A077306 -id:A077306 - cp's OEIS frontend

A075000 Smallest number such that n*a(n) is a concatenation of n consecutive integers; or 0 if no such number exists.

1, 6, 41, 864, 2469, 20576, 493827, 7098637639, 13717421, 1234567891, 82737383012865106529, 10288065758426, 3513762316247164732, 563643651522439401227280, 8230452606740808761

Offset: 1

Amarnath Murthy, Aug 31 2002

Conjecture: For every n there exists a nonzero a(n).

			a(11) = 82737383012865106529 as 11*82737383012865106529 = 910111213141516171819 is the concatenation of 11 numbers from 9 to 19.

Robert G. Wilson v, Table of n, a(n) for n = 1..100

Cf. A074991, A074992, A074993, A074994, A074995, A074996, A036377, A073086, A074999, A075001.

f[ n_ ] := Block[ {id = Range@n}, While[ k = FromDigits@ Flatten@ IntegerDigits@ id/n; !IntegerQ@k, id++ ]; k ]; Array[ f, 16 ] (* Robert G. Wilson v, Oct 19 2007 *)

a(n) = A077306(n)/n. - Amarnath Murthy, Nov 03 2002

More terms from Rick L. Shepherd, Sep 03 2002

A075001 Smallest k such that the concatenation of n consecutive numbers starting with k (from k to n+k-1) is a multiple of n; or 0 if no such number exists.

Original entry on oeis.org

1, 1, 1, 3, 1, 1, 3, 5, 1, 1, 9, 1, 4, 7, 1, 5, 23, 1, 14, 1, 9, 9, 13, 5, 1, 21, 1, 13, 12, 1, 36, 21, 9, 3, 41, 1, 34, 33, 9, 21, 12, 9, 33, 9, 1, 13, 28, 5, 48, 1, 23, 21, 3, 1, 11, 13, 14, 41, 28, 1, 114, 115, 9, 41, 21, 9, 23, 69, 1, 61, 73, 5, 14, 43, 1, 145, 13, 9, 127, 41, 9, 95

Offset: 1

Amarnath Murthy, Aug 31 2002

Conjecture: For every n there exists a k.
First occurrence of k where a(n)=k: 1, 103, 4, 13, 8, 105, 14, 87, 11, 699, 55, 29, 23, 19, 114, 261, 102, 97, 178, 219, 26, 121, 17, 151, 92, ..., . - Robert G. Wilson v
a(n)=1 iff n is in A029455. - Robert G. Wilson v
Increasing a(n)'s: 1, 3, 5, 9, 23, 36, 41, 48, 114, 115, 145, 166, 175, 221, 251, ..., at n = 1, 4, 8, 11, 17, 31, 35, 49, 61, 62, 76, 85, 122, 133, 170, 179, 217, 229, ..., . - Robert G. Wilson v

			a(11) = 9 as 910111213141516171819 the concatenation of 11 numbers from 9 to 19 is divisible by 11 (11*82737383012865106529).

David A. Corneth, Table of n, a(n) for n = 1..10000 (first 2500 terms from Robert G. Wilson v)

Cf. A058183, A074991, A074992, A074993, A074994, A074995, A074996, A036377, A073086, A074999, A075000, A077306, A075000.

Cf. A029455 (indices of 1's).

f[n_] := Block[{c = 1, id = Range@n}, While[k = FromDigits@Flatten@IntegerDigits@id/n; ! IntegerQ@k, id++; c++ ]; c]; Array[f, 82] (* Robert G. Wilson v, Oct 20 2007 *)

/* The following program assumes the conjecture is true. */ /* It has found nonzero a(n) for n up to 500. */ {for(n=1,500, k=0; until(s%n==0,k++; s=0; for(m=k,k+n-1, s=s*(10^length(Str(m)))+m)); print1(k,","))}

a(n) = {my(ld = 1, hd = n, qd, m = Mod(1, n), pow10, qdn = #digits(n), t=log(10*n+.5)\log(10)); qd = n*t+t-10^t\9; pow10 = Mod(10, n)^(qd-1); for(i = 2, n, m = m * Mod(10, n)^#digits(i) + i; ); while(1, if(lift(m) == 0, return(ld)); m -= ld * pow10; hd++; m = m * Mod(10, n)^#digits(hd) + hd; ld++; pow10*=10^(#digits(hd) - #digits(ld)); ) } \\ David A. Corneth, Aug 23 2020

More terms from Rick L. Shepherd, Sep 03 2002

A083468 Smallest multiple of n which is the reverse concatenation of n consecutive numbers.0 if no such number exists.

Original entry on oeis.org

1, 32, 321, 5432, 987645, 765432, 22212019181716, 98765432, 987654321, 19181716151413121110, 2019181716151413121110, 1312111098765432, 19181716151413121110987, 2928272625242322212019181716

Offset: 1

Amarnath Murthy and Meenakshi Srikanth (menakan_s(AT)yahoo.com), May 02 2003

Cf. A077306, A075000, A075001, A083469.

More terms from Sam Alexander, Feb 27 2004

A083469 Smallest k such that n*k is a reverse concatenation of n consecutive natural numbers. a(n) = A083468(n)/n.

Original entry on oeis.org

1, 16, 107, 1358, 197529, 127572, 3173145597388, 12345679, 109739369, 1918171615141312111, 183561974195583011010, 109342591563786, 1475516627031778546999, 209162330374451586572798694

Offset: 1

Amarnath Murthy and Meenakshi Srikanth (menakan_s(AT)yahoo.com), May 02 2003

Cf. A077306, A075000, A075001, A083468.

More terms from Max Alekseyev, Jun 06 2009

cp's OEIS Frontend

A075000 Smallest number such that n*a(n) is a concatenation of n consecutive integers; or 0 if no such number exists.

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A075001 Smallest k such that the concatenation of n consecutive numbers starting with k (from k to n+k-1) is a multiple of n; or 0 if no such number exists.

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A083468 Smallest multiple of n which is the reverse concatenation of n consecutive numbers.0 if no such number exists.

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Crossrefs

Extensions

A083469 Smallest k such that n*k is a reverse concatenation of n consecutive natural numbers. a(n) = A083468(n)/n.

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