A016123 -id:A016123 - cp's OEIS frontend

A218749 a(n) = (46^n - 1)/45.

0, 1, 47, 2163, 99499, 4576955, 210539931, 9684836827, 445502494043, 20493114725979, 942683277395035, 43363430760171611, 1994717814967894107, 91757019488523128923, 4220822896472063930459, 194157853237714940801115, 8931261248934887276851291, 410838017451004814735159387

Offset: 0

M. F. Hasler, Nov 04 2012

Partial sums of powers of 46 (A009990).

Vincenzo Librandi, Table of n, a(n) for n = 0..600
Index entries related to partial sums.
Index entries related to q-numbers.
Index entries for linear recurrences with constant coefficients, signature (47,-46).

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. A009990.

[n le 2 select n-1 else 47*Self(n-1) - 46*Self(n-2): n in [1..20]]; // Vincenzo Librandi, Nov 08 2012

LinearRecurrence[{47, -46}, {0, 1}, 30] (* Vincenzo Librandi, Nov 08 2012 *)
(46^Range[0,20]-1)/45 (* Harvey P. Dale, Aug 17 2017 *)

A218749(n):=(46^n-1)/45$ makelist(A218749(n),n,0,30); /* Martin Ettl, Nov 07 2012 */

PARI
```
A218749(n)=46^n\45
```

From Vincenzo Librandi, Nov 08 2012: (Start)
G.f.: x/((1-x)*(1-46*x)).
a(n) = 47*a(n-1) - 46*a(n-2) with a(0)=0, a(1)=1.
a(n) = 46*a(n-1) + 1 with a(0)=0.
a(n) = floor(46^n/45). (End)
E.g.f.: exp(x)*(exp(45*x) - 1)/45. - Elmo R. Oliveira, Aug 29 2024

A218751 a(n) = (48^n - 1)/47.

Original entry on oeis.org

0, 1, 49, 2353, 112945, 5421361, 260225329, 12490815793, 599559158065, 28778839587121, 1381384300181809, 66306446408726833, 3182709427618887985, 152770052525706623281, 7332962521233917917489, 351982201019228060039473, 16895145648922946881894705, 810966991148301450330945841

Offset: 0

M. F. Hasler, Nov 04 2012

Partial sums of powers of 48 (A009992).

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 (49,-48).

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. A009992.

[n le 2 select n-1 else 49*Self(n-1)-48*Self(n-2): n in [1..20]]; // Vincenzo Librandi, Nov 08 2012

LinearRecurrence[{49, -48}, {0, 1}, 30] (* Vincenzo Librandi, Nov 08 2012 *)

A218751(n):=floor((48^n-1)/47)$ makelist(A218751(n),n,0,30); /* Martin Ettl, Nov 05 2012 */

PARI
```
A218751(n)=48^n\47
```

a(n) = floor(48^n/47).
From Vincenzo Librandi, Nov 08 2012: (Start)
G.f.: x/((1-x)*(1-48*x)).
a(n) = 49*a(n-1) - 48*a(n-2) with a(0)=0, a(1)=1.
a(n) = 48*a(n-1) + 1 with a(0)=0. (End)
E.g.f.: exp(x)*(exp(47*x) - 1)/47. - Elmo R. Oliveira, Aug 29 2024

A086930 Smallest b>1 such that in base b representation the n-th prime is a repunit.

Original entry on oeis.org

2, 4, 2, 10, 3, 16, 18, 22, 28, 2, 36, 40, 6, 46, 52, 58, 60, 66, 70, 8, 78, 82, 88, 96, 100, 102, 106, 108, 112, 2, 130, 136, 138, 148, 150, 12, 162, 166, 172, 178, 180, 190, 192, 196, 198, 14, 222, 226, 228, 232, 238, 15, 250, 256, 262, 268, 270, 276, 280, 282

Offset: 2

Reinhard Zumkeller, Sep 21 2003

From Robert G. Wilson v, Mar 26 2014: (Start)
Obviously the first prime number, 2, can never become a repunit since it is even; therefore this sequence has the offset of 2.
Most terms, a(n), are one less than the n-th prime; e.g., for a(8) the eighth prime is 19_10 = 11_18. Therefore a(n) <= Pi(n)-1.
However there are some terms for which a(n) occurs before Pi(n)-1; e.g., for a(14) the fourteenth prime is 43_10 = 111_6.
Those indices, i, are: 4, 6, 11, 14, 21, 31, 37, 47, 53, 63, 82, 90, ..., . Prime(i) = A085104.
In those cases a(n) is a proper divisor of Prime(n)-1.
(End)

			n=6: A000040(6) = 13 = 1*3^2 + 1*3^1 + 1*3^0: ternary(13)='111' and binary(13)='1101', therefore a(6)=3.

