A050675 Numbers n such that concatenation of numbers from 1 to n is nonsquarefree.

2, 6, 8, 9, 12, 16, 17, 18, 20, 24, 25, 26, 27, 28, 32, 35, 36, 40, 44, 45, 48, 50, 52, 53, 54, 56, 60, 62, 63, 64, 68, 71, 72, 75, 76, 80, 81, 84, 85, 88, 89, 90, 92, 96, 98, 99, 100, 104, 107, 108, 112, 116, 117

Offset

1,1

Comments

All positive multiples of 4 are in this sequence other than 4. Similarly, all numbers which are 0 or 8 mod 9 are in this sequence and all multiples of 25 are in this sequence. - Charles R Greathouse IV, Sep 20 2012

Links

Table of n, a(n) for n=1..53.

Patrick De Geest, Normal Smarandache Concatenated Numbers, Prime factors from 1 up to n

M. Fleuren, Factors and primes of Smarandache sequences.

M. Fleuren, Smarandache Factors and Reverse factors

Carlos Rivera, Puzzle 8. Primes by Listing, The Prime Puzzles & Problems Connection.

Formula

n < a(n) < 2.4n for n > 10. The upper constant can be lowered to 25/11 with more work. - Charles R Greathouse IV, Sep 20 2012

Example

6 is a term because 123456 = 2*2*2*2*2*2*3*643 = 2^6*3*643, not squarefree.

Mathematica

t={}; m=1; Do[m=FromDigits[Flatten[IntegerDigits[{m, n}]]]; If[!SquareFreeQ[m], AppendTo[t, n]], {n, 2, 32}]; t (* Jayanta Basu, May 30 2013 *)
Module[{nn=120, cct}, cct=Table[If[SquareFreeQ[FromDigits[Flatten[IntegerDigits/@Range[ n]]]], 0, 1], {n, nn}]; Position[cct, _?(#==1&)]]//Flatten (* Harvey P. Dale, Sep 20 2023 *)

Prog

(Magma)  a:=[]; c:=1; for n in [2..55] do c:=c*10^#Intseq(n)+n; if not IsSquarefree(c) then Append(~a, n); end if; end for; a; // Marius A. Burtea, Oct 15 2019
(PARI) conc(n) = my(s=""); for(k=1, n, s=Str(s, k)); eval(s); \\ A007908
isok(n) = ! issquarefree(conc(n)); \\ Michel Marcus, Oct 16 2019

Crossrefs

Cf. A013929, A007908, A048342, A050687, A050688.

Sequence in context: A355274 A096507 A288428 * A262981 A336745 A034591

Adjacent sequences: A050672 A050673 A050674 * A050676 A050677 A050678

Keyword

nonn,base,hard,less

Author

Patrick De Geest, Aug 15 1999

Extensions

Corrected by Charles R Greathouse IV, Sep 20 2012

More terms from Sean A. Irvine, Aug 16 2021

Status

approved