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.

Previous Showing 21-30 of 30 results.

A128395 Numbers k such that k^2 divides 15^k-1.

Original entry on oeis.org

1, 2, 4, 7, 8, 14, 16, 28, 56, 112, 136, 272, 452, 812, 904, 952, 1624, 1808, 1904, 3164, 3248, 6328, 11912, 12656, 15368, 18632, 23824, 27608, 30736, 37264, 47908, 55216, 60248, 83384, 91756, 95816, 102604, 107576, 113936, 120496, 130424, 166768
Offset: 1

Views

Author

Alexander Adamchuk, Mar 01 2007

Keywords

Links

Crossrefs

Cf. A127103, A127104, A127105, A127106, A127107, A127102, A127101, A127100, A127092, A128405, A128393, A128395, A128396, A128397, A128398, A128399, A128400, A128401, A128402, A128403, A128404.

Programs

  • Mathematica
    a={1};For[n=1,n<200000,n++,If[PowerMod[15,n,n^2]==1, AppendTo[a, n]]]; a (* Stefan Steinerberger, Jun 10 2007 *)
Join[{1},Select[Range[167000],PowerMod[15,#,#^2]==1&]] (* Harvey P. Dale, Sep 14 2020 *)
  • PARI
    is(k) = Mod(15, k^2)^k == 1; \\ Amiram Eldar, May 21 2024

Extensions

More terms from Stefan Steinerberger, Jun 10 2007

A128402 Numbers k such that k^2 divides 22^k-1.

Original entry on oeis.org

1, 3, 7, 21, 39, 273, 507, 3081, 3549, 21567, 40053, 78117, 280371, 343239, 546819, 1015521, 2056899, 2402673, 5998317, 6171243, 7108647, 8740173, 12338859, 14398293, 18988203, 27115881, 41988219, 43198701, 47727771, 55431363
Offset: 1

Views

Author

Alexander Adamchuk, Mar 01 2007

Keywords

Crossrefs

Cf. A127103, A127104, A127105, A127106, A127107, A127102, A127101, A127100, A127092, A128405, A128393, A128394, A128395, A128396, A128397, A128398, A128399, A128400, A128401, A128403, A128404.

Programs

  • Maple
    select(t -> 22 &^ t - 1 mod t^2 = 0, [seq(2*k+1,k=0..10^6)]); # Robert Israel, Jan 23 2015
  • Mathematica
    a={}; Do[r=(22^n-1)/n^2; If[r==IntegerPart[r], AppendTo[a, n]], {n, 1, 10^3}]; a (* Vladimir Joseph Stephan Orlovsky, Aug 07 2008 *)
  • PARI
    { forstep(m=11,10^8,2, if( Mod(22,m^2)^m==1, print(m) ) ) } \\ Max Alekseyev, Oct 18 2008

Extensions

a(14)-a(30) from Max Alekseyev, Oct 18 2008

A128404 Numbers k such that k^2 divides 24^k-1.

Original entry on oeis.org

1, 23, 1081, 2870377, 7009273, 15954479, 134907719, 329435831, 537539141, 15001987199, 874750261127, 1991103024721, 4272172921319, 4862143429729, 7933540182019, 12816504745411, 41113262272969, 67084347257659
Offset: 1

Views

Author

Alexander Adamchuk, Mar 01 2007

Keywords

Comments

23 divides all terms except the first.

Crossrefs

Cf. A127103, A127104, A127105, A127106, A127107, A127102, A127101, A127100, A127092, A128405, A128393, A128394, A128395, A128396, A128397, A128398, A128399, A128400, A128401, A128402, A128403.

Programs

Extensions

a(5)-a(6) from Farideh Firoozbakht, Mar 05 2007
a(7)-a(10) from Ryan Propper, Feb 23 2008
Terms a(11) onward from Max Alekseyev, May 06 2010

A177243 Numbers k such that k^3 divides 10^(k^2) - 1.

Original entry on oeis.org

1, 3, 9, 111, 333, 1467, 6813, 54279, 84573, 252081, 607947, 665667, 1001001, 1110519, 1823841, 3003003, 3129201, 12914001, 13785399, 22161609, 22957083, 37037037, 41089203, 49235049, 64021761, 92840751, 108503721, 111111111
Offset: 1

Views

Author

Alexander Adamchuk, May 06 2010

Keywords

Crossrefs

Cf. A127100 (k^2 divides 10^k-1), A014950 (k divides 10^k-1).

Programs

  • Mathematica
    Join[{1},Select[Range[1112*10^5],PowerMod[10,#^2,#^3]==1&]] (* Harvey P. Dale, Sep 10 2019 *)

A128456 Quotients A128452(p+1)/p for prime p = A000040(n).

Original entry on oeis.org

2, 7, 311, 127, 23, 157, 7563707819165039903, 75368484119, 47, 9629, 311, 25679, 821, 758771382833029, 12409, 71233, 18438666190697, 2443783, 2939291, 71711, 352883, 181113265579, 167, 105199, 3881, 1314520253, 619, 20759, 117503, 1162660843
Offset: 1

Views

Author

Alexander Adamchuk, Mar 05 2007, Mar 09 2007

Keywords

Comments

a(n) coincides with A128357(n) from n = 2 up to n = 6.

Crossrefs

Cf. A128452, A128357, A128356, A127103, A127104, A127105, A127106, A127107, A127102, A127101, A127100, A127092, A128393, A128394, A128395, A128396, A128397, A128398, A128399, A128400, A128401, A128402, A128403, A128404.

