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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A218741 a(n) = (38^n - 1)/37.

Original entry on oeis.org

0, 1, 39, 1483, 56355, 2141491, 81376659, 3092313043, 117507895635, 4465300034131, 169681401296979, 6447893249285203, 245019943472837715, 9310757851967833171, 353808798374777660499, 13444734338241551098963, 510899904853178941760595, 19414196384420799786902611
Offset: 0

Views

Author

M. F. Hasler, Nov 04 2012

Keywords

Comments

Partial sums of powers of 38 (A009982).

Links

Crossrefs

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. A009982.

Programs

Formula

From Vincenzo Librandi, Nov 07 2012: (Start)
G.f.: x/((1-x)*(1-38*x)).
a(n) = 39*a(n-1) - 38*a(n-2).
a(n) = floor(38^n/37). (End)
E.g.f.: exp(x)*(exp(37*x) - 1)/37. - Elmo R. Oliveira, Aug 29 2024

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

Views

Author

M. F. Hasler, Nov 04 2012

Keywords

Comments

Partial sums of powers of 39 (A009983).

Links

Crossrefs

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.

Programs

Formula

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

A218747 a(n) = (44^n - 1)/43.

Original entry on oeis.org

0, 1, 45, 1981, 87165, 3835261, 168751485, 7425065341, 326702875005, 14374926500221, 632496766009725, 27829857704427901, 1224513738994827645, 53878604515772416381, 2370658598693986320765, 104308978342535398113661, 4589595047071557517001085, 201942182071148530748047741
Offset: 0

Views

Author

M. F. Hasler, Nov 04 2012

Keywords

Comments

Partial sums of powers of 44 (A009988).

Links

Crossrefs

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. A009988.

Programs

Formula

From Vincenzo Librandi, Nov 07 2012: (Start)
G.f.: x/((1-x)*(1-44*x)).
a(n) = 45*a(n-1) - 44*a(n-2).
a(n) = floor(44^n/43). (End)
E.g.f.: exp(x)*(exp(43*x) - 1)/43. - Elmo R. Oliveira, Aug 29 2024

A218748 a(n) = (45^n - 1)/44.

Original entry on oeis.org

0, 1, 46, 2071, 93196, 4193821, 188721946, 8492487571, 382161940696, 17197287331321, 773877929909446, 34824506845925071, 1567102808066628196, 70519626362998268821, 3173383186334922096946, 142802243385071494362571, 6426100952328217246315696
Offset: 0

Views

Author

M. F. Hasler, Nov 04 2012

Keywords

Comments

Partial sums of powers of 45 (A009989).

Links

Crossrefs

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. A009989.

Programs

Formula

G.f.: x/((1-x)*(1-45*x)). - Vincenzo Librandi, Nov 08 2012
a(n) = 46*a(n-1) - 45*a(n-2) with a(0)=0, a(1)=1. - Vincenzo Librandi, Nov 08 2012
a(n) = 45*a(n-1) + 1 with a(0)=0. - Vincenzo Librandi, Nov 08 2012
a(n) = floor(45^n/44). - Vincenzo Librandi, Nov 08 2012
E.g.f.: exp(23*x)*sinh(22*x)/22. - Elmo R. Oliveira, Aug 27 2024

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

Original entry on oeis.org

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

Views

Author

M. F. Hasler, Nov 04 2012

Keywords

Comments

Partial sums of powers of 46 (A009990).

Links

Crossrefs

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.

Programs

Formula

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

Views

Author

M. F. Hasler, Nov 04 2012

Keywords

Comments

Partial sums of powers of 48 (A009992).

Links

Crossrefs

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.

Programs

Formula

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

Views

Author

Reinhard Zumkeller, Sep 21 2003

Keywords

Comments

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)

Examples

			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.

Links

Crossrefs

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

Programs

  • Mathematica
    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]

A125725 Numbers whose base-7 representation is 222....2.

