A086006 -id:A086006 - cp's OEIS frontend

A158476 Lesser of twin pairs in A086006.

107, 71549, 97379, 103067, 136709, 342449, 403829, 686969, 695879, 758267, 795797, 805499, 887399, 941489, 945881, 1023227, 1025747, 1081709, 1081979, 1169027, 1179989, 1261259, 1456919, 1554101, 2110877, 2148659, 2167469, 2433059, 2494439, 2554397, 2623571, 2803121

Offset: 1

Zak Seidov, Mar 20 2009

nonn

Primes p such that both p and p+2 are terms in A086006.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Zak Seidov, First 100 terms.

Cf. A086005, A086006.

Select[Prime[Range[10^6]], PrimeQ[# + 2] && PrimeOmega[2*# - 1] == PrimeOmega[2*# + 1] == 2 && PrimeOmega[2*# + 3] == PrimeOmega[2*# + 5] == 2 &] (* Amiram Eldar, Apr 06 2023 *)

is(p) = (bigomega(2*p+1) == 2) && (bigomega(2*p-1) == 2);
lista(max) = {my(q1 = 0, q2, prev = 0); forprime(p = 2, max, q2 = is(p); if(p == prev + 2 && q1 && q2, print1(prev, ", ")); prev = p; q1 = q2);} \\ Amiram Eldar, Apr 06 2023

More terms from Amiram Eldar, Apr 06 2023

A056809 Numbers k such that k, k+1 and k+2 are products of two primes.

Original entry on oeis.org

33, 85, 93, 121, 141, 201, 213, 217, 301, 393, 445, 633, 697, 841, 921, 1041, 1137, 1261, 1345, 1401, 1641, 1761, 1837, 1893, 1941, 1981, 2101, 2181, 2217, 2305, 2361, 2433, 2461, 2517, 2641, 2721, 2733, 3097, 3385, 3601, 3693, 3865, 3901, 3957, 4285

Offset: 1

Sharon Sela (sharonsela(AT)hotmail.com), May 04 2002

nonn

Each term is the beginning of a run of three 2-almost primes (semiprimes). No runs exist of length greater than three. For the same reason, each term must be odd: If k were even, then so would be k+2. In fact, one of k or k+2 would be divisible by 4, so must indeed be 4 to have only two prime factors. However, neither 2,3,4 nor 4,5,6 is such a run. - Rick L. Shepherd, May 27 2002
k+1, which is twice a prime, is in A086005. The primes are in A086006. - T. D. Noe, May 31 2006
The squarefree terms are listed in A039833. - Jianing Song, Nov 30 2021

			121 is in the sequence because 121 = 11^2, 122 = 2*61 and 123 = 3*41, each of which is the product of two primes.

D. W. Wilson, Table of n, a(n) for n = 1..10000

Intersection of A070552 and A092207.

Cf. A045939, A039833, A086005, A086006, A123255.

f[n_] := Plus @@ Transpose[ FactorInteger[n]] [[2]]; Select[Range[10^4], f[ # ] == f[ # + 1] == f[ # + 2] == 2 & ]
Flatten[Position[Partition[PrimeOmega[Range[5000]],3,1],{2,2,2}]] (* Harvey P. Dale, Feb 15 2015 *)
SequencePosition[PrimeOmega[Range[5000]],{2,2,2}][[;;,1]] (* Harvey P. Dale, Mar 03 2024 *)

forstep(n=1,5000,2, if(bigomega(n)==2 && bigomega(n+1)==2 && bigomega(n+2)==2, print1(n,",")))

is(n)=n%4==1 && isprime((n+1)/2) && bigomega(n)==2 && bigomega(n+2)==2 \\ Charles R Greathouse IV, Sep 08 2015

list(lim)=my(v=List(),t); forprime(p=2,(lim+1)\2, if(bigomega(t=2*p-1)==2 && bigomega(t+2)==2, listput(v,t))); Vec(v) \\ Charles R Greathouse IV, Sep 08 2015

a(n) = A086005(n) - 1 = 2*A086006(n) - 1 = 4*A123255(n) + 1. - Jianing Song, Nov 30 2021

Edited and extended by Robert G. Wilson v, May 04 2002

A124936 Numbers k such that k - 1 and k + 1 are semiprimes.

Original entry on oeis.org

5, 34, 50, 56, 86, 92, 94, 120, 122, 142, 144, 160, 184, 186, 202, 204, 214, 216, 218, 220, 236, 248, 266, 288, 290, 300, 302, 304, 320, 322, 328, 340, 392, 394, 412, 414, 416, 446, 452, 470, 472, 516, 518, 528, 534, 536, 544, 552, 580, 582, 590, 634, 668

Offset: 1

Zak Seidov, Nov 13 2006

nonn

All but the first term are even.

