A111153 -id:A111153 - cp's OEIS frontend

A177221 Numbers k that are the products of two distinct primes such that 2*k + 1 is also the product of two distinct primes.

10, 34, 38, 46, 55, 57, 77, 91, 93, 106, 118, 123, 129, 133, 143, 145, 159, 161, 177, 185, 201, 203, 205, 206, 213, 218, 226, 235, 259, 267, 291, 295, 298, 305, 314, 327, 334, 335, 339, 358, 365, 377, 381, 394, 395, 403, 407, 415, 417, 446, 447, 458, 466, 469

Offset: 1

Vladimir Joseph Stephan Orlovsky, May 05 2010

nonn

			10 is in the sequence because 10 = 2*5 and 2*10+1 = 21 = 3*7.

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

Cf. A006881, A111153, A177210, A177211, A177212, A177213, A177214, A177215, A177216, A177217, A177220.

isA006881:= proc(n) local F;
  F:= ifactors(n)[2];
  nops(F)=2 and F[1, 2]+F[2, 2]=2
end proc:
filter:= n -> andmap(isA006881, [n, 2*n+1]); select(filter, [$1..1000]); # Robert Israel, Nov 09 2017

f[n_]:=Last/@FactorInteger[n]=={1,1}; lst={};Do[If[f[n]&&f[2*n+1],AppendTo[lst,n]],{n,0,3*6!}];lst
Select[Range[500],PrimeNu[#]==PrimeOmega[#]==PrimeNu[2#+1] == PrimeOmega[ 2#+1] == 2&] (* Harvey P. Dale, Feb 22 2018 *)

A113433 Semi-Pierpont semiprimes: products of exactly two Pierpont primes A005109.

Original entry on oeis.org

4, 6, 9, 10, 14, 15, 21, 25, 26, 34, 35, 38, 39, 49, 51, 57, 65, 74, 85, 91, 95, 111, 119, 133, 146, 169, 185, 194, 218, 219, 221, 247, 259, 289, 291, 323, 326, 327, 361, 365, 386, 481, 485, 489, 511, 514, 545, 579, 629, 679, 703, 763, 771, 815, 866, 949, 965

Offset: 1

Jonathan Vos Post, Nov 01 2005

Semiprime both of whose prime factors are Pierpont primes (A005109), which are primes of the form (2^K)*(3^L)+1. Not to be confused with A113432: Pierpont semiprimes [Semiprimes of the form (2^K)*(3^L)+1]. This terminology itself is by analogy to what Tomaszewski used for the Sophie Germain counterparts A111153 and A111206.

			a(1) = 4 = 2^2 = [(2^0)*(3^0)+1]*[(2^1)*(3^0)+1] = A005109(1)*A005109(1).
a(2) = 6 = 2*3 = [(2^0)*(3^0)+1]*[(2^1)*(3^0)+1] = A005109(1)*A005109(2).
a(3) = 9 = 3^2 = [(2^1)*(3^0)+1]*[(2^1)*(3^0)+1] = A005109(2)*A005109(2).
a(4) = 10 = 2*5 = [(2^0)*(3^0)+1]*[(2^2)*(3^0)+1] = A005109(1)*A005109(3).
a(5) = 14 = 2*7 = [(2^0)*(3^0)+1]*[(2^1)*(3^1)+1] = A005109(1)*A005109(4).
a(6) = 15 = 3*5 = [(2^1)*(3^0)+1]*[(2^2)*(3^0)+1] = A005109(2)*A005109(3).

Vincenzo Librandi, Table of n, a(n) for n = 1..498
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, A113432, A113434.

Select[Range[10^3], Plus @@ Last /@ FactorInteger[ # ] == 2 && And @@ (Max @@ First /@ FactorInteger[ # - 1] < 5 &) /@ First /@ FactorInteger[ # ] &] (* Ray Chandler, Jan 24 2006 *)

{a(n)} = Semiprimes A001358 both of whose factors are of the form (2^K)*(3^L)+1. {a(n)} = {A005109(i)*A005109(j) for integers i and j not necessarily distinct}.

A113434 Semi-Pierpont semiprimes which are also Pierpont semiprimes.