Original entry on oeis.org

0, 2, 16, 114, 800, 5602, 39216, 274514, 1921600, 13451202, 94158416, 659108914, 4613762400, 32296336802, 226074357616, 1582520503314, 11077643523200, 77543504662402, 542804532636816, 3799631728457714, 26597422099204000
Offset: 1

Views

Author

Zerinvary Lajos, Feb 02 2007

Keywords

Examples

			base 7.......decimal
0..................0
2..................2
22................16
222..............114
2222.............800
22222...........5602
222222.........39216
2222222.......274514
22222222.....1921600
222222222...13451202
etc...........etc.

Links

Crossrefs

Cf. A007310, A023000, A047522, A165759.
Cf. also A002276, A005610, A020988, A024023, A125831, A125835, A125857 for related or similarly constructed sequences.

Programs

  • GAP
    List([1..25], n-> (7^(n-1) -1)/3); # G. C. Greubel, May 23 2019
  • Magma
    [0] cat [n:n in [1..15000000]| Set(Intseq(n,7)) subset [2]]; // Marius A. Burtea, May 06 2019
  • Magma
    [(7^(n-1)-1)/3: n in [1..25]]; // Marius A. Burtea, May 06 2019
  • Maple
    seq(2*(7^n-1)/6, n=0..25);
  • Mathematica
    FromDigits[#,7]&/@Table[PadLeft[{2},n,2],{n,0,25}]  (* Harvey P. Dale, Apr 13 2011 *)
(7^(Range[25]-1) - 1)/3 (* G. C. Greubel, May 23 2019 *)
  • PARI
    vector(25, n, (7^(n-1)-1)/3) \\ Davis Smith, Apr 04 2019
  • Sage
    [(7^(n-1) -1)/3 for n in (1..25)] # G. C. Greubel, May 23 2019

Formula

a(n) = (7^(n-1) - 1)/3 = 2*A023000(n-1).
a(n) = 7*a(n-1) + 2, with a(1)=0. - Vincenzo Librandi, Sep 30 2010
G.f.: 2*x^2 / ( (1-x)*(1-7*x) ). - R. J. Mathar, Sep 30 2013
From Davis Smith, Apr 04 2019: (Start)
A007310(a(n) + 1) = 7^(n - 1).
A047522(a(n + 1)) = -1*A165759(n). (End)
E.g.f.: (exp(7*x) - 7*exp(x) + 6)/21. - Stefano Spezia, Jan 12 2025

Extensions

Offset corrected by N. J. A. Sloane, Oct 02 2010

A146885 a(n) = 8*Sum_{k=0..n} 7^k.

Original entry on oeis.org

8, 64, 456, 3200, 22408, 156864, 1098056, 7686400, 53804808, 376633664, 2636435656, 18455049600, 129185347208, 904297430464, 6330082013256, 44310574092800, 310174018649608, 2171218130547264, 15198526913830856
Offset: 0

Views

Author

Roger L. Bagula, Nov 02 2008

Keywords

Links

Crossrefs

Cf. A005843, A023000, A068156, A100774, A132583, A146882, A146883, A146885.

Programs

  • Magma
    [n le 2 select 8^n else 8*Self(n-1) -7*Self(n-2): n in [1..41]]; // G. C. Greubel, Oct 12 2022
  • Mathematica
    a[n_]:= Sum[8*7^m, {m,0,n}]; Table[a[n], {n,0,30}]
LinearRecurrence[{8,-7}, {8,64}, 41] (* G. C. Greubel, Oct 12 2022 *)
  • SageMath
    [(4/3)*(7^(n+1)-1) for n in range(41)] # G. C. Greubel, Oct 12 2022

Formula

From G. C. Greubel, Oct 12 2022: (Start)
a(n) = (4/3)*(7^(n+1) - 1).
a(n) = 8*A023000(n+1).
a(n) = 8*a(n-1) - 7*a(n-2).
G.f.: 8/((1-x)*(1-7*x)).
E.g.f.: (4/3)*(7*exp(7*x) - exp(x)). (End)

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

Views

Author

Omar E. Pol, Sep 11 2018

Keywords

Examples

			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.

Crossrefs

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.

Programs

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

Formula

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)).
Previous Showing 61-70 of 81 results. Next

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