cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-9 of 9 results.

A077180 Smallest concatenation of integers starting from 1 that is a multiple of the n-th prime.

Original entry on oeis.org

12, 12, 12345, 1234567891011
Offset: 1

Views

Author

Amarnath Murthy, Nov 01 2002

Keywords

Comments

The next term (a(5)) has 210 digits. - Harvey P. Dale, Aug 15 2022

Crossrefs

Cf. A007908, A077181, A077182.

Programs

  • Mathematica
    sci[p_]:=Module[{k=2},While[!Divisible[FromDigits[Flatten[IntegerDigits/@ Range[ k]]],p],k++];FromDigits[Flatten[IntegerDigits/@Range[k]]]]; sci/@Prime[ Range[ 5]] (* Harvey P. Dale, Aug 15 2022 *)

Extensions

Next term is too large to include, but may be deduced from A077182. - Sascha Kurz, Jan 30 2003
Corrected by Sascha Kurz, Jan 30 2003

A077181 Smallest number k such that k*prime(n) is a concatenation of successive integers starting from 1.

Original entry on oeis.org

6, 4, 2469, 176366841573
Offset: 1

Views

Author

Amarnath Murthy, Nov 01 2002

Keywords

Crossrefs

Cf. A077180, A077182.

Extensions

Next term is too large to include, but may be deduced from A077182. - Sascha Kurz, Jan 30 2003
Corrected by Sascha Kurz, Jan 30 2003

A077183 Smallest number k such that the reverse concatenation of natural numbers from k to 1 is divisible by prime(n), or 0 if no such number exists.

Original entry on oeis.org

0, 2, 0, 2, 14, 15, 9, 5, 16, 4, 25, 21, 40, 67, 78, 66, 25, 111, 161, 49, 30, 15, 27, 20, 63, 98, 102, 3, 99, 92, 296, 71, 22, 367, 4, 48, 50, 91, 45, 241, 137, 258, 23, 28, 212, 40, 96, 408, 456, 110, 16, 731, 403, 667, 90, 130, 111, 458, 146, 18, 577, 276, 708
Offset: 1

Views

Author

Amarnath Murthy, Nov 01 2002

Keywords

Comments

Conjecture: a(n) > 0 for all n > 3, since prime(1) = 2 and prime(3) = 5 are the only primes whose multiples cannot end in 1. - Ryan Propper, Jul 29 2005

Examples

			a(4) = 2 as 21 is divisible by prime(4) = 7.
The smallest reverse concatenation of natural numbers k..1 that is divisible by prime(5) = 11 is 1413121110987654321, so a(5) = k = 14.

Links

Crossrefs

Cf. A000422, A077180, A077181, A077182, A077185.

Programs

  • Mathematica
    Do[p = Prime[n]; k = 1; s = ToString[k]; While[Mod[ToExpression[s], p] > 0, k++; s = ToString[k] <> s]; Print[k], {n, 4, 50}] (* Ryan Propper, Jul 29 2005 *)

Extensions

Corrected and extended by Ralf Stephan, Mar 18 2003
Example clarified by Harvey P. Dale, Aug 22 2013

A077186 Smallest multiple of prime(n) of the form 123...(k-1) k (k-1)...321 ( a concatenation of natural numbers from 1 to k and back to 1), or 0 if no such number exists.

Original entry on oeis.org

0, 12321, 0, 12345654321, 121, 12345654321
Offset: 1

Views

Author

Amarnath Murthy, Nov 01 2002

Keywords

Comments

a(7) has 435 digits (k=109) and is too large to include, a(8) has 399 digits (k=103), k(a(10))=1006, k(a(11))=17, k(a(12))=3 so a(12)=12321, k(a(13))=5 so a(13)=123454321, k(a(14))=103, k(a(17))=1019, k(a(18))=1002, k(a(19))=1001, k(a(20))=16. - R. J. Mathar, May 06 2006

Crossrefs

Cf. A077180, A077181, A077182, A077183, A077184, A077187, A077188, A173426.

Programs

  • PARI
    palid(n)= { local(resul) ; resul=concat("",n) ; forstep(i=n-1,1,-1, resul=concat(i,resul) ; resul=concat(resul,i) ; ) ; return(eval(resul)) ; }
A077186(n)= { local(p) ; if(n==1 || n==3, return(0) ; ) ; p=prime(n) ; for(i=1,1500, if( palid(i)%p ==0, return(i) ; break ; ) ; ) ; return(-1) ; }
for(n=1,20, print("n=",n," k=",A077186(n)) ; ) ;  \\ R. J. Mathar, May 06 2006

Extensions

More terms from R. J. Mathar, May 06 2006

A077188 a(n) = A077186(n)/prime(n).

Original entry on oeis.org

0, 4107, 0, 1763664903, 11, 949665717
Offset: 1

Views

Author

Amarnath Murthy, Nov 01 2002

Keywords

Comments

Next terms are known but too large to include here.
a(5) = 11 is the only example of the form a(n) = prime(n)

Examples

			a(5) = 11 = 121/11.

