A111153 -id:A111153 - cp's OEIS frontend

A175255 Squares in A111153.

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

Offset: 1

Zak Seidov, Mar 15 2010

nonn

All terms == 1 mod 3.

Cf. A001248, A111153, A175256.

a(n) = A175256(n)^2.

A283527 First of three consecutive Sophie Germain semiprimes: n, n+1 and n+2 are all terms of A111153.

Original entry on oeis.org

15117, 17245, 34413, 93453, 143101, 157713, 190621, 208293, 233097, 294301, 323281, 346497, 470341, 501477, 1306113, 1337221, 1346401, 1655853, 1682313, 1774801, 1877613, 1879021, 1933233, 1976041

Offset: 1

Zak Seidov, Mar 09 2017

nonn

All terms are 1 mod 4, see A056809.

Zak Seidov, Table of n, a(n) for n = 1..408

Subsequence of A056809 and of A111153. Cf. A001358.

  po[x_] := PrimeOmega[x];   Select[Range[15117, 200000, 2],
2 == po[#] == po[2*# + 1] ==po[# + 1] == po[2*# + 3] == po[# + 2] ==
po[2*# + 5] &]

{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",")))}

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

A111206 Semi-Sophie Germain semiprimes: semiprimes which are the product of Sophie Germain primes.

Original entry on oeis.org

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

Offset: 1

Christopher M. Tomaszewski (cmt1288(AT)comcast.net), Oct 24 2005

nonn

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.

			a(4) = 10 because 10 is the 4th semiprime both the prime factors of which are Sophie Germain primes.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A111153, A001358, A005384, A111207.

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 *)

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

Extended by Ray Chandler, Oct 31 2005

A113432 Pierpont semiprimes: semiprimes of the form (2^K)*(3^L)+1.

Original entry on oeis.org

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

Offset: 1

Jonathan Vos Post, Nov 01 2005

			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).

Vincenzo Librandi, Table of n, a(n) for n = 1..89
Chris Caldwell, "Pierpont primes." primeform posting, Oct 25, 2005.
Chris Caldwell, "Pierpont primes." primeform posting, Oct 25, 2005. [Cached copy]
Eric Weisstein's World of Mathematics, Pierpont Prime
Eric Weisstein's World of Mathematics, Semiprime

Cf. A001358, A003586, A005109, A055600, A111153, A111206, A113433, A113434.

Select[Range[10^5], Plus @@ Last /@ FactorInteger[ # ] == 2 && Max @@ First /@ FactorInteger[ # - 1] < 5 &] (* Ray Chandler, Jan 24 2006 *)

{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.

A111173 Sophie Germain triprimes: k and 2k + 1 are both the product of 3 primes, not necessarily distinct.

Original entry on oeis.org

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

Offset: 1

Jonathan Vos Post, Oct 21 2005

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.

			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

Zak Seidov, Table of n, a(n) for n = 1..1000
OEIS Wiki, Triprimes

Cf. A005384, A014612, A111153, A111168, A111170, A111171, A111176.

Is3primes:=func; [n: n in [2..1200] | Is3primes(n) and Is3primes(2*n+1)]; // Vincenzo Librandi, Aug 19 2018

fQ[n_]:=PrimeOmega[n] == 3 == PrimeOmega[2 n + 1]; Select[Range@1100, fQ] (* Vincenzo Librandi, Aug 19 2018 *)

is(n)=bigomega(n)==3 && bigomega(2*n+1)==3 \\ Charles R Greathouse IV, Feb 01 2017

{a(n)} = a(n) is an element of A014612 and 2*a(n)+1 is an element of A014612.

Extended by Ray Chandler, Oct 22 2005

Edited by Jon E. Schoenfield, Aug 18 2018

A111176 Sophie Germain 4-almost primes.

Original entry on oeis.org

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

Offset: 1

Jonathan Vos Post, Oct 22 2005

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.

			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

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A005384, A014613, A111153, A111168, A111170, A111171, A111173.

Select[Range[5000],PrimeOmega[#]==PrimeOmega[2#+1]==4&] (* Harvey P. Dale, Nov 09 2011 *)

{a(n)} = a(n) is an element of A014613 and 2*a(n)+1 is an element of A014613.

Extended by Ray Chandler, Oct 22 2005

A111168 Semiprimes n such that 2*n - 1 is also a semiprime.

