A039833 -id:A039833 - cp's OEIS frontend

A242605 Start of a triple of consecutive squarefree numbers which are all semiprimes.

33, 55, 85, 91, 93, 115, 118, 119, 141, 142, 143, 158, 201, 202, 203, 205, 213, 214, 215, 217, 218, 295, 298, 299, 301, 302, 323, 326, 391, 393, 411, 413, 445, 451, 511, 514, 535, 542, 551, 622, 633, 685, 694, 695, 697, 745, 763, 778, 791, 799, 815, 842, 843, 865, 898, 921, 922

Offset: 1

M. F. Hasler, May 18 2014

nonn

Sequence A039833 is a subsequence.

			33 is in the sequence because 33, 34, 35 are all squarefree semiprimes.
55 is in the sequence because 55, 57, 58 (we ignore 56 because it's not squarefree) are all squarefree semiprimes.

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Daniel C. Mayer, Define an "m-triple" to consist of three consecutive squarefree positive integers, each with exactly m prime divisors, Number Theory group on LinkedIn.com

Cf. A242606 (m=3), A242607 (m=4), A242608 (m=5), A242621 (first terms for positive m).

Transpose[Select[Partition[Select[Range[1000],SquareFreeQ],3,1], Union[ PrimeOmega[ #]] =={2}&]][[1]] (* Harvey P. Dale, Feb 07 2016 *)

is_A242605(n,c=2)==issquarefree(n)&&omega(n)==2&&(!c||until(issquarefree(n++),)||is_A242605(n,c-1))

(back(n,c=1)=until(issquarefree(n--)&&c--,);n); for(n=1,999,issquarefree(n)||next;dk==4&&dk==dm&&numdiv(n)==dm&&print1(back(n)",");dk=dm;dm=numdiv(n))

A242804 Integers k such that each of k, k+1, k+2, k+4, k+5, k+6 is the product of two distinct primes.

Original entry on oeis.org

213, 143097, 194757, 206133, 273417, 684897, 807657, 1373937, 1391757, 1516533, 1591593, 1610997, 1774797, 1882977, 1891761, 2046453, 2051493, 2163417, 2163957, 2338053, 2359977, 2522517, 2913837, 3108201, 4221753

Offset: 1

Daniel Constantin Mayer, May 23 2014

nonn

A remarkable gap occurs between the initial two members, and the sequence seems to be rather sparse compared to the related A242805.
Here, the first member k of the sextet is the reference, whereas in A068088 the center k+3 is selected as reference. Observe that k+3 must be divisible by the square 4.
All terms are congruent to 9 (mod 12). - Zak Seidov, Apr 14 2015
From Robert Israel, Apr 15 2015: (Start)
All terms are congruent to 33 (mod 36).
Numbers k in A039833 such that k+4 is in A039833. (End)
From Robert G. Wilson v, Apr 15 2015: (Start)
k is congruent to 33 (mod 36) so one of its factors is 3 and the other is == 11 (mod 12);
k+1 is congruent to 34 (mod 36) so one of its factors is 2 and the other is == 17 (mod 18);
k+2 is congruent to 35 (mod 36) so its factors are == +-1 (mod 6);
k+4 is congruent to 1 (mod 36) so its factors are == +-1 (mod 6);
k+5 is congruent to 2 (mod 36) so one of its factors is 2 and the other is == 1 (mod 18);
k+6 is congruent to 3 (mod 36) so one of its factors is 3 and the other is == 1 (mod 12). (End).
Number of terms < 10^m: 0, 0, 1, 1, 1, 7, 39, 169, 882, 4852, 27479, ...,. - Robert G. Wilson v, Apr 15 2015
Or, numbers k such that k, k+1 and k+2 are terms in A175648. - Zak Seidov, Dec 08 2015

			213=3*71, 214=2*107, 215=5*43, 217=7*31, 218=2*109, 219=3*73.

Zak Seidov and Robert G. Wilson v, Table of n, a(n) for n = 1..10000

Cf. A242793 (minima for two, three and more prime divisors) and A068088 (arbitrary squarefree integers).

Cf. A242805, A039833, A175648, A202319.

f:= t -> numtheory:-issqrfree(t) and (numtheory:-bigomega(t) = 2):
select(t -> andmap(f, [t,t+1,t+2,t+4,t+5,t+6]), [seq(36*k+33,k=0..10^6)]); # Robert Israel, Apr 15 2015

fQ[n_] := PrimeQ[n/3] && PrimeQ[(n + 1)/2] && PrimeQ[(n + 5)/2] && PrimeQ[(n + 6)/3] && PrimeNu[{n + 2, n + 4}] == {2, 2} == PrimeOmega[{n + 2, n + 4}]; k = 33; lst = {}; While[k < 10^8, If[fQ@ k, AppendTo[lst, k]]; k += 36]; lst (* Robert G. Wilson v, Apr 14 2015 and revised Apr 15 2015 after Zak Seidov and Robert Israel *)

default(primelimit, 1000M); i=0; j=0; k=0; l=0; m=0; loc=0; lb=2; ub=9*10^9; o=2; for(n=lb, ub, if(issquarefree(n)&&(o==omega(n)), loc=loc+1; if(1==loc, i=n; ); if(2==loc, if(i+1==n, j=n; ); if(i+1

forstep(x=213,4221753,12, if( isprime(x/3) && isprime((x+1)/2) && 2==omega(x+2) && 2==bigomega(x+2) && 2==omega(x+4) && 2==bigomega(x+4) && isprime((x+5)/2) && isprime((x+6)/3), print1(x", "))) \\ Zak Seidov, Apr 14 2015

a(n) = A202319(n) - 1. - Jon Maiga, Jul 10 2021

A038456 List of pairs of consecutive numbers each with 4 divisors (duplicates removed).