Crossrefs

Cf. A077180, A077181, A077182, A077183, A077184, A077186, A077187.

Extensions

Edited by Max Alekseyev, Sep 14 2009

A077189 Smallest concatenation 123...(k-1) k (k-1)...321 (a concatenation of natural numbers from 1 to k and back to 1) that is a multiple of 2n-1, or 0 if no such number exists.

Original entry on oeis.org

1, 12321, 0, 12345654321, 12321, 121, 12345654321, 0
Offset: 1

Views

Author

Amarnath Murthy, Nov 01 2002

Keywords

Comments

a(5k+3) = 0. Are there any other values of n such that a(n) = 0?
The next value, 2n-1 = 17, has k = 109, for a total of 435 digits and is too large to include. - Franklin T. Adams-Watters, Jun 02 2006

Examples

			a(5) = 12321 is a multiple of 9.

Links

Crossrefs

Cf. A077180, A077181, A077182, A077183, A077184, A077186, A077188, A077190.
Cf. A120008.

Extensions

More terms from Franklin T. Adams-Watters, Jun 02 2006

A077187 Smallest k such that the concatenation 123...(k-1) k (k-1)...321 ( a concatenation of natural numbers from 1 to k and back to 1) is a multiple of prime(n), or 0 if no such number exists.

Original entry on oeis.org

0, 3, 0, 6, 2, 6, 109, 103, 100001, 1006, 17, 3, 5, 103, 1000002, 100012, 1019, 1002, 1001, 16, 8, 105, 1036, 104, 1002, 4, 100000000009, 100004, 52, 156, 10000000012, 1062, 8, 1002, 28, 102, 1011, 1000062, 30, 10001, 118, 52, 43, 10058, 34, 47
Offset: 1

Views

Author

Amarnath Murthy, Nov 01 2002

Keywords

Examples

			a(5) = 2 as 121 is a multiple of 11.

Crossrefs

Cf. A077180, A077181, A077182, A077183, A077184, A077186, A077188.

Programs

  • PARI
    { a(n) = local(p,c10,z,u,v,l,k,lk,L,q); p=prime(n); c10=Mod(10,p); z=znorder(c10); u=v=Mod(1,p); l=1; k=2; L=List(); while(1, lk=1+log(k+0.1)\log(10); if(k==10^(lk-1), L=List()); if( u*c10^(l+lk)+k*c10^l+v==0, return(k)); q=0; t=[u,v,k%p,l%z]; for(j=1,#L,if(t==L[j], q=1+#L-j)); if(q, k+=((10^lk-1-k)\q)*q; L=List(), listput(L,t)); u=u*c10^lk+k; v+=k*c10^l; l+=lk; k++) } \\ Max Alekseyev, Sep 11 2009

Extensions

3 more terms from Erich Friedman, Aug 08 2005
Extended by Max Alekseyev, Sep 11 2009

A077190 a(n) = A077189(n)/(2n-1).

Original entry on oeis.org

1, 4107, 0, 1763664903, 1369, 11, 949665717, 0
Offset: 1

Views

Author

Amarnath Murthy, Nov 01 2002

Keywords

Comments

a(5k+3) = 0. Are there any other values of n so that a(n) = 0?

Examples

			a(5) = 12321/9 = 1369.

Links

Crossrefs

Cf. A077180, A077181, A077182, A077183, A077184, A077186, A077188, A077189.

Extensions

a(7)-a(8) from Jinyuan Wang, Mar 04 2020

A077185 a(n) = A000422(A077183(n))/prime(n).

Original entry on oeis.org

0, 7, 0, 3, 128465555544332211, 11647163162383665717, 58097313, 2859, 702235353091782071927, 149, 814268458452231668262948810358418956591, 5730248154490578733002999666333
Offset: 1

Views

Author

Amarnath Murthy, Nov 01 2002

Keywords

Examples

			a(4) = 3 = 21 /7.

Crossrefs

Cf. A000422, A077180, A077181, A077182, A077183.

Programs

  • Maple
    a077183 := [0, 2, 0, 2, 14, 15, 9, 5, 16, 4, 25, 21, 40, 67, 78, 66, 25, 111, 161, 49, 30, 15, 27, 20, 63, 98, 102, 3, 99, 92, 296, 71, 22, 367, 4, 48, 50, 91, 45, 241, 137, 258, 23, 28, 212, 40, 96, 408, 456, 110] : A055642 := proc(n) floor(log[10](n))+1 ; end : A000422 := proc(n) local resul,i; resul := 0 ; for i from n to 1 by -1 do resul := 10^A055642(i)*resul+i ; od ; end: for n from 1 to nops(a077184) do if op(n,a077184) <> 0 then printf("%a, ",A000422(op(n,a077184))/ithprime(n)) ; else printf("%d, ",0) ; fi ; od ; # R. J. Mathar, Apr 01 2007

Extensions

Corrected and extended by R. J. Mathar, Apr 01 2007
Showing 1-9 of 9 results.

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