A056809 -id:A056809 - cp's OEIS frontend

A045939 Numbers m such that the factorizations of m..m+2 have the same number of primes (including multiplicities).

33, 85, 93, 121, 141, 170, 201, 213, 217, 244, 284, 301, 393, 428, 434, 445, 506, 602, 603, 604, 633, 637, 697, 841, 921, 962, 1041, 1074, 1083, 1084, 1130, 1137, 1244, 1261, 1274, 1309, 1345, 1401, 1412, 1430, 1434, 1448, 1490, 1532, 1556, 1586, 1604

Offset: 1

David W. Wilson

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

Numbers m through m+k have the same number of prime divisors (with multiplicity): A045920 (k=1), this sequence (k=2), A045940 (k=3), A045941 (k=4), A045942 (k=5), A123103 (k=6), A123201 (k=7), A358017 (k=8), A358018 (k=9), A358019 (k=10).
A056809 is a subsequence.
Cf. A006073. - Harvey P. Dale, Apr 19 2011

f[n_]:=Plus@@Last/@FactorInteger[n];lst={};lst={};Do[If[f[n]==f[n+1]==f[n+2],AppendTo[lst,n]],{n,0,7!}];lst (* Vladimir Joseph Stephan Orlovsky, May 12 2010 *)
pd2Q[n_]:=PrimeOmega[n]==PrimeOmega[n+1]==PrimeOmega[n+2]; Select[Range[1700],pd2Q]  (* Harvey P. Dale, Apr 19 2011 *)
SequencePosition[PrimeOmega[Range[1700]],{x_,x_,x_}][[;;,1]] (* Harvey P. Dale, Mar 08 2023 *)

is(n)=my(t=bigomega(n)); bigomega(n+1)==t && bigomega(n+2)==t \\ Charles R Greathouse IV, Sep 14 2015

list(lim)=my(v=List(),a=1,b=1,c); forfactored(n=4,lim\1+2,c=bigomega(n); if(a==b&&a==c, listput(v,n[1]-2)); a=b; b=c); Vec(v) \\ Charles R Greathouse IV, May 07 2020

A039833 Smallest of three consecutive squarefree numbers k, k+1, k+2 of the form p*q where p and q are distinct primes.

Original entry on oeis.org

33, 85, 93, 141, 201, 213, 217, 301, 393, 445, 633, 697, 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, 4413, 4533, 4593, 4881, 5601

Offset: 1

Olivier Gérard

Equivalently: k, k+1 and k+2 all have 4 divisors.
There cannot be four consecutive squarefree numbers as one of them is divisible by 2^2 = 4.
These 3 consecutive squarefree numbers of the form p*q have altogether 6 prime factors always including 2 and 3. E.g., if k = 99985, the six prime factors are {2,3,5,19997,33329,49993}. The middle term is even and not divisible by 3.
Nonsquare terms of A056809. First terms of A056809 absent here are A056809(4)=121=11^2, A056809(14)=841=29^2, A056809(55)=6241=79^2.
Cf. A179502 (Numbers k with the property that k^2, k^2+1 and k^2+2 are all semiprimes). - Zak Seidov, Oct 27 2015
The numbers k, k+1, k+2 have the form 2p-1, 2p, 2p+1 where p is an odd prime.  A195685 gives the sequence of odd primes that generates these maximal runs of three consecutive integers with four positive divisors. - Timothy L. Tiffin, Jul 05 2016
a(n) is always 1 or 9 mod 12. - Charles R Greathouse IV, Mar 19 2022

			33, 34 and 35 all have 4 divisors.
85 is a term as 85 = 17*5, 86 = 43*2, 87 = 29*3.

R. K. Guy, Unsolved Problems in Number Theory, Springer, 1st edition, 1981. See section B18.
David Wells, Curious and interesting numbers, Penguin Books, 1986, p. 114.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Reinhard Zumkeller)
Roberto Conti, Pierluigi Contucci, and Vitalii Iudelevich, Bounds on tree distribution in number theory, arXiv:2401.03278 [math.NT], 2024. See page 13.

Subsequence of A006881 and A056809 and A263990.

