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.

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A204848 Algebraic cofactor of n-th repunit A002275(n).

Original entry on oeis.org

1, 1, 1, 11, 1, 1221, 1, 1111, 333, 122221, 1, 11222211, 1, 12222221, 1233321, 11111111, 1, 111222222111, 1, 112222222211, 123333321, 1344444444431, 1, 1111222222221111, 11111, 12222222222221, 333333333, 1122222222222211, 1, 1011121222222221211101, 1, 1111111111111111, 1233333333321
Offset: 1

Views

Author

N. J. A. Sloane, Jan 19 2012

Keywords

Links

Crossrefs

Cf. A002275, A102380, A204845, A204846, A204847.

Programs

  • PARI
    lista(nn) = {vf = []; vfs = []; for (n=1, nn, if (n==1, print1(n, ", "), rn = (10^n-1)/9; f = factor(rn)[, 1]; vkeep = []; for (k = 1, #f~, if (!vecsearch(vfs, f[k]), vkeep = concat(vkeep, f[k])); ); print1(rn/prod(j=1, #vkeep, vkeep[j]), ", "); vf = concat(vf, vkeep); vfs = Set(vf); ); ); } \\ Michel Marcus, May 20 2018

Formula

Equals A002275(n)/(product of terms in n-th row of A204846).

Extensions

More terms from Michel Marcus, May 20 2018

A177928 Let n be the number whose square n^2 has the decimal expansion { d(1) d(2) ... d(D) }, and let q be the corresponding number whose decimal expansion is { d(2) d(3) ... d(D) d(1)}. Sequence lists numbers n dividing q.

Original entry on oeis.org

1, 2, 3, 9, 27, 33, 66, 99, 123, 246, 271, 333, 351, 407, 429, 462, 481, 518, 546, 567, 666, 693, 702, 715, 777, 814, 819, 924, 936, 999, 1434, 2151, 2868, 3333, 4521, 4818, 6666, 7227, 7373, 7535, 8631, 9042, 9999, 33333, 53658, 54546, 66666, 80487, 81819
Offset: 1

Views

Author

Michel Lagneau, May 15 2010

Keywords

Comments

A178028 is a subsequence of this sequence.
When n divides q, n divides d(D)*(10^D - 1) because q = 10*n^2 - d(D)*(10^D - 1). If n is prime, n divides (10^D - 1); for example, the prime term 271 divides 10^5 - 1 = 99999 = 271*369.

Examples

			429 is in the sequence because 429^2 = 184041 and 840411/429 = 1959.

Links

  • Eric Weisstein's World of Mathematics, Repunits

Crossrefs

Cf. A002275, A003020, A005422, A067063, A102380, A178028.

Programs

  • Maple
    for n from 1 to 10^6 do: d:=convert(n^2, base, 10):n1:=nops(d):s:=sum('d[i]*10^i','i'=1..n1-1)+d[n1]:if irem(s,n)=0 then printf(`%d, `,n):else fi:od:
  • Mathematica
    Select[Range[100000], Mod[FromDigits[RotateLeft[IntegerDigits[#^2]]], #] == 0 &] (* T. D. Noe, Jul 27 2012 *)

A365928 Smallest prime factor of f(n) = 10^(2*n) + (10^n - 1)/9.

Original entry on oeis.org

101, 3, 7, 7, 3, 317, 40637, 3, 7, 7, 3, 1487, 101, 3, 7, 7, 3, 39855301, 641, 3, 7, 7, 3, 162340676822011484150719, 101, 3, 7, 7, 3, 121068683, 47, 3, 7, 7, 3, 107, 71, 3, 7, 7, 3, 67, 695841737, 3, 7, 7, 3, 47, 101, 3, 7, 7, 3, 8933, 677, 3, 7, 7, 3, 10305833206337
Offset: 1

Views

Author

Jean-Marc Rebert, Sep 23 2023

Keywords

Examples

			a(1) = 101, because f(1) = 101 is prime.
a(2) = 3, because the smallest prime factor of f(2) = 10011 = 3 * 337 is 3.

Crossrefs

Cf. A020639, A067063, A365966, A102380.

Programs

  • Mathematica
    a[n_]:=Min[First/@FactorInteger[10^(2*n)+(10^n-1)/9]]; Array[a,59] (* Stefano Spezia, Sep 24 2023 *)
  • PARI
    a(n)=my(x=10^(2*n)+(10^n-1)/9);m=factor(x);return(m[1,1])
  • PARI
    a(n) = my(x=10^(2*n)+(10^n-1)/9, k=10); if (ispseudoprime(x), return(x)); while (1, m=factor(x, k); if (m[1,1]Michel Marcus, Sep 24 2023

Formula

a(3k + 2) = 3, a(6k + 3) = 7, a(6k + 4) = 7.

Extensions

a(60) from Jinyuan Wang, Sep 24 2023

A250210 Irregular triangle read by rows in which row n lists the prime factors of the duodecimal repunit ((12^n-1)/11). (Written in base 10).

Original entry on oeis.org

13, 157, 5, 13, 29, 22621, 7, 13, 19, 157, 659, 4943, 5, 13, 29, 89, 233, 37, 157, 80749, 13, 19141, 22621, 11, 23, 266981089, 5, 7, 13, 19, 29, 157, 20593, 477517, 20369233, 13, 211, 659, 4943, 13063, 61, 157, 661, 9781, 22621, 5, 13, 17, 29, 89, 97, 233, 260753
Offset: 2

Views

Author

Eric Chen, Dec 29 2014

Keywords

Examples

			Triangle begins:
[13]
[157]
[5, 13, 29]
[22621]
[7, 13, 19, 157]
[659, 4943]
[5, 13, 29, 89, 233]
[37, 157, 80749]
[13, 19141, 22621]
...

Crossrefs

Cf. A102380, A004064, A250288.

Programs

  • PARI
    tabf(nn) = for (n=1, nn, print(factor((12^n-1)/11)[,1]~);); \\ Michel Marcus, Dec 29 2014
Previous Showing 11-14 of 14 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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