center straight right along the x-axis in the semiprime spiral of A113688-A113689. Semiprimes from the center straight left along the x-axis in the semiprime spiral are A113692. A113693 contains semiprimes from the center straight up the y-axis in the semiprime spiral. Semiprimes from the center straight down the y-axis in the semiprime spiral are A113691.

Examples

			a(10) = 4*37^2 - 3*37 + 1 = 5366 = 2 * 2683.
a(11) = 4*38^2 - 3*38 + 1 = 5663 = 7 * 809.
a(10) and a(11) are horizontally adjacent in the prime spiral, hence part of a clump and not isolated semiprimes as in A113688.
a(57) = 4*156^2 - 3*156 + 1 = 96877 = 11 * 8807 is the greatest member under 10^5.
		

Crossrefs

Programs

  • Magma
    IsSemiprime:= func; [s: n in [1..120] | IsSemiprime(s) where s is 4*n^2 - 3*n + 1]; // Vincenzo Librandi, Sep 22 2012
  • Mathematica
    Select[Table[4*n^2 - 3*n + 1, {n, 150}], PrimeOmega[#] == 2&] (* Vincenzo Librandi, Sep 22 2012 *)

Formula

{a(n)} = Intersection of A001358 and A054552. Semiprimes of the form 4*k^2 - 3*k + 1.

Extensions

Corrected a(6) by Vincenzo Librandi, Sep 22 2012

A113692 Semiprimes in A054567.

Original entry on oeis.org

6, 69, 106, 265, 334, 411, 589, 799, 1041, 1174, 1315, 1959, 2329, 3394, 4659, 5221, 5815, 7099, 8146, 8511, 10869, 16449, 21979, 23181, 23794, 25681, 26326, 31774, 33949, 35439, 36961, 38515, 40101, 43369, 45051, 48511, 50289, 52099, 54874
Offset: 1

Views

Author

Jonathan Vos Post, Nov 05 2005

Keywords

Comments

This sequence, A113692, contains semiprimes from the center straight left along the x-axis in the semiprime spiral of A113688-A113689. A113690 contains semiprimes from the center straight right along the x-axis in the semiprime spiral. A113691 contains semiprimes from the center straight down the y-axis in the semiprime spiral. A113693 contains semiprimes from the center straight up the y-axis in the semiprime spiral.

Examples

			a(4) = 4*9^2 - 7*9 + 4 = 265 = 5 * 53.
a(5) = 4*10^2 - 7*10 + 4 = 334 = 2 * 167.
a(6) = 4*11^2 - 7*11 + 4 = 411 = 3 * 137.
a(4), a(5) and a(6) are horizontally adjacent in the semiprime spiral, hence part of a clump and not isolated semiprimes as in A113688. a(9), a(10) and a(11) are another such horizontal string of 3 adjacent semiprimes.
a(46) = 4*151^2 - 7*151 + 4 = 90151 = 17 * 5303 is the greatest member under 10^5 (it is coincidence that this integer ends, base 10, with the same 151 that is the index of the quadratic).
		

Crossrefs

Programs

  • Magma
    IsSemiprime:= func; [s: n in [2..120] | IsSemiprime(s) where s is 4*n^2 - 7*n + 4]; // Vincenzo Librandi, Sep 22 2012
  • Mathematica
    Select[Table[4*n^2 - 7*n + 4, {n, 200}], PrimeOmega[#] == 2&] (* Vincenzo Librandi, Sep 22 2012 *)

Formula

{a(n)} = Intersection of A001358 and A054567. Semiprimes of the form 4*k^2 - 7*k + 4.
Showing 1-3 of 3 results.