Cf. A038456, A039832, A008683, A007675, A063736, A063838, A070552, A045939, A195685, A179502.

a039833 n = a039833_list !! (n-1)
a039833_list = f a006881_list where
   f (u : vs@(v : w : xs))
     | v == u+1 && w == v+1 = u : f vs
     | otherwise            = f vs
-- Reinhard Zumkeller, Aug 07 2011

lst = {}; Do[z = n^3 + 3*n^2 + 2*n; If[PrimeOmega[z/n] == PrimeOmega[z/(n + 2)] == 4 && PrimeNu[z] == 6, AppendTo[lst, n]], {n, 1, 5601, 2}]; lst (* Arkadiusz Wesolowski, Dec 11 2011 *)
okQ[n_]:=Module[{cl={n,n+1,n+2}},And@@SquareFreeQ/@cl && Union[ DivisorSigma[ 0,cl]]=={4}]; Select[Range[1,6001,2],okQ] (* Harvey P. Dale, Dec 17 2011 *)
SequencePosition[DivisorSigma[0,Range[6000]],{4,4,4}][[All,1]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Aug 17 2017 *)

is(n)=n%4==1 && factor(n)[,2]==[1,1]~ && factor(n+1)[,2]==[1,1]~ && factor(n+2)[,2]==[1,1]~ \\ Charles R Greathouse IV, Aug 29 2016

is(n)=my(t=n%12); if(t==1, isprime((n+2)/3) && isprime((n+1)/2) && factor(n)[,2]==[1,1]~, t==9 && isprime(n/3) && isprime((n+1)/2) && factor(n+2)[,2]==[1,1]~) \\ Charles R Greathouse IV, Mar 19 2022

A008966(a(n)) * A064911(a(n)) * A008966(a(n)+1) * A064911(a(n)+1) * A008966(a(n)+2) * A064911(a(n)+2) = 1. - Reinhard Zumkeller, Feb 26 2011

Additional comments from Amarnath Murthy, Vladeta Jovovic, Labos Elemer and Benoit Cloitre, May 08 2002

A082919 Numbers k such that k, k+2, k+4, k+6, k+8, k+10, k+12 and k+14 are semiprimes.

Original entry on oeis.org

8129, 9983, 99443, 132077, 190937, 237449, 401429, 441677, 452639, 604487, 802199, 858179, 991289, 1471727, 1474607, 1963829, 1999937, 2376893, 2714987, 3111977, 3302039, 3869237, 4622087, 4738907, 6156137, 7813559, 8090759

Offset: 1

Hugo Pfoertner, Apr 22 2003

nonn

Start of a cluster of 8 consecutive odd semiprimes. Semiprimes in arithmetic progression. All terms are odd, see also A056809.
Note that there cannot exist 9 consecutive odd semiprimes. Out of any 9 consecutive odd numbers, one of them will be divisible by 9. The only multiple of 9 which is a semiprime is 9 itself and it is easy to see that's not part of a solution. - Jack Brennen, Jan 04 2006
For the first 500 terms, a(n) is roughly 40000*n^1.6, so the sequence appears to be infinite. Note that (a(n)+4)/3 and (a(n)+10)/3 are twin primes. - Don Reble, Jan 05 2006
All terms == 11 (mod 18). - Zak Seidov, Sep 27 2012
There is at least one even semiprime between k and k+14 for 1812 of the first 10000 terms. - Donovan Johnson, Oct 01 2012
All terms == {29,47,83} (mod 90). - Zak Seidov, Sep 13 2014
Among the first 10000 terms, from all 80000 numbers a(n)+m, m=0,2,4,6,8,10,12,14, the only square is a(4637) + 2 = 23538003241 = 153421^2 (153421 is prime, of course). - Zak Seidov, Dec 22 2014

			a(1)=8129 because 8129=11*739, 8131=47*173, 8133=3*2711, 8135=5*1627, 8137=79*103, 8139=3*2713, 8141=7*1163, 8143=17*479 are semiprimes.

Author of this sequence is Jack Brennen, who provided the terms up to 991289 in a posting to the seqfan mailing list on April 5, 2003.

Donovan Johnson, Table of n, a(n) for n = 1..10000 (terms a(1001) to a(2000) from Zak Seidov)
Eric Weisstein's World of Mathematics, Semiprime.