Robert G. Wilson v, Table of n, a(n) for n = 2..1001
Eric Weisstein's World of Mathematics, Repunit

Cf. A000225, A003462, A002450, A003463, A003464, A023000, A023001, A002452, A002275, A004022, A016123, A016125, A085104.

f[n_] := Block[{i = 1, d, p = Prime@ n}, d = Rest@ Divisors[p - 1]; While[id = IntegerDigits[p, d[[i]]]; id != Reverse@ id || Union@ id != {1}, i++]; d[[i]]]; Array[f, 60, 2]

A105280 a(0)=0; a(n) = 11*a(n-1) + 11.

Original entry on oeis.org

0, 11, 132, 1463, 16104, 177155, 1948716, 21435887, 235794768, 2593742459, 28531167060, 313842837671, 3452271214392, 37974983358323, 417724816941564, 4594972986357215, 50544702849929376, 555991731349223147, 6115909044841454628, 67274999493256000919, 740024994425816010120

Offset: 0

Alexandre Wajnberg, Apr 25 2005

Harvey P. Dale, Table of n, a(n) for n = 0..960
Index entries for linear recurrences with constant coefficients, signature (12,-11).

Cf. A000918, A016123, A029858, A052379, A052386, A080674.

Row n=11 of A228275.

a:=n-> add(11^j,j=1..n): seq(a(n),n=0..12); # Zerinvary Lajos, Oct 03 2007

NestList[11#+11&,0,20] (* or *) LinearRecurrence[{12,-11},{0,11},20] (* Harvey P. Dale, Dec 02 2023 *)

a(n) = 11^n + a(n-1) (with a(0)=0). - Vincenzo Librandi, Nov 13 2010
From Elmo R. Oliveira, May 24 2025: (Start)
G.f.: 11*x/((x-1)*(11*x-1)).
E.g.f.: 11*exp(x)*(exp(10*x) - 1)/10.
a(n) = 11*(11^n - 1)/10.
a(n) = 12*a(n-1) - 11*a(n-2).
a(n) = A016123(n) - 1. (End)

Corrected by T. D. Noe, Nov 07 2006

A319074 a(n) is the sum of the first n nonnegative powers of the n-th prime.

Original entry on oeis.org

1, 4, 31, 400, 16105, 402234, 25646167, 943531280, 81870575521, 15025258332150, 846949229880161, 182859777940000980, 23127577557875340733, 1759175174860440565844, 262246703278703657363377, 74543635579202247026882160, 21930887362370823132822661921, 2279217547342466764922495586798

Offset: 1

Omar E. Pol, Sep 11 2018

nonn

			For n = 4 the 4th prime is 7 and the sum of the first four nonnegative powers of 7 is 7^0 + 7^1 + 7^2 + 7^3 = 1 + 7 + 49 + 343 = 400, so a(4) = 400.

Main diagonal of A319076.
Cf. A000040, A000203, A006093, A062457, A069459, A093360, A319075.
Cf. A000079, A000244, A000351, A000420, A001020, A001022, A001026, A001029, A009967, A009973, A009975, A009981, A009985, A009987, A009991.
Cf. A126646, A003462, A003463, A023000, A016123, A091030, A091045, A218722, A218726, A218732, A218734, A218740, A218744, A218746, A218750.

a(n) = sum(k=0, n-1, prime(n)^k); \\ Michel Marcus, Sep 13 2018

a(n) = Sum_{k=0..n-1} A000040(n)^k.
a(n) = Sum_{k=0..n-1} A319075(k,n).
a(n) = (A000040(n)^n - 1)/(A000040(n) - 1).
a(n) = (A062457(n) - 1)/A006093(n).
a(n) = A069459(n)/A006093(n).
a(n) = A000203(A000040(n)^(n-1)).
a(n) = A000203(A093360(n)).

A319076 Square array T(n,k) read by antidiagonal upwards in which column k lists the partial sums of the powers of the k-th prime, n >= 0, k >= 1.

Original entry on oeis.org

1, 3, 1, 7, 4, 1, 15, 13, 6, 1, 31, 40, 31, 8, 1, 63, 121, 156, 57, 12, 1, 127, 364, 781, 400, 133, 14, 1, 255, 1093, 3906, 2801, 1464, 183, 18, 1, 511, 3280, 19531, 19608, 16105, 2380, 307, 20, 1, 1023, 9841, 97656, 137257, 177156, 30941, 5220, 381, 24, 1, 2047, 29524, 488281, 960800, 1948717

Offset: 0

Omar E. Pol, Sep 09 2018

T(n,k) is also the sum of the divisors of the n-th nonnegative power of the k-th prime, n >= 0, k >= 1.

