A001022 -id:A001022 - cp's OEIS frontend

A175527 Digit sum of 13^n.

1, 4, 16, 19, 22, 25, 37, 40, 34, 46, 67, 52, 55, 58, 97, 73, 85, 88, 91, 85, 115, 91, 121, 106, 109, 121, 133, 118, 121, 133, 163, 184, 169, 181, 193, 169, 172, 175, 178, 199, 193, 214, 226, 238, 169, 190, 247, 241, 208, 247, 232, 253, 292, 241, 316, 292, 268, 271, 301, 286, 298, 337, 304, 325

Offset: 0

N. J. A. Sloane, Dec 03 2010

It is surprising that many values repeat twice (for 85, 91, 121, 133, 169 this happens at a(n) = a(n+3) (but 169 occurs later for a third time), for 193, 241, 292, ... the second occurrence comes later) while many other values never occur. Is there a simple explanation? - M. F. Hasler, May 18 2017

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Cf. A001022, A175528, A175525.

Cf. sum of digits of k^n: A001370 (k=2), A004166 (k=3), A065713 (k=4), A066001 (k=5), A066002 (k=6), A066003 (k=7), A066004 (k=8), A065999 (k=9), A066005 (k=11), A066006 (k=12), this sequence (k=13).

Table[Total[IntegerDigits[13^k]], {k,0,1000}]

a(n)=sumdigits(13^n); \\ Michel Marcus, Nov 01 2013

a(n) = A007953(A001022(n)). - Michel Marcus, Nov 01 2013

a(n) ~ 4.5*log_10(13)*n ~ 5.0127*n (conjectured). - M. F. Hasler, May 18 2017

A116629 Positive integers k such that 13^k == 3 (mod k).

Original entry on oeis.org

1, 2, 5, 166, 287603, 9241538, 2366680105, 8347156585, 21682897793, 6988245760865, 9045859950329, 10076294257985, 50299408064905, 254874726648713

Offset: 1

Zak Seidov, Feb 19 2006

No other terms below 10^15. - Max Alekseyev, Nov 24 2017

Some larger terms: 1440926367749746685, 76025040962646716305439353859479569558065. - Max Alekseyev, Jun 29 2011

Cf. A116609, A050259, A122780, A130422, A123061, A277554.

Solutions to 13^n == k (mod n): A001022 (k=0), A015963 (k=-1), A116621 (k=1), A116622 (k=2), this sequence (k=3), A116630 (k=4), A116611 (k=5), A116631 (k=6), A116632 (k=7), A295532 (k=8), A116636 (k=9), A116620 (k=10), A116638 (k=11), A116639 (k=15).

Join[{1, 2}, Select[Range[1, 5000], Mod[13^#, #] == 3 &]] (* G. C. Greubel, Nov 19 2017 *)
Join[{1, 2}, Select[Range[10000000], PowerMod[13, #, #] == 3 &]] (* Robert Price, Apr 10 2020 *)

isok(n) = Mod(13, n)^n == 3; \\ Michel Marcus, Nov 19 2017

Two more terms from Ryan Propper, Jan 09 2008

Terms 1,2 are prepended and a(9)-a(14) are added by Max Alekseyev, Jun 29 2011; Nov 24 2017

A116611 Positive integers n such that 13^n == 5 (mod n).

Original entry on oeis.org

1, 2, 4, 44, 82, 236, 25433, 177764, 219244, 86150213, 107218402, 1260236441, 12856300141, 447650116364, 657175627369, 14543842704596, 125035120614917

Offset: 1

Zak Seidov, Feb 19 2006

nonn

No other terms below 10^15.

Some larger terms: 99790373907467602, 846248577183963835642742, 273781047810302314432122404459324, 4174626353309446327489382394518975030641698849116, 211*(13^211-5)/12607932861823674049268705845744 (207 digits). - Max Alekseyev, Jun 29 2011

			44 is in this sequence because 13^44 = 10315908977942302627204470186314316211062255002161 = 234452476771415968800101595143507186615051250049*44 + 5 == 5 (mod 44).

Solutions to 13^n == k (mod n): A001022 (k=0), A015963 (k=-1), A116621 (k=1), A116622 (k=2), A116629 (k=3), A116630 (k=4), this sequence (k=5), A116631 (k=6), A116632 (k=7), A295532 (k=8), A116636 (k=9), A116620(k=10), A116638 (k=11), A116639 (k=15).

Cf. A116609, A128121, A276740, A123091.

Join[{1, 2}, Select[Range[1000000], PowerMod[13, #, #] == 5 &]] (* Robert Price, Apr 10 2020 *)

is(n) = Mod(13,n)^n==5; \\ Charles R Greathouse IV, Jun 08 2015

More terms from Ryan Propper, Apr 01 2006

Terms 1,2,4 are prepended and a(13)-a(17) are added by Max Alekseyev, Jun 29 2011, Nov 27 2017

A116620 Positive integers n such that 13^n == 10 (mod n).