Cf. A001358, A082130, A082131, A056809, A070552, A092207, A092125, A092126, A092127, A092128, A092129, A092209, A217222 (consecutive semiprimes).

PrimeFactorExponentsAdded[n_] := Plus @@ Flatten[Table[ #[[2]], {1}] & /@ FactorInteger[n]]; Select[ Range[3*10^6], PrimeFactorExponentsAdded[ # ] == PrimeFactorExponentsAdded[ # + 2] == PrimeFactorExponentsAdded[ # + 4] == PrimeFactorExponentsAdded[ # + 6] == PrimeFactorExponentsAdded[ # + 8] == PrimeFactorExponentsAdded[ # + 10] == PrimeFactorExponentsAdded[ # + 12] == PrimeFactorExponentsAdded[ # + 14] == 2 &] (* Robert G. Wilson v and Zak Seidov, Feb 24 2004 *)

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

A092207 Semiprimes k such that k+2 is also a semiprime.

Original entry on oeis.org

4, 33, 49, 55, 85, 91, 93, 119, 121, 141, 143, 159, 183, 185, 201, 203, 213, 215, 217, 219, 235, 247, 265, 287, 289, 299, 301, 303, 319, 321, 327, 339, 391, 393, 411, 413, 415, 445, 451, 469, 471, 515, 517, 527, 533, 535, 543, 551, 579, 581, 589, 633, 667

Offset: 1

Robert G. Wilson v and Zak Seidov, Feb 24 2004

nonn

Starting with 33 all terms are odd. First squares are 4, 49, 169, 361, 529, 961, 1369, 2209, 2809, 4489, ... - Zak Seidov, Feb 17 2017

Zak Seidov, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Semiprime

Cf. A056809, A070552, A092125, A092126, A092127, A092128, A092129, A082919, A092209.

PrimeFactorExponentsAdded[n_] := Plus @@ Flatten[Table[ #[[2]], {1}] & /@ FactorInteger[n]]; Select[ Range[ 668], PrimeFactorExponentsAdded[ # ] == PrimeFactorExponentsAdded[ # + 2] == 2 &]
Select[Range[700],PrimeOmega[#]==PrimeOmega[#+2]==2&] (* Harvey P. Dale, Aug 20 2011 *)
SequencePosition[Table[If[PrimeOmega[n]==2,1,0],{n,700}],{1,,1}] [[All,1]] (* Requires Mathematica version 10 or later *) (* _Harvey P. Dale, May 29 2017 *)

is(n)=if(n%2==0, return(n==4)); bigomega(n)==2 && bigomega(n+2)==2 \\ Charles R Greathouse IV, Feb 21 2017

from sympy import factorint
from itertools import count, islice
def agen(): # generator of terms
    yield 4
    nxt = 0
    for k in count(5, 2):
        prv, nxt = nxt, sum(factorint(k+2).values())
        if prv == nxt == 2: yield k
print(list(islice(agen(), 53))) # Michael S. Branicky, Nov 26 2022

A092125 Numbers n such that n, n+2, n+4 are semiprimes.

Original entry on oeis.org

91, 119, 141, 183, 201, 213, 215, 217, 287, 299, 301, 319, 391, 411, 413, 469, 515, 533, 579, 667, 685, 695, 789, 813, 1055, 1077, 1133, 1135, 1137, 1145, 1165, 1203, 1253, 1313, 1343, 1345, 1347, 1383, 1385, 1387, 1389, 1401, 1561, 1639, 1685, 1687, 1761

Offset: 1

Zak Seidov, Feb 22 2004

Semiprimes in arithmetic progression. All terms are odd, see also A056809.

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

Cf. A056809, A070552, A092126, A092127, A092128, A092129.

IsSemiprime:=func< n| &+[ k[2]: k in Factorization(n) ] eq 2 >; [ n: n in [2..4300]|IsSemiprime(n) and IsSemiprime(n+2) and IsSemiprime(n+4)] // Vincenzo Librandi, Dec 16 2010

PrimeFactorExponentsAdded[n_] := Plus @@ Flatten[Table[ #[[2]], {1}] & /@ FactorInteger[n]]; Select[ Range[ 1792], PrimeFactorExponentsAdded[ # ] == PrimeFactorExponentsAdded[ # + 2] == PrimeFactorExponentsAdded[ # + 4] == 2 &] (* Robert G. Wilson v, Feb 24 2004 *)
SequencePosition[Table[If[PrimeOmega[n]==2,1,0],{n,2000}],{1,,1,,1}][[All,1]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Feb 17 2020 *)

A092129 Numbers n such that n, n+2, n+4, n+6, n+8, n+10, n+12 are semiprimes.

