A063838 -id:A063838 - cp's OEIS frontend

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

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

A070268 Numbers k such that mu(k) + mu(k+1) + mu(k+2) = -3.

Original entry on oeis.org

29, 41, 101, 137, 229, 281, 429, 433, 617, 641, 645, 741, 821, 969, 1021, 1085, 1129, 1221, 1309, 1433, 1489, 1581, 1597, 1605, 1613, 1697, 1741, 1877, 1885, 2013, 2053, 2081, 2085, 2109, 2161, 2237, 2265, 2309, 2337, 2377, 2381, 2397, 2409, 2633, 2657, 2665, 2677

Offset: 1

Sharon Sela (sharonsela(AT)hotmail.com), May 09 2002

nonn

All terms == 1 (mod 4). - Robert Israel, Feb 17 2019

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

Cf. A008683 (mu), A063838.

with(numtheory): A070268:=n->`if`(mobius(n)+mobius(n+1)+mobius(n+2)=-3, n, NULL): seq(A070268(n), n=1..10^4); # Wesley Ivan Hurt, Feb 05 2017

seq[max_] := Module[{t = Table[MoebiusMu[n], {n, 1, max}]}, Flatten[Position[Partition[t, 3, 1], ?(Total[#] == -3 &)]]]; seq[2700] (* _Amiram Eldar, Jul 25 2025 *)

isok(n) = moebius(n)+moebius(n+1)+moebius(n+2)==-3;

More terms from Benoit Cloitre, May 13 2002

A083544 a(n) = maximal value of the sum of Mobius function values over a block of n consecutive natural numbers.

Original entry on oeis.org

1, 2, 3, 3, 4, 5, 6, 6, 7, 8, 9, 9, 10, 11, 12, 12, 12, 13, 14, 14, 15, 16, 17, 17, 18, 18, 19, 19, 20, 21, 22, 22, 23, 24, 24, 24, 25, 26, 27, 27, 28, 29, 30, 30, 31, 32, 33, 33, 34, 34, 35, 35, 36, 37, 38, 38, 39, 39, 40, 40, 41, 42, 43, 43, 44, 45, 45, 45, 46, 47, 48, 48, 49, 49, 50, 50

Offset: 1

Yuval Dekel (dekelyuval(AT)hotmail.com), Jun 10 2003

nonn

Comment from Hugh Montgomery (hlm(AT)umich.edu): I do not recall having seen literature on this question. If p is a prime, p < sqrt(k), then there will be a multiple of p^2 in the block and such a number will then contribute 0. Let Q(M, k) denote the numbers of integers between M+1 and M+k (inclusive) that are not divisible by the square of any prime <= sqrt(k). By the sieve of Eratosthenes-Legendre, Q(M,k) = k/zeta(2) +O(sqrt(k)), uniformly in M. Let Q^+(k) = max_M Q(M,k). I expect that the sum of mu(n) over n = M+1..M+k can be as large as Q^+(k) and as small as -Q^+(k). Indeed, I expect that this could be shown to follow from the prime k-tuple conjecture.

The maximum first appears at A225420(n). - T. D. Noe, May 07 2013

Cf. A008683, A063838, A082967, A225420.

a(n) = max sum m=i...(i+n-1) Mobius(m) over i>=1.

Offset corrected by Eric M. Schmidt, May 07 2013

More terms from Don Reble, Apr 21 2021

A082967 Numbers n such that mu(n) + mu(n+1) + mu(n+2) + mu(n+3) + mu(n+4) + mu(n+5) + mu(n+6) = 6.

Original entry on oeis.org

213, 1937, 3093, 3097, 4529, 6401, 7165, 9753, 9933, 10117, 10541, 10577, 11301, 13385, 15005, 18641, 18829, 18965, 25089, 34413, 36285, 37833, 37909, 38157, 38405, 38409, 38413, 39529, 40461, 42981, 44457, 46577, 48741, 51365, 55477

Offset: 1

Yuval Dekel (dekelyuval(AT)hotmail.com), May 27 2003

nonn

The terms up to 18965 were first given by Ignacio Larrosa Cañestro in a thread entitled "Seven consecutive numbers" in the newsgroup sci.math on Apr 29 2003. - Hugo Pfoertner, Aug 26 2006

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Ignacio Larrosa Cañestro, Seven consecutive numbers. Posting in newsgroup sci.math, Apr 29 2003.

Cf. A070268, A063838, A008683.

