A121387 -id:A121387 - cp's OEIS frontend

A080774 Numbers with two prime factors: (4i+1)(4*j+3).

15, 35, 39, 51, 55, 87, 91, 95, 111, 115, 119, 123, 143, 155, 159, 183, 187, 203, 215, 219, 235, 247, 259, 267, 287, 291, 295, 299, 303, 319, 323, 327, 335, 339, 355, 371, 391, 395, 403, 407, 411, 415, 427, 447, 451, 471, 511, 515, 519, 527, 535, 543, 551

Offset: 1

Reinhard Zumkeller, Mar 10 2003

nonn

Semiprimes of the form 4k+3. - Giovanni Teofilatto, Jun 15 2005

There are 971 semiprimes of the form 4k+3 below 10,000.

			a(1) = 15 = 3*5 = (4*1+1)*(4*0+3).

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)
Eric Weisstein's World of Mathematics, Semiprime.
Wikipedia, Semiprime.

Cf. A001358, A006881, A002144, A002145, A107978, A121387.

Select[Range[15, 1000, 2], Last /@ FactorInteger[#] == {1, 1} && IntegerQ[(# - 3)/4] &] (* Vladimir Joseph Stephan Orlovsky, May 07 2011 *)
Select[4*Range[200]+3,PrimeOmega[#]==2&] (* Harvey P. Dale, Nov 02 2024 *)

isok(k) = k % 4 == 3 && bigomega(k) == 2; \\ Amiram Eldar, Jun 15 2025

A009177 Numbers that are the hypotenuses of more than one Pythagorean triangle.

Original entry on oeis.org

25, 50, 65, 75, 85, 100, 125, 130, 145, 150, 169, 170, 175, 185, 195, 200, 205, 221, 225, 250, 255, 260, 265, 275, 289, 290, 300, 305, 325, 338, 340, 350, 365, 370, 375, 377, 390, 400, 410, 425, 435, 442, 445, 450, 455, 475, 481, 485, 493, 500, 505, 507, 510, 520, 525

Offset: 1

David W. Wilson

nonn

Also, hypotenuses of Pythagorean triangles in Pythagorean triples (a, b, c, a < b < c) such that a and b are the hypotenuses of Pythagorean triangles, where the Pythagorean triples (x1, y1, a) and (x2, y2, b) are similar triangles. Sequence gives c values. - Naohiro Nomoto
Any multiple of a term of this sequence is also a term. The primitive elements are the products of two primes, not necessarily distinct, that are == 1 (mod 4): A121387. - Franklin T. Adams-Watters, Dec 21 2015
Numbers appearing more than once in A009000. - Sean A. Irvine, Apr 20 2018

			25^2 = 24^2 + 7^2 = 20^2 + 15^2.
E.g., (a = 15, b = 20, c = 25, a^2 + b^2=c^2); 15 and 20 are the hypotenuses of Pythagorean triangles. The Pythagorean triples (9, 12, 15) and (12, 16, 20) are similar triangles. So c = 25 is in the sequence. - _Naohiro Nomoto_

Robert Israel, Table of n, a(n) for n = 1..10000
Index entries for sequences related to sums of squares

Cf. A004431, A009000, A118882, A121387.

filter:= proc(n) add(`if` (t[1] mod 4 = 1, t[2],0), t = ifactors(n)[2]) >= 2 end proc:
select(filter, [$1..1000]); # Robert Israel, Dec 21 2015

f[n_] := Module[{i = 0, k = 0}, Do[If[Sqrt[n^2 - i^2] == IntegerPart[Sqrt[n^2 - i^2]], k++], {i, n - 1, 1, -1}]; k];
lst = {}; Do[If[f[n] > 2, AppendTo[lst, n]], {n, 4*5!}];
lst (* Vladimir Joseph Stephan Orlovsky, Aug 12 2009 *)

Of the form b(i)*b(j)*k, where b(n) is A004431(n). Numbers whose prime factorization includes at least 2 (not necessarily distinct) primes congruent to 1 mod 4. - Franklin T. Adams-Watters, May 03 2006. [Typo corrected by Ant King, Jul 17 2008]

A131574 Numbers n that are the product of two distinct odd primes and x^2 + y^2 = n has integer solutions.

Original entry on oeis.org

65, 85, 145, 185, 205, 221, 265, 305, 365, 377, 445, 481, 485, 493, 505, 533, 545, 565, 629, 685, 689, 697, 745, 785, 793, 865, 901, 905, 949, 965, 985, 1037, 1073, 1145, 1157, 1165, 1189, 1205, 1241, 1261, 1285, 1313, 1345, 1385, 1405, 1417, 1465, 1469

Offset: 1

Colin Barker, Aug 28 2007, corrected Aug 29 2007

nonn

The two primes are of the form 4*k + 1.

			65 is in the sequence because x^2 + y^2 = 65 = 5*13 has solutions (x,y) = (1,8), (4,7), (7,4) and (8,1).

Ray Chandler, Table of n, a(n) for n = 1..10000 (first 1000 terms from Colin Barker)

Cf. A000415, A121387, A248649, A248712, A264498, A264499

dop(d, nmax) = {
  my(L=List(), v=vector(d,m,1)~, f);
  for(n=1, nmax,
    f=factorint(n);
    if(#f~==d && f[1,1]>2 && f[,2]==v && f[,1]%4==v, listput(L, n))
  );
  Vec(L)
}
dop(2, 3000) \\ Colin Barker, Nov 15 2015

A107978 Products of two primes of the form 4n+3 (A002145).