Original entry on oeis.org

1, 3, 9, 74853, 1275039, 27181907, 31261887, 989255061, 4813809711, 3187842157567, 313768710194691

Offset: 1

Zak Seidov, Feb 19 2006

No other terms below 10^15. - Max Alekseyev, Nov 06 2018

9909932321420413420533943 is a term. - Max Alekseyev, Jun 29 2011

Solutions to 13^n == k (mod n): A001022 (k=0), A015963 (k=-1), A116621 (k=1), A116622 (k=2), A116629 (k=3), A116630 (k=4), A116611 (k=5), A116631 (k=6), A116632 (k=7), A295532 (k=8), A116636 (k=9), this sequence (k=10), A116638 (k=11), A116639 (k=15).

Cf. A116609.

Join[{1, 3, 9}, Select[Range[2000000], PowerMod[13, #, #] == 10 &]] (* Robert Price, Apr 10 2020 *)

More terms from Ryan Propper, Jun 12 2006

Terms 1,3,9 prepended and a(10)-a(11) added by Max Alekseyev, Jun 29 2011, Nov 06 2018

A116630 Positive integers n such that 13^n == 4 (mod n).

Original entry on oeis.org

1, 3, 51, 129, 125869, 158287, 1723647, 1839003, 90808797, 3661886147, 7368982721, 130424652229, 1616928424359, 4003183891851, 66657658685869

Offset: 1

Zak Seidov, Feb 19 2006

No other terms below 10^15. - Max Alekseyev, Nov 26 2017

Some larger terms: 84058689739550643018360088224267, 11083544368708558891212925543084197628431243723. - Max Alekseyev, Jun 26 2011

Solutions to 13^n == k (mod n): A001022 (k=0), A015963(k=-1), A116621 (k=1), A116622 (k=2), A116629(k=3), this sequence (k=4), A116611 (k=5), A116631 (k=6), A116632 (k=7), A295532 (k=8), A116636 (k=9), A116620(k=10), A116638 (k=11), A116639 (k=15)

Cf. A116609, A015921, A125949.

Join[{1, 3}, Select[Range[1, 5000], Mod[13^#, #] == 4 &]] (* G. C. Greubel, Nov 19 2017 *)
Join[{1, 3}, Select[Range[2000000], PowerMod[13, #, #] == 4 &]] (* Robert Price, Apr 10 2020 *)

isok(n) = Mod(13, n)^n == 4; \\ Michel Marcus, Nov 19 2017

More terms from Ryan Propper, Jan 09 2008

Terms 1,3 prepended and a(12)-a(15) added by Max Alekseyev, Jun 26 2011, Nov 26 2017

A116631 Positive integers n such that 13^n == 6 (mod n).

Original entry on oeis.org

1, 7, 8743, 50239, 312389, 8789977, 87453889, 96301009, 3963715129, 5062673539, 6854133309107, 16987071590111, 72278468169733, 411419589731633, 590475819370933

Offset: 1

Zak Seidov, Feb 19 2006

nonn

No other terms below 10^15. A large term: 2455610470454186971078168169. - Max Alekseyev, Dec 19 2017

Solutions to 13^n == k (mod n): A001022 (k=0), A015963 (k=-1), A116621 (k=1), A116622 (k=2), A116629 (k=3), A116630 (k=4), A116611 (k=5), this sequence (k=6), A116632 (k=7), A295532 (k=8), A116636 (k=9), A116620(k=10), A116638 (k=11), A116639 (k=15).

Cf. A116609, A128122, A277628, A277350.

Join[{1}, Select[Range[1, 9000], Mod[13^#, #] == 6 &]] (* G. C. Greubel, Nov 19 2017 *)
Join[{1}, Select[Range[10000000], PowerMod[13, #, #] == 6 &]] (* Robert Price, Apr 10 2020 *)

isok(n) = Mod(13, n)^n == 6; \\ Michel Marcus, Nov 19 2017

More terms from Ryan Propper, Mar 30 2007

Term 1 is prepended and a(11)-a(15) added by Max Alekseyev, Jun 29 2011, Dec 19 2017

A116636 Positive integers k such that 13^k == 9 (mod k).