Original entry on oeis.org

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

Offset: 1

Jonathan Vos Post, Oct 21 2005

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.

			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

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A001358, A111153, A111170, A111171, A111173, A111176.

Select[Range[500],PrimeOmega[#]==PrimeOmega[2#-1]==2&]  (* Harvey P. Dale, Jul 23 2025 *)

is(n)=bigomega(n)==2 && bigomega(2*n-1)==2 \\ Charles R Greathouse IV, Jan 31 2017

{a(n)} = a(n) is an element of A001358 and 2*a(n)-1 is an element of A001358.

Extended by Ray Chandler, Oct 22 2005

A111170 Semiprimes S such that 3*S + 1 is also a semiprime.

Original entry on oeis.org

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

Offset: 1

Jonathan Vos Post, Oct 21 2005

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.

a(n) is either an even semiprime 2*k where k is a prime such that 6*k+1 is a semiprime, or an odd semiprime 2*k+1 where 3*k+2 is a prime. - Robert Israel, Dec 10 2024

			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

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A001358, A111153, A111168, A111171, A111173, A111176.

q:= n-> andmap(x-> 2=numtheory[bigomega](x), [n, 3*n+1]):
select(q, [$4..515])[];  # Alois P. Heinz, May 02 2024
# alternative
N:= 10^4: # to get all terms < (N-1)/3
Primes:= select(isprime, [2, seq(2*k+1, k=1..floor(N/2))]):
SP:={seq(seq(p*q, q=Primes[1..ListTools:-BinaryPlace(Primes, N/p)]), p=Primes)} minus {seq(p^2, p=Primes)}:
sort(convert(SP intersect map(t -> (t-1)/3, SP), list)); # Robert Israel, Dec 10 2024

Select[Range[515],PrimeOmega[#]==2&&PrimeOmega[3*#+1]==2&] (* James C. McMahon, May 01 2024 *)

{a(n)} = a(n) is an element of A001358 and 3*a(n)+1 is an element of A001358.

Extended by Ray Chandler, Oct 22 2005

A111171 Semiprimes S such that 3*S - 1 is also a semiprime.

Original entry on oeis.org

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

Offset: 1

Jonathan Vos Post, Oct 21 2005

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.

			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

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A001358, A111153, A111168, A111170, A111173, A111176.

Select[Range[600],PrimeOmega[#]==PrimeOmega[3#-1]==2&] (* Harvey P. Dale, Jun 20 2018 *)

{a(n)} = a(n) is an element of A001358 and 3*a(n)-1 is an element of A001358.

Corrected and extended by Ray Chandler, Oct 22 2005

A117204 Squarefree positive integers k such that 2*k+1 is also squarefree.

Original entry on oeis.org

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

Offset: 1

Leroy Quet, Mar 02 2006

nonn

The asymptotic density of this sequence is (3/2)*A065474 = 0.4839511484... (Erdős and Ivić, 1987). - Amiram Eldar, Mar 02 2021

			10 and 2*10 +1 = 21 are both squarefree, so 10 is in the sequence.

Ivan Neretin, Table of n, a(n) for n = 1..10000
Paul Erdős and Aleksandar Ivić, The distribution of values of a certain class of arithmetic functions at consecutive integers, Colloq. Math. Soc. János Bolyai, Vol. 51 (1987), pp. 45-91.

Cf. A005117, A117203, A117206.

Cf. A005384, A111153, A065474.

sfQ[n_]:=SquareFreeQ[n]&&SquareFreeQ[2n+1]; Select [Range[200],sfQ] (* Harvey P. Dale, Mar 12 2011 *)

a(n) = (A117203(n) - 1)/2.

More terms from Jonathan Vos Post, Mar 03 2006

Corrected and extended by Harvey P. Dale, Mar 12 2011

cp's OEIS Frontend

A175255 Squares in A111153.

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A283527 First of three consecutive Sophie Germain semiprimes: n, n+1 and n+2 are all terms of A111153.

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A111206 Semi-Sophie Germain semiprimes: semiprimes which are the product of Sophie Germain primes.

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A113432 Pierpont semiprimes: semiprimes of the form (2^K)*(3^L)+1.

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A111173 Sophie Germain triprimes: k and 2k + 1 are both the product of 3 primes, not necessarily distinct.

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A111176 Sophie Germain 4-almost primes.

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A111168 Semiprimes n such that 2*n - 1 is also a semiprime.

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