Formula

a(n) = A128452(A000040(n)+1)/A000040(n).
a(n) = A020639(((p+1)^p - 1)/p^2), i.e., the smallest prime factor of ((p+1)^p - 1)/p^2, where p = A000040(n).

Extensions

Terms a(14) onward from Max Alekseyev, May 05 2010

A128452 Least number k > n such that k^2 divides n^k - 1.

Original entry on oeis.org

4, 21, 6, 1555, 8, 889, 10, 111, 12, 253, 14, 2041, 16, 21, 18, 128583032925805678351, 20, 1432001198261, 22, 39, 24, 1081, 26, 55, 28, 171, 30, 279241, 32, 9641, 34, 1191, 36, 55, 38, 950123, 40, 1641, 42, 33661, 44, 32627169461820247, 46, 63, 48, 583223, 50
Offset: 3

Views

Author

Alexander Adamchuk, Mar 05 2007, Mar 09 2007

Keywords

Comments

For prime p, p divides a(p+1). Quotients a(p+1)/p for prime p = A000040(n) are listed in A128456(n) which coincides with A128357(n) for n from 2 to 6.
a(n) divides n^(n-1) - 1.

Crossrefs

Cf. A128456, A128357, A128356, A127103, A127104, A127105, A127106, A127107, A127102, A127101, A127100, A127092, A128393, A128394, A128395, A128396, A128397, A128398, A128399, A128400, A128401, A128402, A128403, A128404.

Formula

a(2n-1) = 2n.

Extensions

More terms from Alexander Adamchuk, Mar 09 2007
Terms a(22) onward from Max Alekseyev, May 05 2010

A333500 A(n,k) is the n-th number m such that m^2 divides k^m - 1 (or 0 if m does not exist); square array A(n,k), n>=1, k>=1, read by antidiagonals.

Original entry on oeis.org

1, 1, 2, 1, 0, 3, 1, 2, 0, 4, 1, 3, 4, 0, 5, 1, 2, 21, 20, 0, 6, 1, 5, 4, 903, 220, 0, 7, 1, 2, 1555, 6, 2667, 1220, 0, 8, 1, 7, 3, 9673655, 12, 7077, 2420, 0, 9, 1, 2, 889, 4, 187159211791705, 42, 113799, 5060, 0, 10, 1, 3, 4, 2359, 6, 776119592182705, 52, 114681, 13420, 0, 11
Offset: 1

Views

Author

Seiichi Manyama, Mar 24 2020

Keywords

Examples

			Square array A(n,k) begins:
  1, 1,    1,    1,  1,               1, ...
  2, 0,    2,    3,  2,               5, ...
  3, 0,    4,   21,  4,            1555, ...
  4, 0,   20,  903,  6,         9673655, ...
  5, 0,  220, 2667, 12, 187159211791705, ...
  6, 0, 1220, 7077, 42, 776119592182705, ...

Crossrefs

Columns k=1-24 give: A000027, A063524, A127103, A127104, A127105, A127106, A127107, A127102, A127101, A127100, A127092, A128405, A128393, A128394, A128395, A128396, A128397, A128398, A128399, A128400, A128401, A128402, A128403, A128404.
Main diagonal gives A333502.
Cf. A333432.

A292335 Numbers k such that k^2 divides 10^k + 1.

Original entry on oeis.org

1, 11, 253, 11891, 35167, 45023, 96569, 640343, 1035529, 1652849, 2221087, 19588019, 30096121, 48669863, 74901409, 88023001, 89007677, 104391089, 143938531, 154261943, 308731093, 395256917, 674608561, 725434237, 920636893, 1737332531, 2620923899, 4137081047, 4183360819, 5621571197, 6765110957, 9090909091, 10411295851, 14510361371
Offset: 1

Views

Author

Max Alekseyev, Sep 15 2017

Keywords

Links

Crossrefs

Subsequence of A015958.
Cf. A127100, A128683.

A333502 a(n) is the n-th number m such that m^2 divides n^m - 1 (or 0 if m does not exist).

Original entry on oeis.org

1, 0, 4, 903, 12, 776119592182705, 12, 42931441, 136, 27486820443, 60, 107342336783, 84
Offset: 1

Views

Author

Seiichi Manyama, Mar 24 2020

Keywords

Crossrefs

Main diagonal of A333500.
Cf. A127106, A127100, A128405, A333433.

Programs

  • PARI
    {a(n) = if(n==2, 0, my(cnt=0, k=0); while(cnt

Formula

a(n) = A333500(n,n).

A177164 a(n) = (n^r - 1)/r^2, where r = (n^(n-1) - 1)/(n-1).

Original entry on oeis.org

1, 5, 9972894583, 449853889404077636694265177903207995382439448590987815041588427345865911961016023550064137351211162870609
Offset: 2

Views

Author

Alexander Adamchuk, May 04 2010

Keywords

Comments

The next term has 1204 digits.
r = (n^(n-1) - 1)/(n-1) = A060072(n) is the (n-1)-digit repunit in base n.
r^2 divides n^r - 1 for all bases n > 1.

Examples

			a(10) = (10^111111111 - 1)/111111111^2.

Crossrefs

Cf. A060072, A127100, A127101, A127102, A127103, A127104, A127105, A127106, A127107, A127092.

Programs

  • Mathematica
    Table[(n^((n^(n - 1) - 1)/(n - 1)) - 1)/((n^(n - 1) - 1)/(n - 1))^2, {n, 2, 6}]

Formula

a(n) = (n^((n^(n-1) - 1)/(n-1)) - 1)/((n^(n-1) - 1)/(n-1))^2.
a(n) = (n^A060072(n) - 1)/A060072(n)^2.
Previous Showing 21-30 of 30 results.

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