Original entry on oeis.org

1, 2, 4, 8, 10, 172, 296, 332, 410, 872, 1048, 1070, 1544, 2830, 3470, 7486, 9196, 22184, 90892, 121174, 299816, 575206, 885112, 1329388, 1386430, 2518994, 4167272, 5600212, 8475016, 9180370, 12348446, 18483076, 19185890, 20806274, 28984094, 37114141

Offset: 1

Zak Seidov, Feb 19 2006

nonn

For k > 9 in this sequence, A116609(k) = 9. - Iain Fox, Nov 20 2017

Max Alekseyev, Table of n, a(n) for n = 1..112

Solutions to 13^n == k (mod n): A001022 (k=0), A015963 (k=-1), A116621 (k=1), A116622 (k=2), A116629 (k=3), A116630 (k=4), A116611 (k=5), A116631 (k=6), A116632 (k=7), A295532 (k=8), this sequence (k=9), A116620 (k=10), A116638 (k=11), A116639 (k=15).

Cf. A116609.

Join[{1, 2, 4, 8}, Select[Range[1, 9000], Mod[13^#, #] == 9 &]] (* G. C. Greubel, Nov 19 2017 *)
Join[{1,2,4,8},Select[Range[38*10^6],PowerMod[13,#,#]==9&]] (* Harvey P. Dale, Jul 06 2025 *)

isok(n) = Mod(13, n)^n == 9; \\ Michel Marcus, Nov 19 2017

More terms from Ryan Propper, Nov 05 2006

Terms 1,2,4,8 prepended by Max Alekseyev, Jun 28 2011

A153651 Triangle T(n, k) = T(n-1, k) + T(n-1, k-1) + (j+5)prime(j)T(n-2, k-1) with j=6, read by rows.

Original entry on oeis.org

2, 13, 13, 2, 334, 2, 2, 2195, 2195, 2, 2, 2483, 52152, 2483, 2, 2, 2771, 368520, 368520, 2771, 2, 2, 3059, 726360, 8194776, 726360, 3059, 2, 2, 3347, 1125672, 61619496, 61619496, 1125672, 3347, 2, 2, 3635, 1566456, 166614648, 1295091960, 166614648, 1566456, 3635, 2

Offset: 1

Roger L. Bagula, Dec 30 2008

			Triangle begins as:
   2;
  13,   13;
   2,  334,       2;
   2, 2195,    2195,         2;
   2, 2483,   52152,      2483,          2;
   2, 2771,  368520,    368520,       2771,         2;
   2, 3059,  726360,   8194776,     726360,      3059,       2;
   2, 3347, 1125672,  61619496,   61619496,   1125672,    3347,    2;
   2, 3635, 1566456, 166614648, 1295091960, 166614648, 1566456, 3635, 2;

G. C. Greubel, Rows n = 1..50 of the triangle, flattened

Sequences with variable (p,q,j): A153516 (0,1,2), A153518 (0,1,3), A153520 (0,1,4), A153521 (0,1,5), A153648 (1,0,3), A153649 (1,1,4), A153650 (1,4,5), this sequence (1,5,6), A153652 (2,1,7), A153653 (2,1,8), A153654 (2,1,9), A153655 (2,1,10), A153656 (2,3,9), A153657 (2,7,10).

Cf. A001022 (powers of 13).

f:= func< n,j | Round(((3-(-1)^n)/2)*NthPrime(j)^(n-1) - 2^((3-(-1)^n)/2)) >;
function T(n,k,p,q,j)
  if n eq 2 then return NthPrime(j);
  elif (n eq 3 and k eq 2 or n eq 4 and k eq 2 or n eq 4 and k eq 3) then return f(n,j);
  elif (k eq 1 or k eq n) then return 2;
  else return T(n-1,k,p,q,j) + T(n-1,k-1,p,q,j) + (p*j+q)*NthPrime(j)*T(n-2,k-1,p,q,j);
  end if; return T;
end function;
[T(n,k,1,5,6): k in [1..n], n in [1..12]]; // G. C. Greubel, Mar 06 2021

T[n_, k_, p_, q_, j_]:= T[n,k,p,q,j]= If[n==2, Prime[j], If[n==3 && k==2 || n==4 && 2<=k<=3, ((3-(-1)^n)/2)*Prime[j]^(n-1) -2^((3-(-1)^n)/2), If[k==1 || k==n, 2, T[n-1,k,p,q,j] + T[n-1,k-1,p,q,j] + (p*j+q)*Prime[j]*T[n-2,k-1,p,q,j] ]]];
Table[T[n,k,1,5,6], {n,12}, {k,n}]//Flatten (* modified by G. C. Greubel, Mar 06 2021 *)

@CachedFunction
def f(n,j): return ((3-(-1)^n)/2)*nth_prime(j)^(n-1) - 2^((3-(-1)^n)/2)
def T(n,k,p,q,j):
    if (n==2): return nth_prime(j)
    elif (n==3 and k==2 or n==4 and 2<=k<=3): return f(n,j)
    elif (k==1 or k==n): return 2
    else: return T(n-1,k,p,q,j) + T(n-1,k-1,p,q,j) + (p*j+q)*nth_prime(j)*T(n-2,k-1,p,q,j)
flatten([[T(n,k,1,5,6) for k in (1..n)] for n in (1..12)]) # G. C. Greubel, Mar 06 2021