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

Cf. A092207 (k and k+2 are semiprimes), A086005 (k-1, k, k+1 are semiprimes), A086006 (primes p such that 2*p-1 and 2*p+1 are semiprimes), A082130 (2*k-1 and 2*k+1 are semiprimes).

IsSemiprime:=func< n | &+[k[2]: k in Factorization(n)] eq 2 >; [ n: n in [1..700] | IsSemiprime(n+1) and IsSemiprime(n-1)]; // Vincenzo Librandi, Mar 30 2015

lst={};Do[If[Plus@@Last/@FactorInteger[n-1]==2&&Plus@@Last/@FactorInteger[n+1]==2,AppendTo[lst,n]],{n,7!}];lst (* Vladimir Joseph Stephan Orlovsky, Feb 01 2009 *)
Select[Range[2, 700], PrimeOmega[# + 1] == PrimeOmega[# - 1] == 2 &] (* Vincenzo Librandi, Mar 30 2015 *)

list(lim)=if(lim<5,return([])); my(v=List([5]),x=1,y=1); forfactored(z=7,lim\1+1, if(vecsum(z[2][,2])==2 && vecsum(x[2][,2])==2, listput(v,z[1]-1)); x=y; y=z); Vec(v) \\ Charles R Greathouse IV, May 22 2018

from sympy import factorint
from itertools import count, islice
def agen(): # generator of terms
    yield 5
    nxt = 0
    for k in count(6, 2):
        prv, nxt = nxt, sum(factorint(k+1).values())
        if prv == nxt == 2: yield k
print(list(islice(agen(), 53))) # Michael S. Branicky, Nov 26 2022

a(n) = A092207(n) + 1; at n>=2, a(n) = 2*A082130(n-1).

A086005 Semiprimes sandwiched between semiprimes.

Original entry on oeis.org

34, 86, 94, 122, 142, 202, 214, 218, 302, 394, 446, 634, 698, 842, 922, 1042, 1138, 1262, 1346, 1402, 1642, 1762, 1838, 1894, 1942, 1982, 2102, 2182, 2218, 2306, 2362, 2434, 2462, 2518, 2642, 2722, 2734, 3098, 3386, 3602, 3694, 3866, 3902, 3958, 4286, 4414

Offset: 1

Reinhard Zumkeller, Jul 07 2003

nonn

These are some of the balanced semiprimes (see A213025). - Alonso del Arte, Jun 04 2012

			94 = 47*2: 94 - 1 = 3*31 and 94 + 1 = 5*19, therefore 94 is in the sequence.

T. D. Noe, Table of n, a(n) for n=1..10000
Eric Weisstein's World of Mathematics, Semiprime

Cf. A001358, A056809, A064911, A100484, A123255.

a086005 n = a086005_list !! (n-1)
a086005_list = filter
   (\x -> a064911 (x - 1) == 1 && a064911 (x + 1) == 1) a100484_list
-- Reinhard Zumkeller, Aug 08 2013, Jun 10 2012

u[n_]:=Plus@@Last/@FactorInteger[n]==2;lst={};Do[If[u[n],sp=n;If[u[sp-1]&&u[sp+1],AppendTo[lst,sp]]],{n,8!}];lst  (* Vladimir Joseph Stephan Orlovsky, Nov 16 2009 *)
(* First run program for A109611 to define semiPrimeQ *) Select[Range[4000], Union[{semiPrimeQ[# - 1], semiPrimeQ[#], semiPrimeQ[# + 1]}] == {True} &] (* Alonso del Arte, Jun 03 2012 *)
Select[Partition[Range@ 4000, 3, 1], Union@ PrimeOmega@ # == {2} &][[All, 2]] (* Michael De Vlieger, Jun 14 2017 *)

from itertools import count, islice
from sympy import factorint, isprime
def agen(): # generator of terms
    nxt = 0
    for k in count(2, 2):
        prv, nxt = nxt, sum(factorint(k+1).values())
        if prv == nxt == 2 and isprime(k//2): yield k
print(list(islice(agen(), 46))) # Michael S. Branicky, Nov 26 2022

a(n) = 2*A086006(n).

a(n) = A056809(n)+1. - Zak Seidov, Sep 30 2012

A109373 Semiprimes of the form semiprime + 1.