Original entry on oeis.org

4, 9, 10, 25, 49, 65, 289

Offset: 1

Jonathan Vos Post, Nov 01 2005

Semiprimes both of whose prime factors are Pierpont primes (A005109), which are primes of the form (2^K)*(3^L)+1 and where the semiprime is itself of the form (2^K)*(3^L)+1.
No more under 10^50; what is the next element of this sequence?
No more terms <= 10^100. - Robert Israel, Mar 10 2017
This sequence is complete, see Links. - Charlie Neder, Feb 04 2019

			a(1) = 4 = 2^2 = [(2^0)*(3^0)+1]*[(2^0)*(3^0)+1] = (2^0)*(3^1)+1.
a(2) = 9 = 3^2 = [(2^1)*(3^0)+1]*[(2^1)*(3^0)+1] = (2^3)*(3^0)+1.
a(3) = 10 = 2*5 = [(2^0)*(3^0)+1]*[(2^2)*(3^0)+1] = (2^0)*(3^2)+1.
a(4) = 25 = 5^2 = [(2^2)*(3^0)+1]*[(2^2)*(3^0)+1] = (2^3)*(3^1)+1.
a(5) = 49 = 7^2 = [(2^1)*(3^1)+1]*[(2^1)*(3^1)+1] = (2^4)*(3^1)+1.
a(6) = 65 = 5*13 = [(2^2)*(3^0)+1]*[(2^2)*(3^1)+1] = (2^6)*(3^0)+1.
a(7) = 289 = 17^2 = [(2^4)*(3^0)+1]*[(2^4)*(3^0)+1] = (2^5)*(3^2)+1.

Chris Caldwell, "Pierpont primes." primeform posting, Oct 25, 2005.
Chris Caldwell, "Pierpont primes." primeform posting, Oct 25, 2005. [Cached copy]
Charlie Neder, Proof of the completeness of this sequence
Eric Weisstein's World of Mathematics, Pierpont Prime
Eric Weisstein's World of Mathematics, Semiprime

Cf. A001358, A003586, A005109, A055600, A111153, A111206, A113432, A113433.

N:= 10^100: # to get all terms <= N
PP:= select(isprime, {seq(seq(1+2^i*3^j, i=0..ilog2((N-1)/3^j)),j=0..floor(log[3](N-1)))}):
SP:= select(t -> t <= N and t = 1+2^padic:-ordp(t-1,2)*3^padic:-ordp(t-1,3), [seq(seq(PP[i]*PP[j], j=1..i),i=1..nops(PP))]):
sort(convert(SP,list)); # Robert Israel, Mar 10 2017

{a(n)} = intersection of A113432 and A113433. {a(n)} = Semiprimes A001358 of the form (2^K)*(3^L)+1 both of whose factors are of the form (2^K)*(3^L)+1. {a(n)} = {integers P such that, for nonnegative integers I, J, K, L, m, n there is a solution to (2^I)*(3^J)+1 = [(2^K)*(3^L)+1]*[(2^m)*(3^n)+1] where both [(2^K)*(3^L)+1] and [(2^m)*(3^n)+1] are prime}.

A175256 a(n) = sqrt(A175255(n)).

Original entry on oeis.org

2, 5, 13, 17, 19, 23, 31, 53, 71, 89, 113, 127, 157, 163, 167, 181, 197, 229, 347, 373, 401, 409, 419, 449, 487, 503, 509, 523, 541, 563, 571, 577, 599, 647, 751, 769, 773, 823, 827, 883, 919, 937, 941, 967, 971, 977, 1009, 1013, 1031, 1039, 1171, 1201, 1223

Offset: 1

Zak Seidov, Mar 15 2010

nonn

Zak Seidov, Table of n, a(n) for n=1..10000

Cf. A001248, A111153, A175255.

A176896 Semiprimes s such that (s-1)/2 is also semiprime.

Original entry on oeis.org

9, 21, 51, 69, 77, 93, 111, 115, 155, 183, 187, 213, 237, 247, 259, 267, 287, 291, 319, 323, 339, 355, 371, 403, 407, 411, 413, 427, 437, 453, 471, 519, 535, 579, 583, 591, 597, 611, 629, 655, 669, 671, 679, 717, 723, 731, 755, 763, 789, 791, 807, 815, 831

Offset: 1

Juri-Stepan Gerasimov, Apr 28 2010

nonn

Semiprime analog of safe primes (A005385). - Zak Seidov, Feb 01 2017

			a(1)=9 because (3*3 - 1)/2 = 4 = 2*2; a(2)=21 because (3*7 - 1)/2 = 10 = 2*5.

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

Cf. A001358, A005385, A111153.

Select[Range[1000],PrimeOmega[#]==PrimeOmega[(#-1)/2]==2&] (* Harvey P. Dale, Jan 28 2016 *)

a(n) = 2*A111153(n) + 1. - Zak Seidov, Sep 10 2012

Corrected (497 and 745 removed) by R. J. Mathar, May 01 2010

A211162 Sophie Germain 5-almost primes.

