A009983 -id:A009983 - cp's OEIS frontend

A218743 a(n) = (40^n - 1)/39.

0, 1, 41, 1641, 65641, 2625641, 105025641, 4201025641, 168041025641, 6721641025641, 268865641025641, 10754625641025641, 430185025641025641, 17207401025641025641, 688296041025641025641, 27531841641025641025641, 1101273665641025641025641, 44050946625641025641025641

Offset: 0

M. F. Hasler, Nov 04 2012

Partial sums of powers of 40 (A009983).

Vincenzo Librandi, Table of n, a(n) for n = 0..600
Index entries related to partial sums.
Index entries for linear recurrences with constant coefficients, signature (41,-40).

Cf. similar sequences of the form (k^n-1)/(k-1): A000225, A003462, A002450, A003463, A003464, A023000, A023001, A002452, A002275, A016123, A016125, A091030, A135519, A135518, A131865, A091045, A218721, A218722, A064108, A218724-A218734, A132469, A218736-A218753, A133853, A094028, A218723.

Cf. A009983.

[n le 2 select n-1 else 41*Self(n-1) - 40*Self(n-2): n in [1..20]]; // Vincenzo Librandi, Nov 07 2012

LinearRecurrence[{41, -40}, {0, 1}, 30] (* Vincenzo Librandi, Nov 07 2012 *)

A218743(n):=floor(40^n/39)$ makelist(A218743(n),n,0,30); /* Martin Ettl, Nov 05 2012 */

PARI
```
a(n)=40^n\39
```

a(n) = floor(40^n/39).
From Vincenzo Librandi, Nov 07 2012: (Start)
G.f.: x/((1-x)*(1-40*x)).
a(n) = 41*a(n-1) - 40*a(n-2). (End)
E.g.f.: exp(x)*(exp(39*x) - 1)/39. - Elmo R. Oliveira, Aug 29 2024

A100401 Digital root of 3^n.

Original entry on oeis.org

1, 3, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9

Offset: 0

Cino Hilliard, Dec 30 2004

This sequence also gives the digital root of 12^n, 21^n, 30^n, 39^n, 48^n, 57^n, ... (any k^n where k is congruent to 3 mod 9). - Timothy L. Tiffin, Dec 02 2023

			For n=14, the digits of 3^14 = 4782969 sum to 45, whose digits sum to 9. So, a(14) = 9.

Index entries for linear recurrences with constant coefficients, signature (1).

Cf. A000244, A001021, A007953, A009965, A009974, A009983, A009992, A010888, A100403, A225374.

Table[PowerMod[3, n, 18], {n, 0, 100}] (* Timothy L. Tiffin, Dec 03 2023 *)
PadRight[{1,3}, 100, 9] (* Timothy L. Tiffin, Dec 03 2023 *)

a(n) = if( n<2, [1,3][n+1], 9); \\ Joerg Arndt, Dec 03 2023

[power_mod(3,n,18) for n in range(105)] # Zerinvary Lajos, Nov 25 2009

a(n) = 3^n mod 18. - Zerinvary Lajos, Nov 25 2009
From Timothy L. Tiffin, Nov 30 2023: (Start)
a(n) = 9 for n >= 2.
G.f.: (1+2x+6x^2)/(1-x).
a(n) = A100403(n) for n <> 1. (End)
a(n) = A010888(A000244(n)). - Michel Marcus, Dec 01 2023
a(n) = A010888(A001021(n)) = A010888(A009965(n)) = A010888(A009974(n)) = A010888(A009983(n)) = A010888(A009992(n)) = A010888(A225374(n)). - Timothy L. Tiffin, Dec 02 2023
E.g.f.: 9*exp(x) - 6*x - 8. - Elmo R. Oliveira, Aug 08 2024
a(n) = A007953(3*a(n-1)) = A010888(3*a(n-1)). - Stefano Spezia, Mar 20 2025

A218742 a(n) = (39^n - 1)/38.

Original entry on oeis.org

0, 1, 40, 1561, 60880, 2374321, 92598520, 3611342281, 140842348960, 5492851609441, 214221212768200, 8354627297959801, 325830464620432240, 12707388120196857361, 495588136687677437080, 19327937330819420046121, 753789555901957381798720, 29397792680176337890150081

Offset: 0

M. F. Hasler, Nov 04 2012

Partial sums of powers of 39 (A009983).

