

A078143


Smallest term of a run of at least 9 consecutive integers which are not squarefree.


11



8870024, 33908368, 49250144, 69147868, 70918820, 111500620, 112931372, 164786748, 167854344, 200997948, 203356712, 207543320, 211014920, 216785256, 221167422, 221167423, 221167424, 236645624, 240574368, 262315467, 262315468
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OFFSET

1,1


COMMENTS

The sequence includes an infinite family of arithmetic progressions. Such AP's can be constructed to each term, with large differences [like squares of primorials, A061742(7)]. It is necessary to solve suitable systems of linear Diophantine equations. E.g.: arithmetic progression subsequences of starting 9chains is {mk+69147868+j} where j=0..8, m=510510^2 because square prime factors of a(4)+j=68147868+j are 4, 49, 121, 169, 4, 9, 289, 25, 4 resp. for j=0..8; k goes to infinity; 7th primorial is sufficient, 9th is not necessary. Construction is provable for arbitrary long [>9] chains.  Labos Elemer, Nov 25 2002
More precisely, if in one run {a(n)+j, j=0..8} the maximum smallest square factor is p^2, then an infinite subsequence is given by {a(n)+(p#)^2*k, k=0..oo}, where p# = A034386(p). One may get a smaller step taking the least L^2 which has a square factor in common with each of the 9 consecutive terms.  M. F. Hasler, Feb 03 2016


LINKS

Donovan Johnson, Table of n, a(n) for n = 1..1000


FORMULA

A078143 = { A077647[k]  A077647[k+1] = A077647[k]+1 } = { A077640[k]  A077640[k+2] = A077640[k]+2 } = { A078144[k]  A078144[k+4] = A078144[k]+4 } etc. Note that A049535 is defined differently.  M. F. Hasler, Feb 01 2016
a(n) < 4666864390*n. With more work this bound can be decreased significantly.  Charles R Greathouse IV, Nov 05 2017


MATHEMATICA

s9[x_] := Apply[Plus, Table[Abs[MoebiusMu[x+j]], {j, 0, 8}]]; Do[If[Equal[s9[n], 0], Print[n]], {n, 8000000, 1000000000}]


PROG

(PARI) is(n)=for(i=n, n+8, if(!issquarefree(i), return(0))); 1 \\ Charles R Greathouse IV, Nov 05 2017


CROSSREFS

Cf. A013929, A045882 (first of the kchains), A051681.
Cf. A068781 (2chains), A070258 (3chains), A070284 (4chains), A078144 (5chains), A049535 (6chains), A077640 (7chains), A077647 (8chains), A078143 (9chains), A268313 (10chains), A268314 (11chains).
Sequence in context: A346334 A203669 A233476 * A216003 A293374 A141645
Adjacent sequences: A078140 A078141 A078142 * A078144 A078145 A078146


KEYWORD

nonn


AUTHOR

Labos Elemer, Nov 22 2002


EXTENSIONS

a(6)a(21) from Donovan Johnson, Nov 26 2008


STATUS

approved