Original entry on oeis.org

688, 1552, 3496, 4360, 5008, 6352, 6952, 7546, 7672, 9256, 9625, 9712, 10062, 10300, 10840, 11632, 11875, 12112, 12136, 12460, 12712, 13432, 13648, 13744, 13912, 14152, 14812, 14920, 15484, 16562, 17050, 17104, 17272, 17608, 17752, 18130, 18232, 18616, 18952, 19062, 19624, 19792, 21100, 21136, 21352

Offset: 1

Jonathan Vos Post and Robert G. Wilson v, Jan 30 2013

nonn

Numbers n that are products of exactly 5 primes, such that 2*n + 1 are also products of exactly 5 primes. By analogy with A111153 Sophie Germain semiprimes: semiprimes n such that 2n+1 is also a semiprime; A111173 Sophie Germain 3-almost primes; A111176 Sophie Germain 4-almost primes.
From Zak Seidov, Jan 30 2013: (Start)
First integers n such that both n and 2n+1 are Sophie Germain 5-almost primes are: 54708, 103812, 111952, 113368, 117328, 134312, 159568, 160062, 165462, 199048, 205812.
First integers n such that n, 2n+1 and 4n+3 all are Sophie Germain 5-almost primes are: 159568, 301812, 431068, 444388, 564718, 1144468, 1420468, 1653162, 1687768, 1794568.
First integers n such that n, 2n+1, 4n+3 and 8n+7 all are Sophie Germain 5-almost primes are: 2991345, 4553367, 7760616, 9145318, 9332368, 12919266, 14283535, 14659746, 15144118.
First integers n such that n, 2n+1, 4n+3, 8n+7 and 16n+15 all are Sophie Germain 5-almost primes are: 15144118, 18515752, 41092024, 60406662, 71783890, 87353512, 94144212
First integers n such that n, 2n+1, 4n+3, 8n+7, 16n+15 and 32n+31 all are Sophie Germain 5-almost primes are: 211457337, 237572475, 245071092, 352015408, 415695462, 433833417.
First integers n such that n, 2n+1, 4n+3, 8n+7, 16n+15, 32n+31 and 64n+63 all are Sophie Germain 5-almost primes are: 433833417, 463078210, 648871975. (End)

			a(1) = 688 because 688 = 2^4 * 43, and 2*688 + 1 = 1377 = 3^4 * 17.

Vincenzo Librandi, Table of n, a(n) for n = 1..2000

Cf. A014614, A111153, A111173, A111176.

Is5primes:=func; [n: n in [2..22000] | Is5primes(n) and Is5primes(2*n+1)]; // Bruno Berselli, Jan 30 2013

fQ[n_] := PrimeOmega[n] == 5 == PrimeOmega[2 n + 1]; Select[Range@ 100000, fQ] (* Robert G. Wilson v *)

is(n)=bigomega(n)==5 && bigomega(2*n+1)==5 \\ Charles R Greathouse IV, Feb 01 2017

{n in A014614 such that 2*n + 1 is in A014614}.

A111207 Numbers that are both Sophie Germain semiprimes and semi-Sophie Germain semiprimes.

Original entry on oeis.org

4, 10, 25, 46, 55, 106, 123, 145, 159, 205, 226, 267, 339, 358, 415, 466, 529, 573, 583, 718, 753, 843, 865, 979, 1077, 1195, 1243, 1257, 1293, 1366, 1405, 1465, 1473, 1486, 2098, 2157, 2206, 2427, 2455, 2545, 2563, 2581, 2599, 2629, 2809, 2818, 2998, 3057

Offset: 1

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

nonn

This is the intersection of the sequence of Sophie Germain semiprimes (A111153) and semi-Sophie Germain semiprimes (A111206).

			a(4)=46 because 46 is the 4th semiprime such that 2*46+1=93 is a semiprime and both of its factors are Sophie Germain primes: 2*2+1=5 and 2*23+1=47.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A111153, A001358, A005384, A111206.

seqQ[n_] := AllTrue[{n, 2*n + 1}, PrimeOmega[#] == 2 &] && AllTrue[First /@ FactorInteger[n], PrimeQ[2*# + 1] &]; Select[Range[3000], seqQ] (* Amiram Eldar, May 10 2020 *)

Extended by Ray Chandler, Oct 31 2005

A115170 Non-cubefree numbers k such that 2k+1 is also non-cubefree (A046099).

Original entry on oeis.org

40, 256, 312, 472, 688, 904, 1120, 1200, 1312, 1336, 1552, 1768, 1984, 2187, 2200, 2312, 2416, 2456, 2632, 2848, 2875, 3064, 3280, 3312, 3429, 3496, 3712, 3928, 3944, 4144, 4312, 4360, 4576, 4792, 5008, 5224, 5312, 5440, 5562, 5656, 5872, 6088, 6250, 6304, 6312, 6345, 6520, 6655, 6688, 6736, 6952

Offset: 1

Jonathan Vos Post, Mar 03 2006

The probability that a random integer is cubefree is 1/zeta(3) (see A088453).