Position[Partition[MoebiusMu[Range[60000]],7,1],?(Total[#] == 6&),1,Heads->False]//Flatten (* _Harvey P. Dale, Jan 23 2019 *)

isok(n) = moebius(n) + moebius(n+1) + moebius(n+2) + moebius(n+3) + moebius(n+4) + moebius(n+5) + moebius(n+6) == 6 \\ Michel Marcus, Aug 06 2013

More terms from Rick L. Shepherd, May 28 2003

A114180 Numbers n with mu(n) = mu(n+1) = mu(n+2).

Original entry on oeis.org

29, 33, 41, 48, 85, 93, 98, 101, 124, 137, 141, 201, 213, 217, 229, 242, 243, 281, 301, 342, 350, 393, 423, 429, 433, 445, 475, 548, 603, 617, 633, 641, 645, 697, 724, 741, 774, 821, 844, 845, 846, 869, 921, 969, 1021, 1024, 1041, 1085, 1129, 1137, 1189

Offset: 1

Franklin T. Adams-Watters, Feb 03 2006

nonn

Any sequence of 4 or more consecutive numbers with the same value for mu must all have mu(n)=0 (n divisible by a proper square) since at least one of every 4 consecutive numbers is divisible by 4.

A261890(a(n)) = 0. - Reinhard Zumkeller, Sep 05 2015

			mu(n)=1 for 33,34,35; 85,86,87; 93,94,95; ...
mu(n)=-1 for 29,30,31; 41,42,43; 101,102,103; ...
mu(n)=0 for 48,49,50; 98,99,100; 124,125,126; ...

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

Union of A070258, A063838 and A070268. Cf. A008683, A070284.

Cf. A261890.

a114180 n = a114180_list !! (n-1)
a114180_list = filter ((== 0) . a261890) [1..]
-- Reinhard Zumkeller, Sep 05 2015

SequencePosition[MoebiusMu[Range[1200]],{x_,x_,x_}][[;;,1]] (* Harvey P. Dale, Jul 23 2023 *)

A063736 Patterns of possible squarefree triples of 3 consecutive numbers {4k+1, 4k+2, 4k+3} are coded as follows: compute A008966(x) getting one of {000, 001, 010, 011, 100, 101, 110, 111} and convert to decimal.

Original entry on oeis.org

7, 7, 3, 7, 5, 7, 2, 7, 7, 7, 7, 3, 1, 5, 7, 6, 7, 7, 6, 7, 3, 7, 5, 7, 4, 7, 7, 7, 7, 3, 3, 1, 7, 6, 7, 7, 6, 5, 3, 7, 5, 7, 2, 6, 7, 7, 7, 3, 7, 5, 7, 6, 7, 7, 7, 7, 3, 7, 5, 7, 4, 3, 5, 7, 7, 3, 7, 5, 6, 6, 7, 7, 3, 5, 3, 7, 5, 7, 6, 7, 7, 3, 7, 3, 5, 4, 7, 4, 7, 7, 2, 7, 3, 6, 5, 7, 6, 7, 7, 7, 7, 3, 7, 5, 7

Offset: 0

Labos Elemer, Aug 24 2001

nonn

All code values arise corresponding to 8 classes of patterns. E.g., the first nonsquarefree triple (000 pattern, code=0) appears at 844, [845, 846, 847], 848 as a middle part of a nonsquarefree 5-tuple. Start values of code=7 triples are listed in A063238.

			a(0) = 4*A008966(1)+2*A008966(2)+A008966(3) = 4+2+1 = 7.
a(11) = 4*A008966(45)+2*A008966(46)+A008966(47) = 0+2+1 = 3.
a(12) = 4*A008966(49)+2*A008966(50)+A008966(51) = 0+0+1 = 1.
a(13) = 4*A008966(53)+2*A008966(54)+A008966(55) = 4+0+1 = 5.
a(14) = 4*A008966(57)+2*A008966(58)+A008966(59) = 4+2+1 = 7.

Cf. A007675, A063838, A008966.

a(n) = 4*A008966(4n+1)+2*A008966(4n+2)+A008966(4n+3).

A063848 Numbers k such that mu(k) + mu(k+1) + mu(k+2) = 2.

Original entry on oeis.org

14, 20, 32, 34, 38, 55, 56, 84, 86, 91, 92, 94, 117, 118, 121, 122, 132, 133, 140, 142, 143, 144, 145, 158, 159, 176, 183, 200, 202, 203, 204, 205, 208, 212, 214, 215, 216, 218, 219, 235, 247, 252, 297, 298, 299, 300, 302, 303, 319, 325, 326, 327, 328, 333

Offset: 1

Jason Earls, Aug 26 2001

Harry J. Smith, Table of n, a(n) for n = 1..1000

Cf. A063838.

