Displaying 1-10 of 26 results found.
4, 25, 169, 289, 361, 529, 961, 2809, 5041, 7921, 12769, 16129, 24649, 26569, 27889, 32761, 38809, 52441, 120409, 139129, 160801, 167281, 175561, 201601, 237169, 253009, 259081, 273529, 292681, 316969, 326041, 332929, 358801, 418609, 564001
First of three consecutive Sophie Germain semiprimes: n, n+1 and n+2 are all terms of A111153.
+20
1
15117, 17245, 34413, 93453, 143101, 157713, 190621, 208293, 233097, 294301, 323281, 346497, 470341, 501477, 1306113, 1337221, 1346401, 1655853, 1682313, 1774801, 1877613, 1879021, 1933233, 1976041
COMMENTS
All terms are 1 mod 4, see A056809.
MATHEMATICA
po[x_] := PrimeOmega[x]; Select[Range[15117, 200000, 2],
2 == po[#] == po[2*# + 1] ==po[# + 1] == po[2*# + 3] == po[# + 2] ==
po[2*# + 5] &]
PROG
(PARI) {bo(x)=bigomega(x)
forstep(n=15117, 2000000, 2, if(
2 == bo(n) && 2 == bo(n+1) && 2 == bo(n+2) && 2 == bo(2*n+1) &&
2 == bo(2*n+3) && 2 == bo(2*n+5), print1(n", ")))}
(PARI) list(lim)=lim\=1; my(v=List(), x=2*lim+5, u=vectorsmall(x)); forprime(p=2, x\2, forprime(q=2, min(lim\p, p), u[p*q]=1)); forstep(n=15117, lim, 4, if(u[n] && u[n+1] && u[n+2] && u[2*n+1] && u[2*n+3] && u[2*n+5], listput(v, n))); Vec(v) \\ Charles R Greathouse IV, Mar 10 2017
Semi-Sophie Germain semiprimes: semiprimes which are the product of Sophie Germain primes.
+10
17
4, 6, 9, 10, 15, 22, 25, 33, 46, 55, 58, 69, 82, 87, 106, 115, 121, 123, 145, 159, 166, 178, 205, 226, 249, 253, 262, 265, 267, 319, 339, 346, 358, 382, 393, 415, 445, 451, 466, 478, 502, 519, 529, 537, 562, 565, 573, 583, 586, 655, 667, 699, 717, 718, 753, 838
COMMENTS
Define an n-almost Sophie Germain almost-prime to be an n-almost prime all the prime factors of which are Sophie Germain primes. Note the contrast between this terminology and that of Sophie Germain n-almost primes, they are different.
EXAMPLE
a(4) = 10 because 10 is the 4th semiprime both the prime factors of which are Sophie Germain primes.
MATHEMATICA
lst={}; Do[If[Plus@@Last/@FactorInteger[n]==2, a=First/@FactorInteger[n]; b=a[[1]]; k=0; If[Length[a]==2, c=a[[2]]; If[ !PrimeQ[2*c+1], k=1]]; If[PrimeQ[2*b+1]&&k==0, AppendTo[lst, n]]], {n, 7!}]; lst (* Vladimir Joseph Stephan Orlovsky, Feb 27 2009 *) Module[{nn=100, sgp}, sgp=Select[Prime[Range[100]], PrimeQ[2#+1]&]; Select[ Union[ Times@@@Tuples[sgp, 2]], #<=10nn&]] (* Harvey P. Dale, May 08 2019 *)
PROG
(PARI) list(lim)=my(v=List(), u=v, t); forprime(p=2, lim\2, if(isprime(2*p+1), listput(u, p))); for(i=1, #u, for(j=1, i, t=u[i]*u[j]; if(t>lim, break); listput(v, t))); Set(v) \\ Charles R Greathouse IV, Feb 05 2017
AUTHOR
Christopher M. Tomaszewski (cmt1288(AT)comcast.net), Oct 24 2005
Pierpont semiprimes: semiprimes of the form (2^K)*(3^L)+1.
