A066509 -id:A066509 - cp's OEIS frontend

A215217 Smaller member of a pair of sphenic twins, consecutive integers, each the product of three distinct primes.

230, 285, 429, 434, 609, 645, 741, 805, 902, 969, 986, 1001, 1022, 1065, 1085, 1105, 1130, 1221, 1245, 1265, 1309, 1310, 1334, 1406, 1434, 1442, 1462, 1490, 1505, 1533, 1581, 1598, 1605, 1614, 1634, 1729, 1742, 1833, 1885, 1886, 1946, 2013, 2014, 2054, 2085

Offset: 1

Martin Renner, Aug 06 2012

nonn

455 is not a term of the sequence, since 455 = 5*7*13 is sphenic, i.e., the number of distinct prime factors is 3, though 456 = 2^3*3*19 has 3 distinct prime factors but is not sphenic, because the number of prime factors with repetition is 5 > 3.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A007304, A066509, A140077.

twinLow [] = []
twinLow [_] = []
twinLow (n : (m : ns))
    | m == n + 1 = n : twinLow (m : ns)
    | otherwise = twinLow (m : ns)
a215217 n = (twinLow a007304_list) !! (n - 1)
-- Peter Dolland, May 31 2019

Sphenics:= select(t -> (map(s->s[2],ifactors(t)[2])=[1,1,1]), {$1..10000}):
Sphenics intersect map(`-`,Sphenics,1); # Robert Israel, Aug 13 2014

Select[Range[2500], (PrimeNu[#] == PrimeOmega[#] == PrimeNu[#+1] == PrimeOmega[#+1] == 3)&] (* Jean-François Alcover, Apr 11 2014 *)
SequencePosition[Table[If[PrimeNu[n]==PrimeOmega[n]==3,1,0],{n,2500}],{1,1}][[All,1]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Aug 02 2017 *)

is_a033992(n) = omega(n)==3 && bigomega(n)==3
is(n) = is_a033992(n) && is_a033992(n+1) \\ Felix Fröhlich, Jun 10 2019

A242621 Start of the least triple of consecutive squarefree numbers each of which has exactly n distinct prime factors.

Original entry on oeis.org

2, 33, 1309, 27962, 3323705, 296602730, 41704979953

Offset: 1

M. F. Hasler, May 18 2014

nonn

As the example of a(4)=27962 shows, "consecutive squarefree numbers" means consecutive elements of A005117, not necessarily consecutive integers that (additionally) are squarefree; this would be a more restrictive condition.

a(8) <= 102099792179229 because A093550 - 1 is an upper bound of the present sequence.

			The two squarefree numbers following a(4)=27962, namely, 27965 and 27966, also have 4 prime divisors just as a(4).

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

See A242605-A242608 for triples of consecutive squarefree numbers with m=2,..,5 prime factors.
See A246470 for the quadruplet and A246548 for the 5-tuple versions of this sequence.
See A039833, A066509, A176167 and A192203 for triples of consecutive numbers which are squarefree and have m=2,..,5 prime factors.

Edited and a(6)-a(7) added by Hans Havermann, Aug 27 2014

A248202 Sphenic numbers (A007304) whose neighbors are sphenic.

Original entry on oeis.org

1310, 1886, 2014, 2666, 3730, 5134, 6062, 6214, 6306, 6478, 6854, 6986, 7258, 7954, 8394, 8534, 8786, 9214, 9454, 9822, 9878, 10282, 10946, 11606, 12454, 12566, 12802, 12858, 12994, 13054, 14134, 14314, 14330, 14466, 14818, 15086, 15266, 15806, 16114, 16134

Offset: 1

James G. Merickel, Oct 03 2014

nonn

Subsequence of A169834 and offset by 1 from the values in A066509.

			1309, 1310 and 1311 factor as 7*11*17, 2*5*131 and 3*19*23, respectively.  No smaller such trio exists, so a(1)=1310.