Original entry on oeis.org

9, 21, 33, 49, 57, 69, 77, 93, 121, 129, 133, 141, 161, 177, 201, 209, 213, 217, 237, 249, 253, 301, 309, 321, 329, 341, 361, 381, 393, 413, 417, 437, 453, 469, 473, 489, 497, 501, 517, 529, 537, 553, 573, 581, 589, 597, 633, 649, 669, 681, 713, 717, 721, 737

Offset: 1

Jonathan Vos Post, Jun 12 2005

Every odd semiprime must be in one of three disjoint sets: the product of two primes of the form 4n+1 (A121387), the product of two primes of the form 4n+3 (A107978), or the product of a prime of the form 4n+1 and a prime of the form 4n+3 (A080774).

T. D. Noe, Table of n, a(n) for n=1..1000
Eric Weisstein's World of Mathematics, Semiprime.

Cf. A001358, A002145, A080774, A121387.
Union of A131574 and A080109.
Third row of A121388.

p = Select[ Prime@ Range@ 60, Mod[ #, 4] == 3 &]; Take[ Sort@ Flatten@ Table[ p[[i]] p[[j]], {j, 30}, {i, j}], 54] (* or *)
fQ[n_] := Block[{fi = FactorInteger@ n}, Plus @@ Last /@ fi == 2 && Union@ Mod[ First /@ fi, 4] == {3}]; Select[ Range@ 748, fQ@# &] (* Robert G. Wilson v, May 20 2010 *)

{a(n)} = {p*q: p and q both elements of A002145}.

Edited by N. J. A. Sloane, May 20 2010

A108181 Semiprimes of the form 4n + 1.

Original entry on oeis.org

9, 21, 25, 33, 49, 57, 65, 69, 77, 85, 93, 121, 129, 133, 141, 145, 161, 169, 177, 185, 201, 205, 209, 213, 217, 221, 237, 249, 253, 265, 289, 301, 305, 309, 321, 329, 341, 361, 365, 377, 381, 393, 413, 417, 437, 445, 453, 469, 473, 481, 485, 489, 493, 497

Offset: 1

Giovanni Teofilatto, Jun 14 2005

Either a(n)=(4*i+1)*(4*j+1) or a(n)=(4*i+3)*(4*j+3); - Reinhard Zumkeller, Jun 15 2005

A107978 is a subsequence. - Reinhard Zumkeller, Jun 15 2005

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
K. Ford, Jason Sneed, Chebyshev's Bias for products of two primes, Exper. Math. 19 (4) (2010) 385-398

Cf. A080774, A001358, A002144, A002145, A107978, A121387.

IsSemiprime:= func; [s: n in [2..150] | IsSemiprime(s) where s is 4*n + 1]; // Vincenzo Librandi, Sep 22 2012

Select[4Range[0, 150] + 1, PrimeOmega[#] == 2&] (* Vincenzo Librandi, Sep 22 2012 *)

Corrected and extended by Reinhard Zumkeller, Jun 15 2005

A121388 Table read by antidiagonals which classifies the proper odd numbers by form (i,j) where i and j are the number of prime factors of the form 4k-1 and 4k+1 respectively.

Original entry on oeis.org

3, 5, 7, 9, 13, 11, 15, 21, 17, 19, 25, 35, 33, 29, 23, 27, 65, 39, 49, 37, 31, 45, 63, 85, 51, 57, 41, 43, 75, 105, 99, 145, 55, 69, 53, 47, 125, 175, 117, 147, 169, 87, 77, 61, 59, 81, 325, 195, 153, 171, 185, 91, 93, 73, 67, 135, 189, 425, 255, 165, 207, 205, 95, 121, 89

Offset: 1

Alford Arnold, Jul 26 2006

			Table begins:
Row(i,j)
1: (1,0) 3,7,11,19,23,31,43,47,59,67,71,79,83,103,... (A002145)
2: (0,1) 5,13,17,29,37,41,53,61,73,89,97,101,109,... (A002144)
3: (2,0) 9,21,33,49,57,69,77,93,121,129,133,141,... (A107978)
4: (1,1) 15,35,39,51,55,87,91,95,111,115,119,... (A080774)
5: (0,2) 25,65,85,145,169,185,205,221,265,289,... (A121387)
6: (3,0) 27,63,99,147,171,207,231,279,343,...
7: (2,1) 45,105,117,153,165,245,261,273,...
8: (1,2) 75,175,195,255,275,435,455,...
9: (0,3) 125,325,425,725,845,925,...
10:(4,0) 81,189,297,441,513,...
11:(3,1) 135,315,351,459,...
12:(2,2) 225,525,585,...
13:(1,3) 375,875,...
14:(0,4) 625,...

Cf. A120027. Subsequences begin A002145 A002144 A107978 A080774 A121387.

Edited, corrected and extended by Ray Chandler, Aug 08 2010

cp's OEIS Frontend

A080774 Numbers with two prime factors: (4*i+1)*(4*j+3).

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A009177 Numbers that are the hypotenuses of more than one Pythagorean triangle.

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A131574 Numbers n that are the product of two distinct odd primes and x^2 + y^2 = n has integer solutions.

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A107978 Products of two primes of the form 4n+3 (A002145).

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A108181 Semiprimes of the form 4n + 1.

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A121388 Table read by antidiagonals which classifies the proper odd numbers by form (i,j) where i and j are the number of prime factors of the form 4k-1 and 4k+1 respectively.

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A080774 Numbers with two prime factors: (4i+1)(4*j+3).