+10
10
4, 9, 10, 25, 33, 49, 55, 65, 82, 129, 145, 217, 289, 649, 865, 973, 1537, 1945, 2049, 2305, 3073, 4097, 4609, 5833, 6145, 6913, 8193, 8749, 9217, 11665, 13123, 15553, 20737, 23329, 24577, 27649, 31105, 34993, 41473, 62209, 69985, 73729, 78733
LINKS
Eric Weisstein's World of Mathematics, Semiprime
FORMULA
{a(n)} = Intersection of {(2^K)*(3^L)+1} A055600 and semiprimes A001358. a(n) is in this sequence iff there exist nonnegative integers K and L such that Omega((2^K)*(3^L)+1) = 2.
EXAMPLE
a(1) = 4 = (2^0)*(3^1)+1 = 2^2 hence the semiprime A001358(1). a(2) = 9 = (2^3)*(3^0)+1 = 3^2 hence the semiprime A001358(3). a(3) = 10 = (2^0)*(3^2)+1 = 2 * 5 hence the semiprime A001358(4). a(4) = 25 = (2^3)*(3^1)+1 = 5^2 hence the semiprime A001358(9). a(5) = 33 = (2^5)*(3^0)+1 = 3 * 11 hence the semiprime A001358(11). a(6) = 49 = (2^4)*(3^1)+1 = 7^2 hence the semiprime A001358(17). a(7) = 55 = (2^1)*(3^3)+1 = 5 * 11 hence the semiprime A001358(19).
MATHEMATICA
Select[Range[10^5], Plus @@ Last /@ FactorInteger[ # ] == 2 && Max @@ First /@ FactorInteger[ # - 1] < 5 &] (* Ray Chandler, Jan 24 2006 *)
Sophie Germain triprimes: k and 2k + 1 are both the product of 3 primes, not necessarily distinct.
+10
9
52, 76, 130, 171, 172, 212, 238, 318, 322, 325, 332, 357, 370, 387, 388, 402, 423, 430, 436, 442, 465, 507, 508, 556, 604, 610, 654, 665, 670, 710, 722, 747, 759, 762, 772, 775, 786, 790, 805, 814, 822, 826, 847, 874, 885, 902, 906, 916, 927, 942, 987, 1004
COMMENTS
There should also be triprime chains of length j analogous to Cunningham chains of the first kind and Tomaszewski chains of the first kind. A triprime chain of length j is a sequence of triprimes a(1) < a(2) < ... < a(j) such that a(i+1) = 2*a(i) + 1 for i = 1, ..., j-1. The first of these are: Length 3: 332, 665, 1331 = 11^3; 387, 775, 1551 = 3 * 11 * 47.
FORMULA
{a(n)} = a(n) is an element of A014612 and 2*a(n)+1 is an element of A014612.
EXAMPLE
n k = a(n) 2k + 1
= ================ ================
1 52 = 2^2 * 13 105 = 3 * 5 * 7
2 76 = 2^2 * 19 153 = 3^2 * 17
3 130 = 2 * 5 * 13 261 = 3^2 * 29
4 171 = 3^2 * 19 343 = 7^3
5 172 = 2^2 * 43 345 = 3 * 5 * 23
6 212 = 2^2 * 53 425 = 5^2 * 17
MATHEMATICA
fQ[n_]:=PrimeOmega[n] == 3 == PrimeOmega[2 n + 1]; Select[Range@1100, fQ] (* Vincenzo Librandi, Aug 19 2018 *)
PROG
(Magma) Is3primes:=func<i|&+[d[2]: d in Factorization(i)] eq 3>; [n: n in [2..1200] | Is3primes(n) and Is3primes(2*n+1)]; // Vincenzo Librandi, Aug 19 2018
Sophie Germain 4-almost primes.