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Wikipedia, Sphenic number

Cf. A007304, A169834, A066509, A248201, A248203, A248204, A259349, A259350, A259801.

a248202[n_Integer] := Select[Range[n],
  And[And[PrimeNu[#] == 3, PrimeNu[# - 1] == 3, PrimeNu[# + 1] == 3], And[PrimeOmega[#] == 3, PrimeOmega[# - 1] == 3, PrimeOmega[# + 1] == 3]] &]; a248202[20166](* Michael De Vlieger, Nov 06 2014 *)
f[n_]:=Last/@FactorInteger[n]=={1, 1, 1}; lst={}; Do[If[f[n]&&f[n+1]&&f[n+2], AppendTo[lst, n + 1]], {n, 17000}]; lst (* Vincenzo Librandi, Jul 24 2015 *)
Mean/@SequencePosition[Table[If[PrimeNu[n]==PrimeOmega[n]==3,1,0],{n,20000}],{1,1,1}] (* Harvey P. Dale, Dec 08 2024 *)

sq(n)=bigomega(n)==3 && omega(n)==3;
for(n=3,10^5,if(sq(n-1)&&sq(n)&&sq(n+1),print1(n,", ")));
\\ Joerg Arndt, Oct 18 2014

a(n) = A066509(n)+1.

A248203 Numbers n such that n-1, n, and n+1 are the product of 4 distinct primes.

Original entry on oeis.org

203434, 214490, 225070, 258014, 294594, 313054, 315722, 352886, 389390, 409354, 418846, 421630, 452354, 464386, 478906, 485134, 500906, 508046, 508990, 526030, 528410, 538746, 542270, 542794, 548302, 556870, 559690, 569066, 571234, 579886, 582406, 588730

Offset: 1

James G. Merickel, Oct 28 2014

nonn

A subsequence of A066509 and offset by one from A176167.

			203433 factors as 3*19*43*83, 203434 factors as 2*7*11*1321 and 203435 factors as 5*23*29*61; and with no similar smaller trio a(1)=203434. [Corrected by _James G. Merickel_, Jul 23 2015]

Anders Hellström, Table of n, a(n) for n = 1..300

Cf. A066509, A176167, A248201, A248202, A248204, A259349, A259350, A259801.

f1[n_]:=Last/@FactorInteger[n]=={1, 1, 1, 1}; f2[n_]:=Max[Last/@FactorInteger[n]]; lst={}; Do[If[f1[n]&&f1[n + 1]&&f1[n+2], AppendTo[lst, n + 1]], {n, 2 8!, 4 9!}]; lst (* Vincenzo Librandi, Aug 02 2015 *)

{
\\ Initialized at A093550(4) (3rd term there, w/offset=2). If this \\
\\ program is to run from a different starting value of n, it must not \\
\\ be congruent to -1, 0 or 1 modulo 9 (in addition to being congruent \\
\\ to 2 modulo 4), and either u or the vector s needs to be brought into \\
\\ agreement. \\
n=203434;s=[4,4,8,8,8,4];u=1;
while(1,
  if(issquarefree(n) &&
    issquarefree(n-1) &&
    issquarefree(n+1) &&
    omega(n)==4 &&
    omega(n-1)==4 &&
    omega(n+1)==4,
    print1(n, ", "));
  n+=s[u];if(u==6,u=1,u++))
} \\ James G. Merickel, Jul 23 2015

is_ok(n)=(n>1&&omega(n-1)==4&&omega(n)==4&&omega(n+1)==4&&issquarefree(n-1)&&issquarefree(n)&&issquarefree(n+1));
first(m)=my(v=vector(m),i,t=2);for(i=1,m,while(!is_ok(t),t++);v[i]=t;t++);v; /* Anders Hellström, Aug 01 2015 */

a(n) = A176167(n)+1.

A248204 Middle values in trios of products of 5 distinct primes.

Original entry on oeis.org

16467034, 18185870, 21134554, 21374354, 21871366, 22247554, 22412534, 22721586, 24845314, 25118094, 25228930, 25435334, 25596934, 26217246, 27140114, 29218630, 29752346, 30323734, 30563246, 31943066, 32663266, 33367894, 36055046, 38269022, 39738062, 40547066

Offset: 1

James G. Merickel, Oct 28 2014

nonn

A subsequence of A066509 and offset by 1 from A192203.