			The corner of the square array is as follows:
         A126646 A003462 A003463  A023000    A016123    A091030     A091045
A000012        1,      1,      1,       1,         1,         1,          1, ...
A008864        3,      4,      6,       8,        12,        14,         18, ...
A060800        7,     13,     31,      57,       133,       183,        307, ...
A131991       15,     40,    156,     400,      1464,      2380,       5220, ...
A131992       31,    121,    781,    2801,     16105,     30941,      88741, ...
A131993       63,    364,   3906,   19608,    177156,    402234,    1508598, ...
.......      127,   1093,  19531,  137257,   1948717,   5229043,   25646167, ...
.......      255,   3280,  97656,  960800,  21435888,  67977560,  435984840, ...
.......      511,   9841, 488281, 6725601, 235794769, 883708281, 7411742281, ...
...

Rows 0-5: A000012, A008864, A060800, A131991, A131992, A131993.
Columns 1-15: A126646, A003462, A003463, A023000, A016123, A091030, A091045, A218722, A218726, A218732, A218734, A218740, A218744, A218746, A218750.
Main diagonal gives A319074.
Cf. A000079, A000244, A000351, A000420, A001020, A001022, A001026, A001029, A009967, A009973, A009975, A009981, A009985, A009987, A009991.
Cf. A000040, A000203, A006093, A319075.

T(n, k) = sigma(prime(k)^n); \\ Michel Marcus, Sep 13 2018

T(n,k) = A000203(A000040(k)^n).
T(n,k) = Sum_{j=0..n} A000040(k)^j.
T(n,k) = Sum_{j=0..n} A319075(j,k).
T(n,k) = (A000040(k)^(n+1) - 1)/(A000040(k) - 1).
T(n,k) = (A000040(k)^(n+1) - 1)/A006093(k).

A361475 Array read by ascending antidiagonals: A(n, k) = (k^n - 1)/(k - 1), with k >= 2.

Original entry on oeis.org

0, 1, 0, 3, 1, 0, 7, 4, 1, 0, 15, 13, 5, 1, 0, 31, 40, 21, 6, 1, 0, 63, 121, 85, 31, 7, 1, 0, 127, 364, 341, 156, 43, 8, 1, 0, 255, 1093, 1365, 781, 259, 57, 9, 1, 0, 511, 3280, 5461, 3906, 1555, 400, 73, 10, 1, 0, 1023, 9841, 21845, 19531, 9331, 2801, 585, 91, 11, 1, 0

Offset: 0

Stefano Spezia, Mar 13 2023

			The array begins:
   0,  0,  0,   0,   0, ...
   1,  1,  1,   1,   1, ...
   3,  4,  5,   6,   7, ...
   7, 13, 21,  31,  43, ...
  15, 40, 85, 156, 259, ...
  ...

Cf. A003992, A361291 (k=2*n+1), A361476 (antidiagonal sums).

Cf. A000225 (k=2), A003462 (k=3), A002450 (k=4), A003463 (k=5), A003464 (k=6), A023000 (k=7), A023001 (k=8), A002452 (k=9), A002275 (k=10), A016123 (k=11).

A[n_,k_]:=(k^n-1)/(k-1); Flatten[Table[A[n-k+2,k],{n,0,10},{k,2,n+2}]]

E.g.f. of column k: exp(x)*(exp((k-1)*x) - 1)/(k - 1).
E.g.f. of column k: 2*exp((k+1)*x/2)*sinh((k-1)*x/2)/(k - 1).
A(n, k) = Sum_{i=0..n-1} k^i.

A268839 a(n) = Sum_{j=1..10^n-1} 2^f(j) where f(j) is the number of zero digits in the decimal representation of j.

Original entry on oeis.org

9, 108, 1197, 13176, 144945, 1594404, 17538453, 192922992, 2122152921, 23343682140, 256780503549, 2824585539048, 31070440929537, 341774850224916, 3759523352474085, 41354756877214944, 454902325649364393, 5003925582143008332, 55043181403573091661

Offset: 1

Michel Lagneau, Feb 14 2016

We calculate the number of integers between 1 and 10^n - 1 having k zeros in their decimal representation. To form a such number consisting of m digits (k < m), place k zeros in m-1 possible positions, then we must choose m-k digits different from zero. Thus, the number of integers between 1 and 10^n - 1 having k zeros in their decimal representation is: Sum_{m=k+1..n} binomial(m-1, k)*9^(m-k).