+10
9
40, 220, 580, 712, 808, 812, 904, 940, 1062, 1192, 1444, 1592, 1612, 1690, 1812, 1876, 2002, 2152, 2212, 2236, 2254, 2488, 2502, 2562, 2650, 2662, 2788, 3010, 3052, 3064, 3112, 3162, 3208, 3258, 3272, 3352, 3448, 3550, 3580, 3820, 3832, 3892, 3910, 4012
COMMENTS
4-almost primes P such that 2*P + 1 are also 4-almost primes. There should also be 4-almost prime chains of length k analogous to Cunningham chains of the first kind and Tomaszewski chains of the first kind. A 4-almost prime chain of length k is a sequence of 4-almost primes a(1) < a(2) < ... < a(k) such that a(i+1) = 2*a(i) + 1 for i = 1, ..., k-1. There are no such chains beginning with integers under 1200.
FORMULA
{a(n)} = a(n) is an element of A014613 and 2*a(n)+1 is an element of A014613.
EXAMPLE
n p 2*p+1
1 40 = 2^3 * 5 81 = 3^4
2 220 = 2^2 * 5 * 11 441 = 3^2 * 7^2
3 580 = 2^2 * 5 * 29 1161 = 3^3 * 43
4 712 = 2^3 * 89 1425 = 3 * 5^2 * 19
5 808 = 2^3 * 101 1617 = 3 * 7^2 * 11
6 812 = 2^2 * 7 * 29 1625 = 5^3 * 13
MATHEMATICA
Select[Range[5000], PrimeOmega[#]==PrimeOmega[2#+1]==4&] (* Harvey P. Dale, Nov 09 2011 *)
Semiprimes n such that 2*n - 1 is also a semiprime.
+10
7
25, 26, 33, 35, 39, 46, 58, 62, 65, 85, 93, 94, 111, 118, 119, 133, 134, 145, 146, 155, 161, 178, 183, 202, 206, 209, 214, 219, 226, 235, 237, 247, 249, 253, 259, 265, 267, 287, 291, 295, 299, 334, 335, 341, 361, 362, 377, 382, 386, 391, 393, 395, 407, 422
COMMENTS
Define an m-th degree Tomaszewski n-chain of the first (second) kind and length k to be a sequence of n-almost primes p(1) < p(2) < ... < p(k) such that s(i+1) = m*s(i) +(-) 1 for i = 1, ..., k-1. Notice that a 2nd degree Tomaszewski 1-chain of the first (second) kind is the familiar Cunningham chain of the first (second) kind.
FORMULA
{a(n)} = a(n) is an element of A001358 and 2*a(n)-1 is an element of A001358.
EXAMPLE
n s(n) s*2-1
1 25 = 5^2 49 = 7^2
2 26 = 2 * 13 51 = 3 * 17
3 33 = 3 * 11 65 = 5 * 13
4 35 = 5 * 7 69 = 3 * 23
5 39 = 3 * 13 77 = 7 * 11
MATHEMATICA
Select[Range[500], PrimeOmega[#]==2&&PrimeOmega[2#-1]==2&] (* Harvey P. Dale, Aug 30 2015 *)
Semiprimes S such that 3*S - 1 is also a semiprime.
+10
6
9, 21, 22, 25, 26, 49, 62, 65, 69, 74, 85, 93, 121, 122, 129, 133, 141, 146, 158, 161, 166, 178, 185, 194, 205, 209, 221, 249, 253, 262, 265, 289, 298, 302, 305, 309, 346, 358, 361, 365, 381, 382, 386, 413, 446, 466, 473, 485, 489, 493, 501, 505, 514, 526, 553
COMMENTS
This is analogous to Sophie Germain semiprimes A111153 and the chains shown are analogous to Cunningham chains of the second kind and Tomaszewski chains of the second kind. Define a 3n-1 semiprime chain of length k. This is a sequence of semiprimes s(1) < s(2) < ... < s(k) such that s(i+1) = 3*s(i) - 1 for i = 1, ..., k-1. Length 3: 9, 26, 77; 49, 146, 437; 65, 194, 581; 129, 386, 1157; 158, 473, 1418; 187, 562, 1685. Length 4: 74, 221, 662, 1985; 122, 365, 1094, 3281. Length 5: 21, 62, 185, 554, 1661.