T(n, k) = T(n-1, k) + T(n-1, k-1) + (j+5)*prime(j)*T(n-2, k-1) with j=6.
From G. C. Greubel, Mar 06 2021: (Start)
T(n,k,p,q,j) = T(n-1,k,p,q,j) + T(n-1,k-1,p,q,j) + (p*j+q)*prime(j)*T(n-2,k-1,p,q,j) with T(2,k,p,q,j) = prime(j), T(3,2,p,q,j) = 2*prime(j)^2 -4, T(4,2,p,q,j) = T(4,3,p,q,j) = prime(j)^2 -2, T(n,1,p,q,j) = T(n,n,p,q,j) = 2 and (p,q,j) = (1,5,6).
Sum_{k=0..n} T(n,k,p,q,j) = 2*prime(j)^(n-1) for j=6, = 2*A001022(n-1). (End)

Edited by G. C. Greubel, Mar 06 2021

A116638 Positive integers n such that 13^n == 11 (mod n).

Original entry on oeis.org

1, 2, 158, 301823, 1851103, 8616098, 60528467, 1087582634, 1628818307, 16205558969

Offset: 1

Zak Seidov, Feb 19 2006

No other terms below 10^15. - Max Alekseyev, Nov 07 2018

Large terms: 38763675625170712166, 527929122195463909516681113715859203.

Solutions to 13^n == k (mod n): A001022 (k=0), A015963 (k=-1), A116621 (k=1), A116622 (k=2), A116629 (k=3), A116630 (k=4), A116611 (k=5), A116631 (k=6), A116632 (k=7), A295532 (k=8), A116636 (k=9), A116620 (k=10), this sequence (k=11), A116639 (k=15).

Cf. A116609.

Join[{1, 2}, Select[Range[1, 9000], Mod[13^#, #] == 11 &]] (* G. C. Greubel, Nov 20 2017 *)
Join[{1, 2}, Select[Range[10000000], PowerMod[13, #, #] == 11 &]] (* Robert Price, Apr 10 2020 *)

More terms from Ryan Propper, Jan 11 2008

Edited by Max Alekseyev, May 04 2010

A008573 Digits of powers of 13.

Original entry on oeis.org

1, 1, 3, 1, 6, 9, 2, 1, 9, 7, 2, 8, 5, 6, 1, 3, 7, 1, 2, 9, 3, 4, 8, 2, 6, 8, 0, 9, 6, 2, 7, 4, 8, 5, 1, 7, 8, 1, 5, 7, 3, 0, 7, 2, 1, 1, 0, 6, 0, 4, 4, 9, 9, 3, 7, 3, 1, 3, 7, 8, 5, 8, 4, 9, 1, 8, 4, 9, 1, 7, 9, 2, 1, 6, 0, 3, 9, 4, 0, 3, 7, 2, 3, 2, 9, 8, 0, 8, 5, 1, 2, 2, 4, 8, 1, 3, 0, 2, 8

Offset: 0

N. J. A. Sloane

The constant whose decimal expansion is this sequence is irrational (Mahler, 1981). - Amiram Eldar, Mar 23 2025

			Triangle begins:
  1;
  1, 3;
  1, 6, 9;
  2, 1, 9, 7;
  2, 8, 5, 6, 1;
  3, 7, 1, 2, 9, 3;
  4, 8, 2, 6, 8, 0, 9;
  ...

Seiichi Manyama, Rows n = 0..100, flattened
Kurt Mahler, On some irrational decimal fractions, Journal of Number Theory, Vol. 13, No. 2 (1981), pp. 268-269.

Cf. A001022.

Cf. A000455, A008564, A008565, A008566, A008567, A008568, A008569, A008570, A008571, A008572, A010054.

IntegerDigits/@(13^Range[0,20])//Flatten (* Harvey P. Dale, Oct 24 2017 *)

cp's OEIS Frontend

A175527 Digit sum of 13^n.

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A116629 Positive integers k such that 13^k == 3 (mod k).

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A116611 Positive integers n such that 13^n == 5 (mod n).

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A116620 Positive integers n such that 13^n == 10 (mod n).

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A116630 Positive integers n such that 13^n == 4 (mod n).

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A116631 Positive integers n such that 13^n == 6 (mod n).

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A116636 Positive integers k such that 13^k == 9 (mod k).

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A153651 Triangle T(n, k) = T(n-1, k) + T(n-1, k-1) + (j+5)*prime(j)*T(n-2, k-1) with j=6, read by rows.

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A153651 Triangle T(n, k) = T(n-1, k) + T(n-1, k-1) + (j+5)prime(j)T(n-2, k-1) with j=6, read by rows.