Original entry on oeis.org

14, 15, 21, 22, 26, 27, 33, 34, 35, 38, 39, 57, 58, 85, 86, 87, 93, 94, 95, 118, 119, 122, 123, 133, 134, 141, 142, 143, 145, 146, 158, 159, 177, 178, 201, 202, 203, 205, 206, 213, 214, 215, 217, 218, 219, 253, 254, 298, 299, 301, 302, 303, 326, 327, 334, 335, 381, 382, 393, 394, 395, 445, 446, 447, 453, 454

Offset: 1

Felice Russo

nonn

			14 and 15 because both have 4 as number of divisors and are consecutive.

D. Wells, Curious and interesting numbers, Penguin Books.

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

Cf. A039832, A039833.

Union[Flatten[Select[Partition[Range[500],2,1],DivisorSigma[0,First[#]] == DivisorSigma[0,Last[#]]==4&]]] (* Harvey P. Dale, Jul 22 2012 *)
SequencePosition[DivisorSigma[0,Range[500]],{4,4}]//Flatten//Union (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Aug 15 2016 *)

isA038456(n) = (numdiv(n)==4) && ((numdiv(n+1)==4) || (numdiv(n-1)==4)) \\ Michael B. Porter, Feb 03 2010

Corrected and extended by Olivier Gérard

Corrected by Rick L. Shepherd, Jun 07 2002

A364307 Numbers k such that k, k+1 and k+2 have exactly 2 distinct prime factors.

Original entry on oeis.org

20, 33, 34, 38, 44, 50, 54, 55, 56, 74, 75, 85, 86, 91, 92, 93, 94, 98, 115, 116, 117, 122, 133, 134, 141, 142, 143, 144, 145, 146, 158, 159, 160, 175, 176, 183, 187, 200, 201, 205, 206, 207, 212, 213, 214, 215, 216, 217, 224, 235, 247, 248, 295, 296

Offset: 1

R. J. Mathar, Jul 18 2023

nonn

			44 = 2^2*11 has 2 distinct prime factors, and so has 45 = 3^2*5 and so has 46 = 2*23, so 44 is in the sequence.

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

Subsequence of A006073 and of A074851.
Cf. A364308 (3 factors), A364309 (4 factors), A364266 (5 factors), A364265 (6 factors), A001221.
A039833 is a subsequence.

q[n_] := q[n] = PrimeNu[n] == 2; Select[Range[300], q[#] && q[#+1] && q[#+2] &] (* Amiram Eldar, Oct 01 2024 *)

{k: A001221(k) = A001221(k+1) = A001221(k+2) = 2}.

A075039 Smallest of three consecutive squarefree numbers having equal numbers of prime factors.

Original entry on oeis.org

33, 85, 93, 141, 201, 213, 217, 301, 393, 445, 633, 697, 921, 1041, 1137, 1261, 1309, 1345, 1401, 1641, 1761, 1837, 1885, 1893, 1941, 1981, 2013, 2101, 2181, 2217, 2305, 2361, 2433, 2461, 2517, 2641, 2665, 2721, 2733, 3097, 3385, 3601, 3693, 3729, 3865

Offset: 1

Amarnath Murthy, Sep 03 2002

nonn

			33 is a member as 33, 34 and 35 are of the form p*q where p and q are primes.

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

Cf. A001221, A001222, A005117, A052214, A086263, A039833.

Select[Range[4000], AllTrue[# + Range[0, 2], SquareFreeQ] && Equal @@ PrimeNu[# + Range[0, 2]] &] (* Amiram Eldar, Feb 24 2021 *)

A001221(a(n)) = A001222(a(n)) = A001221(a(n)+1) = A001222(a(n)+1).

More terms from Matthew Conroy, Sep 08 2002
Edited by Reinhard Zumkeller, Jul 14 2003
Offset corrected by Amiram Eldar, Feb 24 2021

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

A242492 For any integer m > 1, the m-th term of the sequence is the minimal squarefree integer x with exactly m prime divisors such that x+1 and x+2 are also squarefree integers with exactly m prime divisors.

Original entry on oeis.org

33, 1309, 203433, 16467033, 1990586013, 41704979953, 102099792179229

Offset: 2

Daniel Constantin Mayer, May 16 2014

The five terms for m = 2,3,4,5,6 were computed with the aid of PARI/GP. But it seems to be rather difficult to compute higher terms, if they exist at all.