FORMULA
{a(n)} = a(n) is an element of A001358 and 3*a(n)-1 is an element of A001358.
EXAMPLE
n s(n) 3 *s -1
1 9 = 3^2 26 = 2 * 13
2 21 = 3 * 7 62 = 2 * 31
3 22 = 2 * 11 65 = 5 * 13
4 25 = 5^2 74 = 2 * 37
5 26 = 2 * 13 77 = 7 * 11
6 49 = 7^2 146 = 2 * 73
MATHEMATICA
Select[Range[600], PrimeOmega[#]==PrimeOmega[3#-1]==2&] (* Harvey P. Dale, Jun 20 2018 *)
Squarefree positive integers k such that 2*k+1 is also squarefree.
+10
6
1, 2, 3, 5, 6, 7, 10, 11, 14, 15, 17, 19, 21, 23, 26, 29, 30, 33, 34, 35, 38, 39, 41, 42, 43, 46, 47, 51, 53, 55, 57, 59, 61, 65, 66, 69, 70, 71, 74, 77, 78, 79, 82, 83, 86, 89, 91, 93, 95, 97, 101, 102, 105, 106, 107, 109, 110, 111, 113, 114, 115, 118, 119
COMMENTS
The asymptotic density of this sequence is (3/2)* A065474 = 0.4839511484... (Erdős and Ivić, 1987). - Amiram Eldar, Mar 02 2021
EXAMPLE
10 and 2*10 +1 = 21 are both squarefree, so 10 is in the sequence.
MATHEMATICA
sfQ[n_]:=SquareFreeQ[n]&&SquareFreeQ[2n+1]; Select [Range[200], sfQ] (* Harvey P. Dale, Mar 12 2011 *)
Semiprimes S such that 3*S + 1 is also a semiprime.
+10
5
15, 35, 38, 39, 55, 62, 82, 86, 87, 91, 106, 111, 115, 118, 119, 134, 142, 155, 159, 178, 187, 194, 218, 226, 235, 254, 259, 267, 278, 287, 295, 298, 299, 314, 319, 326, 327, 334, 335, 339, 371, 382, 386, 391, 395, 398, 411, 422, 427, 446, 451, 454, 502, 515
COMMENTS
This is analogous to Sophie Germain semiprimes A111153 and the chains shown are analogous to Cunningham chains of the first kind and Tomaszewski chains of the first kind. Define a 3n+1 semiprime chain of length k. This is a sequence of semiprimes s(1) < s(2) < ... < s(k) such that s(i+1) = 3*s(i) + 1 for i = 1, ..., k-1. Length 3: 111, 334, 1003; 142, 427, 1282. Length 4: 35, 106, 319, 958; 86, 259, 778, 2335; 187, 562, 1687, 5062.
FORMULA
{a(n)} = a(n) is an element of A001358 and 3*a(n)+1 is an element of A001358.
EXAMPLE
n s(n) 3*s + 1
1 15 = 3 * 5 46 = 2 * 23
2 35 = 5 * 7 106 = 2 * 53
3 38 = 2 * 19 115 = 5 * 23
4 39 = 3 * 13 118 = 2 * 59
5 55 = 5 * 11 166 = 2 * 83
6 62 = 2 * 31 187 = 11 * 17
MAPLE
q:= n-> andmap(x-> 2=numtheory[bigomega](x), [n, 3*n+1]):
MATHEMATICA
Select[Range[515], PrimeOmega[#]==2&&PrimeOmega[3*#+1]==2&] (* James C. McMahon, May 01 2024 *)
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