Original entry on oeis.org

10, 15, 22, 26, 34, 35, 39, 58, 86, 87, 94, 95, 119, 122, 123, 134, 142, 143, 146, 159, 178, 202, 203, 206, 214, 215, 218, 219, 254, 299, 302, 303, 327, 335, 362, 382, 394, 395, 446, 447, 454, 482, 502, 515, 527, 538, 543, 554, 566, 623, 634, 635, 695, 698

Offset: 1

Jonathan Vos Post, Aug 24 2005

			a(1) = 10 because (3*3+1)=(2*5) = 10.
a(2) = 15 because (2*7+1)=(3*5) = 15.
a(3) = 22 because (3*7+1)=(2*11) = 22.
a(4) = 26 because (5*5+1)=(2*13) = 26.
a(5) = 34 because (3*11+1)=(2*17) = 34.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Primes are in A000040. Semiprimes are in A001358.
Primes of the form semiprime + 1 are in A005385 (safe primes).
Semiprimes of the form semiprime + 1 are in this sequence.
3-almost primes of the form semiprime + 1 are in A109067.
4-almost primes of the form semiprime + 1 are in A109287.
5-almost primes of the form semiprime + 1 are in A109383.
Least n-almost prime of the form semiprime + 1 are in A128665.
Cf. A056809, A077065, A079148, A086005, A086006, A092307.
Subsequence of A088707; A064911.

a109373 n = a109373_list !! (n-1)
a109373_list = filter ((== 1) . a064911) a088707_list
-- Reinhard Zumkeller, Feb 20 2012

fQ[n_] := Plus @@ Last /@ FactorInteger[n] == 2; Select[ Range[ 700], fQ[ # - 1] && fQ[ # ] &] (* Robert G. Wilson v *)
With[{sps=Select[Range[700],PrimeOmega[#]==2&]},Transpose[Select[ Partition[ sps,2,1],#[[2]]-#[[1]]==1&]][[2]]] (* Harvey P. Dale, Sep 05 2012 *)

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

a(n) is in this sequence iff a(n) is in A001358 and (a(n)-1) is in A001358.

a(n) = A070552(n) + 1.

Extended by Ray Chandler and Robert G. Wilson v, Aug 25 2005

Edited by Ray Chandler, Mar 20 2007

A195685 Primes p for which tau(2p-1) = tau(2p+1) = 4.

Original entry on oeis.org

17, 43, 47, 71, 101, 107, 109, 151, 197, 223, 317, 349, 461, 521, 569, 631, 673, 701, 821, 881, 919, 947, 971, 991, 1051, 1091, 1109, 1153, 1181, 1217, 1231, 1259, 1321, 1361, 1367, 1549, 1693, 1801, 1847, 1933, 1951, 1979, 2143, 2207, 2267, 2297, 2441, 2801

Offset: 1

Timothy L. Tiffin, Sep 22 2011

nonn

Sequence terms are a subset of those listed in A086006 and A068497.

The numbers 2p-1, 2p, 2p+1 form a run (indeed, a maximal run) of three consecutive integers each with four positive divisors. The first two examples are 33, 34, 35 and 85, 86, 87. A039833 gives the first number in these maximal 3-integer runs. - Timothy L. Tiffin, Jul 05 2016

			tau(2*17-1) = tau(33) = tau(3*11) = 4 = tau(5*7) = tau(35) = tau(2*17+1) and tau(2*43-1) = tau(85) = tau(5*17) = 4 = tau(3*29) = tau(87) = tau(2*43+1). - _Timothy L. Tiffin_, Jul 05 2016

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

Cf. A039833, A068497, A086006, A248201.

with(numtheory):
q:= p-> isprime(p) and tau(2*p-1)=4 and tau(2*p+1)=4:
select(q, [$1..3000])[];  # Alois P. Heinz, Apr 18 2019

Select[Prime[Range[500]], DivisorSigma[0, 2 # - 1] == DivisorSigma[0, 2 # + 1] == 4 &] (* T. D. Noe, Sep 22 2011 *)
Select[Mean[#]/2&/@SequencePosition[DivisorSigma[0,Range[6000]],{4,,4}],PrimeQ] (* _Harvey P. Dale, Nov 26 2021 *)

lista(nn) = forprime(p=2, nn, if ((numdiv(2*p-1) == 4) && (numdiv(2*p+1) == 4), print1(p, ", "))); \\ Michel Marcus, Jul 06 2016

a(n) = A248201(n)/2. - Torlach Rush, Jun 25 2021

A115393 Numbers n such that n, n-1 and n-2 are semiprimes.