The distribution of squarefree integers with exactly m prime factors is given in the book by Montgomery and Vaughan, Multiplicative Number Theory, but I do not have access to it and do not know whether it also addresses the problem of three consecutive numbers of this kind.

			33 = 3*11, 34 = 2*17, 35 = 5*7;
1309 = 7*11*17, 1310 = 2*5*131, 1311 = 3*19*23;
203433 = 3*19*43*83, 203434 = 2*7*11*1321, 203435 = 5*23*29*61;
16467033 = 3*11*17*149*197, 16467034 = 2*19*23*83*227, 16467035 = 5*13*37*41*167; (CPU time 48 seconds)
1990586013 = 3*13*29*67*109*241, 1990586014 = 2*23*37*43*59*461, 1990586015 = 5*11*17*19*89*1259. (CPU time 2 hours and 34 minutes)

Hugh L. Montgomery and Robert C. Vaughan: "Multiplicative Number Theory: 1. Classical Theory", Cambridge studies in advanced mathematics, vol. 97, Cambridge University Press (2007)

Math Overflow, Asymptotics of special square-free numbers, Mar 20 2014
Daniel Constantin Mayer, PARI/GP script "AdjacentSquareFree.gp"
Daniel Constantin Mayer, PARI/GP script "SquareFreeTriplets.gp"
Prime puzzles & problems connection, Square-free triples

Cf. A007675 (any m), A039833 (m=2), A066509 (m=3), A176167 (m=4), A192203 (m=5), A068088 (sextets with gap).

Cf. A242605-A242608 for start of triples of consecutive squarefree numbers with m=2,...,5 prime factors, A242621 for the analog of the present sequence in that spirit.

{default(primelimit,2M); lb=2; ub=2*10^9; m=1; i=0; j=0; loc=0; while(m<6, m=m+1; for(n=lb,ub, if(issquarefree(n)&&(m==omega(n)), loc=loc+1; if(1==loc, i=n; ); if(2==loc, if(i+1==n, j=n; ); if(i+1

a(n) = A093550(n)-1. - M. F. Hasler, May 20 2014

A359746 Numbers k such that k, k+1 and k+2 have the same ordered prime signature.

Original entry on oeis.org

33, 85, 93, 141, 201, 213, 217, 301, 393, 445, 633, 697, 921, 1041, 1137, 1261, 1309, 1345, 1401, 1641, 1761, 1837, 1885, 1893, 1941, 1981, 2013, 2101, 2181, 2217, 2305, 2361, 2433, 2461, 2517, 2641, 2665, 2721, 2733, 3097, 3385, 3601, 3693, 3729, 3865, 3901, 3957

Offset: 1

Amiram Eldar, Jan 13 2023

nonn

First differs from its subsequence A039833 at n = 17, and from its subsequence A075039 at n = 53.
The ordered prime signature of a number n is the list of exponents of the distinct prime factors in the prime factorization of n, in the order of the prime factors (A124010).
Can 4 consecutive integers have the same ordered prime signature? There are no such quadruples below 10^9.
The answer to the question above is no. Two out of every four consecutive numbers are even and their powers of 2 are different. - Ivan N. Ianakiev, Jan 13 2023

			33 is a term since 33 = 3^1 * 11^1, 34 = 2^1 * 17^1, and 35 = 5^1 * 7^1 have the same ordered prime signature, (1, 1).
4923 is a term since 4923 = 3^2 * 547^1, 4924 = 2^2 * 1231^1, and 4925 = 5^2 * 197^1 have the same ordered prime signature, (2, 1).
603 is a term of A052214 but not a term of this sequence, since 603 = 3^2 * 67^1, 604 = 2^2 * 151^1, and 605 = 5^1 * 11^2 have different ordered prime signatures, (2, 1) or (1, 2).

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

Subsequence of A052214 and A359745.
Subsequences: A039833, A075039.
Cf. A124010, A175590.

q[n_] := SameQ @@ (FactorInteger[#][[;; , 2]]& /@ (n + {0, 1, 2})); Select[Range[2, 4000], q]

lista(nmax) = {my(e1 = [], e2 = factor(2)[,2]); for(n = 3, nmax, e3 = factor(n)[,2]; if(e1 == e2 && e2 == e3, print1(n-2, ", ")); e1 = e2; e2 = e3); }

cp's OEIS Frontend

A242605 Start of a triple of consecutive squarefree numbers which are all semiprimes.

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A242804 Integers k such that each of k, k+1, k+2, k+4, k+5, k+6 is the product of two distinct primes.

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A038456 List of pairs of consecutive numbers each with 4 divisors (duplicates removed).

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A364307 Numbers k such that k, k+1 and k+2 have exactly 2 distinct prime factors.

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A075039 Smallest of three consecutive squarefree numbers having equal numbers of prime factors.

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

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A242492 For any integer m > 1, the m-th term of the sequence is the minimal squarefree integer x with exactly m prime divisors such that x+1 and x+2 are also squarefree integers with exactly m prime divisors.

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