Hence the sum: Sum_{m=1..n} Sum_{k=0..m-1} binomial(m-1,k)*9^(m-k)*2^k = Sum_{m=1..n} 9^m*(11/9)*(m-1) = (9/10)*(11^n - 1).

			a(1) = 9 because 2^f(1) + 2^f(2) + ... + 2^f(9) = 2^0 + 2^0 + ... + 2^0 = 9;
a(2) = 108 because 2^f(1) + 2^f(2) + ... + 2^f(99) = 9*10 + 2*9 = 108, where f(10) = f(20) = ... = f(90) = 1 and f(i) = 0 otherwise.

Colin Barker, Table of n, a(n) for n = 1..950
Index entries for linear recurrences with constant coefficients, signature (12,-11).

Cf. A016123, A033713, A033714, A055641, A119291.

[(9/10)*(11^n-1): n in [1..20]]; // Vincenzo Librandi, Feb 15 2016

for n from 1 to 100 do: x:=(9/10)*(11^n-1):printf(‘%d, ‘,x):od:

Table[Table[(9/10) (11^n - 1), {n, 1, 20}]] (* Bruno Berselli, Feb 15 2016 *)
CoefficientList[Series[9/((1 - 11 x) (1 - x)), {x, 0, 33}], x] (* Vincenzo Librandi, Feb 15 2016 *)

Vec(9*x/((1-11*x)*(1-x)) + O(x^30)) \\ Colin Barker, Feb 22 2016

a(n) = (9/10)*(11^n-1) = 9*A016123(n-1).
From Vincenzo Librandi, Feb 15 2016: (Start)
G.f.: (9*x)/((1-11*x)*(1-x)).
a(n) = 11*a(n-1) + 9. (End)
E.g.f.: 9*exp(x)*(exp(10*x) - 1)/10. - Stefano Spezia, Sep 13 2023

Name edited by Jon E. Schoenfield, Sep 13 2017

A016198 Expansion of g.f. 1/((1-x)(1-2x)(1-5x)).

Original entry on oeis.org

1, 8, 47, 250, 1281, 6468, 32467, 162590, 813461, 4068328, 20343687, 101722530, 508620841, 2543120588, 12715635707, 63578244070, 317891351421, 1589457019248, 7947285620527, 39736429151210, 198682147853201, 993410743460308, 4967053725690147, 24835268645227950

Offset: 0

N. J. A. Sloane, Dec 11 1999

Index entries for linear recurrences with constant coefficients, signature (8,-17,10).

Cf. A000225, A000392, A002275, A002452, A003462, A003463, A003464, A016123, A016125, A016208, A016209, A016218, A016256, A023000, A023001.

Cf. A016127.

a:=n->sum((5^(n-j)-2^(n-j))/3,j=0..n): seq(a(n), n=1..20); # Zerinvary Lajos, Jan 04 2007

Join[{a=1,b=8},Table[c=7*b-10*a+1;a=b;b=c,{n,60}]] (* Vladimir Joseph Stephan Orlovsky, Feb 06 2011 *)

a(n) = (25*5^n - 16*2^n + 3)/12. - Bruno Berselli, Feb 09 2011
a(n) = [(5^0-2^0) + (5^1-2^1) + ... + (5^n-2^n)]/3. - r22lou(AT)cox.net, Nov 14 2005
a(0)=1, a(n) = 5*a(n-1) + 2^(n+1) - 1. - Vincenzo Librandi, Feb 07 2011
From Elmo R. Oliveira, Mar 26 2025: (Start)
E.g.f.: exp(x)*(25*exp(4*x) - 16*exp(x) + 3)/12.
a(n) = 8*a(n-1) - 17*a(n-2) + 10*a(n-3).
a(n) = A016127(n+1) - A003463(n+2). (End)

More terms from Wesley Ivan Hurt, May 05 2014

cp's OEIS Frontend

A218749 a(n) = (46^n - 1)/45.

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Magma

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A218751 a(n) = (48^n - 1)/47.

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A086930 Smallest b>1 such that in base b representation the n-th prime is a repunit.

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A105280 a(0)=0; a(n) = 11*a(n-1) + 11.

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A319074 a(n) is the sum of the first n nonnegative powers of the n-th prime.

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A319076 Square array T(n,k) read by antidiagonal upwards in which column k lists the partial sums of the powers of the k-th prime, n >= 0, k >= 1.

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A361475 Array read by ascending antidiagonals: A(n, k) = (k^n - 1)/(k - 1), with k >= 2.

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A268839 a(n) = Sum_{j=1..10^n-1} 2^f(j) where f(j) is the number of zero digits in the decimal representation of j.

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A016198 Expansion of g.f. 1/((1-x)(1-2x)(1-5x)).