Original entry on oeis.org

3091, 8129, 8131, 9983, 9985, 55559, 92603, 99443, 99445, 112709, 132077, 132079, 182749, 190937, 190939, 209479, 237449, 237451, 239089, 249689, 296779, 300449, 313411, 401429, 401431, 441677, 441679, 452639, 452641, 547157, 604487, 604489

Offset: 1

Zak Seidov, Feb 22 2004

Semiprimes in arithmetic progression. All terms are odd, see also A056809.

Cf. A056809, A070552, A092125, A092126, A092127, A092128.

PrimeFactorExponentsAdded[n_] := Plus @@ Flatten[Table[ #[[2]], {1}] & /@ FactorInteger[n]]; Select[ Range[ 631200], PrimeFactorExponentsAdded[ # ] == PrimeFactorExponentsAdded[ # + 2] == PrimeFactorExponentsAdded[ # + 4] == PrimeFactorExponentsAdded[ # + 6] == PrimeFactorExponentsAdded[ # + 8] == PrimeFactorExponentsAdded[ # + 10] == PrimeFactorExponentsAdded[ # + 12] == 2 &] (* Robert G. Wilson v, Feb 24 2004 *)
Select[Range[610000],Union[PrimeOmega[#+Range[0,12,2]]]=={2}&] (* Harvey P. Dale, Oct 14 2018 *)

More terms from Don Reble, Feb 23 2004

More terms from Robert G. Wilson v, Feb 24 2004

A055212 Number of composite divisors of n.

Original entry on oeis.org

0, 0, 0, 1, 0, 1, 0, 2, 1, 1, 0, 3, 0, 1, 1, 3, 0, 3, 0, 3, 1, 1, 0, 5, 1, 1, 2, 3, 0, 4, 0, 4, 1, 1, 1, 6, 0, 1, 1, 5, 0, 4, 0, 3, 3, 1, 0, 7, 1, 3, 1, 3, 0, 5, 1, 5, 1, 1, 0, 8, 0, 1, 3, 5, 1, 4, 0, 3, 1, 4, 0, 9, 0, 1, 3, 3, 1, 4, 0, 7, 3, 1, 0, 8, 1, 1, 1, 5, 0, 8, 1, 3, 1, 1, 1, 9, 0, 3, 3, 6, 0, 4, 0, 5, 4

Offset: 1

Leroy Quet, Jun 23 2000

Trivially, there is only one run of three consecutive 0's. However, there are infinitely many runs of three consecutive 1's and they are at positions A056809(n), A086005(n), and A115393(n) for n >= 1. - Timothy L. Tiffin, Jun 21 2021

			a[20] = 3 because the composite divisors of 20 are 4, 10, 20.

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

Complement of A083399.
Cf. A000005, A001221, A008578 (indices of 0's), A033273, A056809, A086005, A115393, A137944, A137945, A344713.
Cf. A001620, A077761.

a055212 = subtract 1 . a033273  -- Reinhard Zumkeller, Sep 15 2015

Table[ Count[ PrimeQ[ Divisors[n] ], False] - 1, {n, 1, 105} ]
Table[Count[Divisors[n],?CompositeQ],{n,120}] (* Requires Mathematica version 10 or later *) (* _Harvey P. Dale, Jul 09 2018 *)
a[n_] := DivisorSigma[0, n] - PrimeNu[n] - 1; Array[a, 100] (* Amiram Eldar, Jun 18 2022 *)

a(n) = numdiv(n) - omega(n) - 1; \\ Michel Marcus, Oct 17 2015

a(n) = A033273(n) - 1.
a(n) = tau(n)-omega(n)-1, where tau=A000005 and omega=A001221. - Reinhard Zumkeller, Jun 13 2003
G.f.: -x/(1 - x) + Sum_{k>=1} (x^k - x^prime(k))/((1 - x^k)*(1 - x^prime(k))). - Ilya Gutkovskiy, Mar 21 2017
Sum_{k=1..n} a(k) ~ n*log(n) - n*log(log(n)) + (2*gamma - 2 - B)*n, where gamma is Euler's constant (A001620) and B is Mertens's constant (A077761). - Amiram Eldar, Dec 07 2023

A092126 Numbers n such that n, n+2, n+4, n+6 are semiprimes.