			312 is there because 2^3 divides 312 and 5^3 divides 312*2+1.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..700 from R. J. Mathar)

Cf. A005384 (n and 2n+1 prime), A046099, A088453, A111153 (n and 2n+1 semiprime), A117204 (n and 2n+1 squarefree), A115228.

noncfQ[n_] := AnyTrue[FactorInteger[n][[;; , 2]], # > 2 &]; Select[Range[7000], noncfQ[#] && noncfQ[2*# + 1] &] (* Amiram Eldar, May 25 2025 *)

Edited by Don Reble, Mar 05 2006

2875 inserted by R. J. Mathar, Dec 08 2015

A133123 Double Sophie Germain semiprimes: semiprimes s such that s1=2s+1 and s2=2s1+1 are also semiprimes.

Original entry on oeis.org

38, 46, 106, 129, 133, 145, 169, 201, 203, 235, 289, 291, 298, 334, 335, 381, 407, 417, 458, 489, 497, 529, 538, 579, 583, 597, 623, 626, 649, 685, 689, 694, 707, 758, 767, 781, 815, 898, 899, 921, 926, 959, 995, 1073, 1079, 1082, 1094, 1099, 1139, 1142

Offset: 1

Zak Seidov, Sep 19 2007

nonn

Numbers n such that both n and 2n+1 are in A111153.

If 29+30*k, 39+40*k and 47+48*k are all primes then 58+60*k is in the sequence. Thus Dickson's conjecture implies this sequence is infinite. - Robert Israel, Mar 17 2019

			38=2*19, 2*38+1=77=7*11 and 2*77+1=155=5*31;
129=3*43, 2*129+1=259=7*37 and 2*259+1=519=3*173.

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

Cf. A111153.

filter:= n -> andmap(numtheory:-bigomega=2, [n,2*n+1,4*n+3]):
select(filter, [$1..2000]); # Robert Israel, Mar 17 2019

fQ[n_]:=2==Plus@@Last/@FactorInteger[n];Select[Range[2000],fQ[ # ]&&fQ[2#+1]&&fQ[4#+3]&]

n, n1=2n+1 and n2=2n1+1 are semiprimes.

A188059 Numbers k with the property that k, k+1 and 2*k+1 are all semiprimes.

Original entry on oeis.org

25, 34, 38, 57, 93, 118, 133, 145, 177, 201, 205, 213, 218, 298, 334, 361, 381, 394, 446, 501, 633, 694, 698, 842, 865, 878, 898, 921, 1114, 1141, 1226, 1285, 1293, 1465, 1513, 1654, 1713, 1726, 1761, 1857, 1893, 1941, 1981, 2018, 2041, 2217, 2306, 2426, 2433, 2577, 2581, 2734, 2746, 2901, 2973, 3133, 3193, 3214, 3241, 3386, 3578, 3661, 3693, 3746, 3754, 3777, 3826, 3957

Offset: 1

Zak Seidov, Mar 20 2011

nonn

Numbers k such that 2k+1 is a semiprime and the sum of two consecutive semiprimes (k and k+1).

			25 is a term: k = 25 = 5*5, k+1 = 26 = 2*13, 2k+1 = 51 = 3*17.

Zak Seidov, Table of n, a(n) for n = 1..1309 (terms < 200000)

Cf. A001358 (semiprimes).

Cf. A176896 (safe semiprimes), A111153 (Sophie Germain semiprimes), A070552.

Select[Range[4000],Union[PrimeOmega[{#,#+1,2 #+1}]]=={2}&] (* Harvey P. Dale, May 11 2012 *)

Equals A111153 intersect A070552. - M. F. Hasler, Mar 20 2011

cp's OEIS Frontend

A177221 Numbers k that are the products of two distinct primes such that 2*k + 1 is also the product of two distinct primes.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

A113433 Semi-Pierpont semiprimes: products of exactly two Pierpont primes A005109.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A113434 Semi-Pierpont semiprimes which are also Pierpont semiprimes.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Formula

A175256 a(n) = sqrt(A175255(n)).

Views

Author

Keywords

Links

Crossrefs

A176896 Semiprimes s such that (s-1)/2 is also semiprime.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A211162 Sophie Germain 5-almost primes.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A111207 Numbers that are both Sophie Germain semiprimes and semi-Sophie Germain semiprimes.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Extensions

A115170 Non-cubefree numbers k such that 2k+1 is also non-cubefree (A046099).

Views

Author

Keywords

Comments