Flatten[Position[Partition[MoebiusMu[Range[350]],3,1],?(Total[#]==2&),{1},Heads->False]] (* _Harvey P. Dale, Nov 26 2015 *)

m(n) = moebius(n)+moebius(n+1)+moebius(n+2);
j=[]; for(n=1,1000, if(m(n)==2,j=concat(j,n))); j

M(n) = moebius(n) + moebius(n + 1) + moebius(n + 2)
{ n=0; for (m=1, 10^9, if(M(m)==2, write("b063848.txt", n++, " ", m); if (n==1000, break)) ) } \\ Harry J. Smith, Sep 01 2009

A063849 Numbers k such that mu(k) + mu(k+1) + mu(k+2) = 1.

Original entry on oeis.org

8, 13, 21, 24, 25, 26, 37, 44, 49, 50, 54, 57, 62, 63, 74, 75, 80, 90, 115, 116, 119, 120, 123, 134, 146, 157, 160, 175, 177, 185, 187, 206, 207, 209, 224, 234, 248, 253, 259, 260, 265, 274, 278, 287, 294, 295, 296, 304, 314, 321, 323, 324, 329, 332, 338, 340

Offset: 1

Jason Earls, Aug 26 2001

Harry J. Smith, Table of n, a(n) for n = 1..1000

Cf. A063838.

m(n)=moebius(n)+moebius(n+1)+moebius(n+2);
j=[]; for(n=1,1000, if(m(n)==1,j=concat(j,n))); j

M(n) = moebius(n) + moebius(n + 1) + moebius(n + 2)
{ n=0; for (m=1, 10^9, if(M(m)==1, write("b063849.txt", n++, " ", m); if (n==1000, break)) ) } \\ Harry J. Smith, Sep 01 2009

A063684 Numbers k such that m(k!) = 2, where m(k) = mu(k) + mu(k+1) + mu(k+2).

Original entry on oeis.org

8, 13, 14, 18, 19, 20, 25, 36, 38, 43, 48, 51, 52, 54, 60, 71, 74, 75, 78, 80, 87, 91, 92, 105, 108, 110, 112, 114

Offset: 1

Jason Earls, Aug 22 2001

Equivalently, k such that m(k!) = 2, where m(k) = mu(k+1) + mu(k+2), as mu(k!)=0 for all k >= 4 (because 4=2^2 divides k!). - Rick L. Shepherd, Aug 20 2003

127 belongs to the sequence. - Serge Batalov, Feb 17 2011

			8 is a term: 8! = 40320; mu(40320) = 0, mu(40321) = 1, mu(40322) = 1, 0+1+1 = 2.
98 is not a term because 98! + 2 = 2 * 31003012014959 * 114951592532951 * 2015644865638913835753087050212028452990938458387 * P78 has an odd number of factors. - _Sean A. Irvine_, Feb 03 2010

Dario A. Alpern, Factorization using the Elliptic Curve Method.
Paul Leyland, Factors of n!+1 (updated 1 Oct 2006).

Cf. A063838, A008683.

Cf. A084846 (mu(n!+1)).

m(n) = moebius(n)+moebius(n+1)+moebius(n+2); for(n=1,10^4, if(m(n!)==2,print(n)))

More terms from Rick L. Shepherd, Aug 20 2003
Two more terms from Sean A. Irvine, Feb 03 2010, Feb 08 2010
Two new terms, 105 and 108, from Daniel M. Jensen, Feb 19 2011, Mar 02 2011
Two more terms, 110 and 112, from Serge Batalov, Mar 04-05 2011
One more term, 114, from Sean A. Irvine, May 25 2015

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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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A070268 Numbers k such that mu(k) + mu(k+1) + mu(k+2) = -3.

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A083544 a(n) = maximal value of the sum of Mobius function values over a block of n consecutive natural numbers.

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A082967 Numbers n such that mu(n) + mu(n+1) + mu(n+2) + mu(n+3) + mu(n+4) + mu(n+5) + mu(n+6) = 6.

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A114180 Numbers n with mu(n) = mu(n+1) = mu(n+2).

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A063736 Patterns of possible squarefree triples of 3 consecutive numbers {4k+1, 4k+2, 4k+3} are coded as follows: compute A008966(x) getting one of {000, 001, 010, 011, 100, 101, 110, 111} and convert to decimal.

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A063848 Numbers k such that mu(k) + mu(k+1) + mu(k+2) = 2.

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A063849 Numbers k such that mu(k) + mu(k+1) + mu(k+2) = 1.

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