			16467033=3*11*17*149*197,
16467034=2*19*23*83*227, and
16467035=5*13*37*41*167,
with no smaller similar trio. So a(1)=16467034. [Corrected by _James G. Merickel_, Jul 23 2015]

James G. Merickel, Table of n, a(n) for n = 1..26

Cf. A066509, A192203, A248201, A248202, A248203, A259349, A259350, A259801.

{
\\ This program checks all consecutives with elements not divisible \\
\\ by the squares of 2 or 3. More efficiency is required if enormous \\
\\ numbers of terms are sought and for the analog sequences beyond \\
\\ 6 prime factors. The start value is A093550(5). If a start other \\
\\ than this is chosen, one must be sure that (one of) s or u is \\
\\ adjusted if it needs to be. \\
n=16467034;s=[8,4,4,4,8,8];u=1;
while(1,
  if(issquarefree(n) && issquarefree(n-1) && issquarefree(n+1) && omega(n)==5 && omega(n-1)==5 && omega(n+1)==5, print1(n" "));
  n+=s[u];
  if(u==6,u=1,u++)
)
} \\ James G. Merickel, Jul 23 2015

a(n) = A192203(n) + 1.

A192203 Numbers k such that k, k+1, and k+2 are each the product of exactly 5 distinct primes.

Original entry on oeis.org

16467033, 18185869, 21134553, 21374353, 21871365, 22247553, 22412533, 22721585, 24845313, 25118093, 25228929, 25345333, 25596933, 26217245, 27140113, 29218629, 29752345, 30323733, 30563245, 31943065, 32663265, 33367893, 36055045, 38269021, 39738061, 40547065

Offset: 1

Gil Broussard, Jun 25 2011

nonn

Numbers k such that k, k+1, and k+2 are all members of A046387. - N. J. A. Sloane, Jul 17 2024
A subsequence of A242608 intersect A016813. - M. F. Hasler, May 19 2014
All terms are congruent to 1 mod 4. - Zak Seidov, Dec 22 2014

			a(1)=16467033 because it is the product of 5 distinct primes (3,11,17,149,197), and so are a(1)+1: 16467034 (2,19,23,83,227), and a(1)+2: 16467035 (5,13,37,41,167).

Zak Seidov, Table of n, a(n) for n = 1..154

Cf. A046387, A140079. Subsequence of A318964 and of A364266.

Cf. A039833, A066509, A176167.

SequencePosition[Table[If[PrimeNu[n]==PrimeOmega[n]==5,1,0],{n,164*10^5,406*10^5}],{1,1,1}][[;;,1]]+164*10^5-1 (* Harvey P. Dale, Jul 17 2024 *)

forstep(n=1+10^7,1e8,4, for(k=n,n+2,issquarefree(k)||next(2)); for(k=n,n+2,omega(k)==5||next(2));print1((n)", ")) \\ M. F. Hasler, May 19 2014

A176167 First of a triple of consecutive integers, each the product of 4 distinct primes.

Original entry on oeis.org

203433, 214489, 225069, 258013, 294593, 313053, 315721, 352885, 389389, 409353, 418845, 421629, 452353, 464385, 478905, 485133, 500905, 508045, 508989, 526029, 528409, 538745, 542269, 542793, 548301, 556869, 559689, 569065, 571233, 579885

Offset: 1

Vladimir Joseph Stephan Orlovsky, Apr 10 2010

nonn

A subsequence of A242607 and A016813. - M. F. Hasler, May 19 2014

			203433 is a term: 203433 = 3*19*43*83, 203434 = 2*7*11*1321, 203435 = 5*23*29*61.