Original entry on oeis.org

35, 87, 95, 123, 143, 203, 215, 219, 303, 395, 447, 635, 699, 843, 923, 1043, 1139, 1263, 1347, 1403, 1643, 1763, 1839, 1895, 1943, 1983, 2103, 2183, 2219, 2307, 2363, 2435, 2463, 2519, 2643, 2723, 2735, 3099, 3387, 3603, 3695, 3867, 3903, 3959, 4287

Offset: 1

Zak Seidov, Mar 08 2006

nonn

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

Cf. A056809, A086005, A086006.

filter:= proc(n) local k;
   k:= n mod 3;
   if k = 0 then isprime(n/3) and isprime((n-1)/2) and numtheory:-bigomega(n-2)=2
   elif k= 1 then false
   else isprime((n-2)/3) and isprime((n-1)/2) and numtheory:-bigomega(n)=2
   fi
end proc:
select(filter, [seq(i,i=3..10000,4)]); # Robert Israel, Jun 11 2020

upto=5000;p=Prime[Range[PrimePi[upto/2]]];lim=Floor[Sqrt[upto]]; sp={};k=0; While[k++;p[[k]]<=lim,sp=Join[sp,p[[k]] *Take[p,{k,PrimePi[upto/p[[k]]]}]]]; sp=Sort[sp];Transpose[Select [Partition[sp,3,1], Last[#]-#[[2]]==#[[2]]-First[#]==1&]][[3]] (* Harvey P. Dale, Mar 21 2011 -- semiprime generating portion of program from A001358 *)

a(n)=A056809(n)+2=A086005(n)+1=2*A086006(n)+1.

A115394 List of triples of semiprimes: each three numbers are consecutive semiprimes.

Original entry on oeis.org

33, 34, 35, 85, 86, 87, 93, 94, 95, 121, 122, 123, 141, 142, 143, 201, 202, 203, 213, 214, 215, 217, 218, 219, 301, 302, 303, 393, 394, 395, 445, 446, 447, 633, 634, 635, 697, 698, 699, 841, 842, 843, 921, 922, 923, 1041, 1042, 1043, 1137, 1138, 1139, 1261

Offset: 1

Zak Seidov, Mar 08 2006

nonn

Cf. A056809, A086005, A086006.

Flatten[Select[Partition[Union[Times@@@Tuples[Prime[Range[150]],{2}]],3,1],#[[3]]-#[[2]]==#[[2]]-#[[1]]==1&]]  (* Harvey P. Dale, Jan 22 2011 *)

a(3n-2) = A056809(n), a(3n-1) = A086005(n), a(3n) = A086005(n) + 1 = 2*A086006(n) + 1, n=1,2,...

A328058 Primes p such that 2*p-1 is a semiprime.

Original entry on oeis.org

5, 11, 13, 17, 29, 43, 47, 61, 67, 71, 73, 89, 101, 103, 107, 109, 127, 151, 181, 191, 197, 223, 227, 241, 251, 269, 277, 283, 317, 349, 359, 373, 397, 409, 421, 433, 457, 461, 467, 487, 521, 541, 569, 571, 631, 643, 647, 659, 673, 701, 709, 719, 733, 739, 751, 757, 769, 821, 857, 859, 881, 883

Offset: 1

J. M. Bergot and Robert Israel, Oct 03 2019

nonn

			a(3)=13 is in the sequence because it is prime and 2*13-1=5^2 is a semiprime.

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

Cf. A000040, A001358. Includes A067756 and A162336.

Cf. A086006, A234095.

[p: p in PrimesUpTo(1000)| &+[d[2]: d in Factorization(2*p-1)] eq 2]; // Marius A. Burtea, Oct 03 2019

select(t -> isprime(t) and numtheory:-bigomega(2*t-1)=2, [2,seq(i,i=3..10000,2)]);

Select[Prime@ Range@ 153, PrimeOmega[2 # - 1] == 2 &] (* Michael De Vlieger, Oct 03 2019 *)

isok(p) = isprime(p) && (bigomega(2*p-1) == 2); \\ Michel Marcus, Oct 04 2019

cp's OEIS Frontend

A158476 Lesser of twin pairs in A086006.

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A056809 Numbers k such that k, k+1 and k+2 are products of two primes.

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A124936 Numbers k such that k - 1 and k + 1 are semiprimes.

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Magma

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A086005 Semiprimes sandwiched between semiprimes.

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A109373 Semiprimes of the form semiprime + 1.

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A195685 Primes p for which tau(2p-1) = tau(2p+1) = 4.

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A115393 Numbers n such that n, n-1 and n-2 are semiprimes.