Original entry on oeis.org

213, 215, 299, 411, 1133, 1135, 1343, 1345, 1383, 1385, 1387, 1685, 1793, 1835, 1837, 1891, 1937, 1939, 1957, 2045, 2315, 2317, 2513, 2567, 2807, 2809, 2929, 3091, 3093, 3095, 3097, 3147, 3149, 3647, 3957, 3977, 3979, 4115, 4313, 4315, 4411, 4529, 4531

Offset: 1

Zak Seidov, Feb 22 2004

Semiprimes in arithmetic progression. All terms are odd, see also A056809.

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

Cf. A056809, A070552, A092125, A092127, A092128, A092129.

PrimeFactorExponentsAdded[n_] := Plus @@ Flatten[Table[ #[[2]], {1}] & /@ FactorInteger[n]]; Select[ Range[ 4626], PrimeFactorExponentsAdded[ # ] == PrimeFactorExponentsAdded[ # + 2] == PrimeFactorExponentsAdded[ # + 4] == PrimeFactorExponentsAdded[ # + 6] == 2 &] (* Robert G. Wilson v, Feb 24 2004 *)
Transpose[SequencePosition[Table[If[PrimeOmega[n]==2,1,0],{n,5000}],{1,,1,,1,,1}]][[1]] (* The program uses the SequencePosition function from Mathematica version 10 *) (* _Harvey P. Dale, Feb 10 2016 *)

A092128 Numbers n such that n, n+2, n+4, n+6, n+8, n+10 are semiprimes.

Original entry on oeis.org

1383, 3091, 3093, 5609, 8129, 8131, 8133, 9753, 9983, 9985, 9987, 10401, 11013, 12053, 13637, 16499, 22457, 30991, 43339, 45803, 49083, 53761, 55559, 55561, 58277, 63047, 63951, 64829, 69603, 71727, 76803, 80633, 92603, 92605, 98493

Offset: 1

Zak Seidov, Feb 22 2004

Semiprimes in arithmetic progression. All terms are odd, see also A056809.

Cf. A056809, A070552, A092125, A092126, A092127, A092129.

PrimeFactorExponentsAdded[n_] := Plus @@ Flatten[Table[ #[[2]], {1}] & /@ FactorInteger[n]]; Select[ Range[ 99210], PrimeFactorExponentsAdded[ # ] == PrimeFactorExponentsAdded[ # + 2] == PrimeFactorExponentsAdded[ # + 4] == PrimeFactorExponentsAdded[ # + 6] == PrimeFactorExponentsAdded[ # + 8] == PrimeFactorExponentsAdded[ # + 10] == 2 &] (* Robert G. Wilson v, Feb 24 2004 *)
spQ[n_]:=PrimeOmega[n]==2; Select[Range[100000],AllTrue[#+{0,2,4,6,8,10},spQ]&] (* Harvey P. Dale, Dec 19 2021 *)

More terms from Don Reble, Feb 23 2004

More terms from Robert G. Wilson v, Feb 24 2004

cp's OEIS Frontend

A045939 Numbers m such that the factorizations of m..m+2 have the same number of primes (including multiplicities).

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A039833 Smallest of three consecutive squarefree numbers k, k+1, k+2 of the form p*q where p and q are distinct primes.

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A082919 Numbers k such that k, k+2, k+4, k+6, k+8, k+10, k+12 and k+14 are semiprimes.

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A086005 Semiprimes sandwiched between semiprimes.

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A092207 Semiprimes k such that k+2 is also a semiprime.

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A092125 Numbers n such that n, n+2, n+4 are semiprimes.

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A092129 Numbers n such that n, n+2, n+4, n+6, n+8, n+10, n+12 are semiprimes.

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