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

Cf. A039833, A066509, A192203. Subsequence of A140078 and of A318896.

f1[n_]:=Last/@FactorInteger[n]=={1,1,1,1};f2[n_]:=Max[Last/@FactorInteger[n]];lst={};Do[If[f1[n]&&f1[n+1]&&f1[n+2],AppendTo[lst,n]],{n,5*8!,7*9!}];lst

forstep(n=1+10^5,10^7,4, for(k=n,n+2,issquarefree(k)||next(2)); for(k=n,n+2,omega(k)==4||next(2));print1((n)",")) \\ M. F. Hasler, May 19 2014

A242606 Start of a triple of consecutive squarefree numbers each of which has exactly 3 distinct prime factors.

Original entry on oeis.org

1309, 1442, 1885, 2013, 2091, 2665, 2694, 2714, 3243, 3422, 3655, 3729, 3854, 3855, 4430, 4431, 4503, 4921, 5034, 5035, 5133, 5282, 5678, 5795, 5882, 5883, 5943, 5954, 6054, 6061, 6094, 6213, 6302, 6303, 6305, 6306, 6477, 6851, 6853, 6873, 6985, 7202, 7257, 7334, 7383, 7682, 7730, 7802, 7842, 7922, 7953, 8238, 8239

Offset: 1

M. F. Hasler, May 18 2014

nonn

Sequence A066509 is a subsequence.

			The two squarefree numbers following a(1)=1309=7*11*17 are 1310=2*5*131 and 1311=3*19*23, all three have 3 prime divisors.
The same is true for a(2)=1442, 1443 and the next squarefree number which is 1446.
Further examples are provided by the first "sphenic triples" (1309, 1310, 1311), (1885, 1886, 1887) and (2013, 2014, 2015).

Harvey P. Dale, Table of n, a(n) for n = 1..10000
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

See A242605-A242608 for triples of consecutive squarefree numbers (A005117) with m=2,...,5 prime factors; A242621 (first terms for positive m).

Transpose[Select[Partition[Select[Range[10000],SquareFreeQ],3,1], Union[ PrimeNu[ #]] == {3}&]][[1]] (* Harvey P. Dale, Apr 29 2016 *)

(back(n)=for(i=1,2,until(issquarefree(n--),));n);for(n=1,9999,issquarefree(n)||next;ndk==ndm&&omega(n)==ndm&&ndk==3&&print1(back(n)",");ndk=ndm;ndm=omega(n))

Minor edit by Hans Havermann, Aug 19 2014

A215015 Number of sphenic twins up to 10^n.

Original entry on oeis.org

0, 0, 11, 337, 4206, 43330, 417479, 3917508, 36358375, 336046778, 3105465308, 28739218426

Offset: 1

Martin Renner, Aug 06 2012

The sphenic twin pairs {230, 231}, {285, 286}, ... are counted once at a time.

			a(3) = 11 since there are 11 sphenic twins below 10^3 whose smaller members are 230, 285, 429, 434, 609, 645, 741, 805, 902, 969, 986.

Lucas A. Brown, Python program.

Cf. A007304, A215217, A066509, A215218, A215152.

sphQ[n_]:= FactorInteger[n][[;;,2]] == {1, 1, 1}; c = 0; p = 10; q1 = 0; seq = {}; Do[q2 = sphQ[k]; If[q1 && q2, c++]; If[k == p, AppendTo[seq, c]; p*=10]; q1 = q2, {k, 2, 10^5}]; seq (* Amiram Eldar, Dec 26 2019 *)

a(8)-a(10) from Amiram Eldar, Dec 26 2019

a(11)-a(12) from Lucas A. Brown, Feb 12 2024

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

cp's OEIS Frontend

A215217 Smaller member of a pair of sphenic twins, consecutive integers, each the product of three distinct primes.

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A242621 Start of the least triple of consecutive squarefree numbers each of which has exactly n distinct prime factors.

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A248202 Sphenic numbers (A007304) whose neighbors are sphenic.

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A248203 Numbers n such that n-1, n, and n+1 are the product of 4 distinct primes.

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A248204 Middle values in trios of products of 5 distinct primes.

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A192203 Numbers k such that k, k+1, and k+2 are each the product of exactly 5 distinct primes.

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A176167 First of a triple of consecutive integers, each the product of 4 distinct primes.

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