Vincenzo Librandi, Table of n, a(n) for n = 0..600
Index entries related to partial sums.
Index entries for linear recurrences with constant coefficients, signature (40,-39).

Cf. similar sequences of the form (k^n-1)/(k-1): A000225, A003462, A002450, A003463, A003464, A023000, A023001, A002452, A002275, A016123, A016125, A091030, A135519, A135518, A131865, A091045, A218721, A218722, A064108, A218724-A218734, A132469, A218736-A218753, A133853, A094028, A218723.

Cf. A009983.

[n le 2 select n-1 else 40*Self(n-1) - 39*Self(n-2): n in [1..20]]; // Vincenzo Librandi, Nov 07 2012

LinearRecurrence[{40, -39}, {0, 1}, 30] (* Vincenzo Librandi, Nov 07 2012 *)
(39^Range[0,20]-1)/38 (* Harvey P. Dale, Mar 05 2023 *)

A218742(n):=(39^n-1)/38$
makelist(A218742(n),n,0,30); /* Martin Ettl, Nov 07 2012 */

PARI
```
a(n)=39^n\38
```

a(n) = floor(39^n/38).
From Vincenzo Librandi, Nov 07 2012: (Start)
G.f.: x/((1-x)*(1-39*x)).
a(n) = 40*a(n-1) - 39*a(n-2). (End)
E.g.f.: exp(20*x)*sinh(19*x)/19. - Elmo R. Oliveira, Aug 29 2024

A063941 a(n) = 17*39^n.

Original entry on oeis.org

17, 663, 25857, 1008423, 39328497, 1533811383, 59818643937, 2332927113543, 90984157428177, 3548382139698903, 138386903448257217, 5397089234482031463, 210486480144799227057, 8208972725647169855223, 320149936300239624353697, 12485847515709345349794183

Offset: 0

Deborah Florez (deborahf(AT)bcn.net), Sep 01 2001

Mervine Edwards and Siegfried Haenisch, New Views in Algebra, 1: An Integrated Approach, Educational Design, Inc, 1999 (ISBN #0-87694-578-7); p. 9, #16.

Harry J. Smith, Table of n, a(n) for n = 0..100
Tanya Khovanova, Recursive Sequences.
Index entries for linear recurrences with constant coefficients, signature (39).

Cf. A009983.

for n from 0 to 30 do printf(`%d,`, 17*39^n) od;

NestList[39*# &, 17, 20] (* Paolo Xausa, Jul 03 2025 *)

a(n) = { 17*39^n } \\ Harry J. Smith, Sep 03 2009

From Philippe Deléham, Nov 24 2008: (Start)
a(n) = 39*a(n-1), n > 0; a(0)=17.
G.f.: 17/(1-39*x).
a(n) = 17*A009983(n). (End)
E.g.f.: 17*exp(39*x). - Elmo R. Oliveira, Jul 09 2025

Better description from Brian Galebach, Sep 05 2001

More terms from James Sellers, Sep 25 2001

A165860 Totally multiplicative sequence with a(p) = 39.

Original entry on oeis.org

1, 39, 39, 1521, 39, 1521, 39, 59319, 1521, 1521, 39, 59319, 39, 1521, 1521, 2313441, 39, 59319, 39, 59319, 1521, 1521, 39, 2313441, 1521, 1521, 59319, 59319, 39, 59319, 39, 90224199, 1521, 1521, 1521, 2313441, 39, 1521, 1521, 2313441, 39, 59319, 39

Offset: 1

Jaroslav Krizek, Sep 28 2009

G. C. Greubel, Table of n, a(n) for n = 1..10000

39^PrimeOmega[Range[100]] (* G. C. Greubel, Apr 15 2016 *)

a(n) = 39^bigomega(n); \\ Altug Alkan, Apr 15 2016

a(n) = A009983(A001222(n)) = 39^bigomega(n) = 39^A001222(n).

cp's OEIS Frontend

A218743 a(n) = (40^n - 1)/39.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

Maxima

PARI

Formula

A100401 Digital root of 3^n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Sage

Formula

A218742 a(n) = (39^n - 1)/38.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

Maxima

PARI

Formula

A063941 a(n) = 17*39^n.

Views

Author

Keywords

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

Extensions

A165860 Totally multiplicative sequence with a(p) = 39.

Views

Author

Keywords

Links

Programs

Mathematica

PARI

Formula