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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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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)).

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

Views

Author

Omar E. Pol, Sep 09 2018

Keywords

Comments

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.

Examples

			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, ...
...

Crossrefs

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.

Programs

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

Formula

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).

A272199 Expansion of 1/(1 - 2*x + 13*x^2).

Original entry on oeis.org

1, 2, -9, -44, 29, 630, 883, -6424, -24327, 34858, 385967, 318780, -4380011, -12904162, 31131819, 230017744, 55321841, -2879586990, -6478357913, 24477915044, 133174482957, -51863929658, -1834996137757, -2995761189960, 17863427410921, 74671750291322
Offset: 0

Views

Author

Wolfdieter Lang, Apr 27 2016

Keywords

Comments

a(n) gives the coefficient c(13^n) of (eta(z^6))^4, a modular cusp form of weight 2, when expanded in powers of q = exp(2*Pi*i*z), Im(z) > 0, assuming alpha-multiplicativity (not valid for p = 2 and 3) with alpha(x) = x (weight 2) and input c(13) = +2. Eta is the Dedekind function. See the Apostol reference, p. 138, eq. (54) for alpha-multiplicativity and p. 130, eq. (39) with k=2. See also A000727 where a(n)=c(13^n) = A000727((13^n-1)/6)=A000727(2*A091030(n)), n >= 0. For the proof that alpha-multiplicativity leads to the recurrence involving Chebyshev's S polynomials see a comment on A168175, and the Apostol reference, Exercise 6., p. 139.

Examples

			a(2) = c(13^2) = A000727(2*A091030(2)) =
A000727(28) = -9.

References

  • Tom M. Apostol, Modular Functions and Dirichlet Series in Number Theory, Second edition, Springer, 1990, pp. 130, 138 - 139.

Links

Crossrefs

Cf. A023000, A049310, A168175.

Programs

  • Magma
    I:=[1,2]; [n le 2 select I[n] else 2*Self(n-1)-13*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Nov 25 2016
  • Mathematica
    CoefficientList[Series[1/(1 - 2 x + 13 x^2), {x, 0, 25}], x] (* Michael De Vlieger, Apr 27 2016 *)
LinearRecurrence[{2, -13}, {1, 2}, 30] (* Vincenzo Librandi, Nov 25 2016 *)
  • PARI
    Vec(1/(1-2*x+13*x^2) + O(x^99)) \\ Altug Alkan, Apr 28 2016

Formula

G.f.: 1/(1 - 2*x + 13*x^2).
a(n) = 2*a(n-1) - 13*a(n-2), a(-1) = 0, a(0) = 1.
a(n) = sqrt(13)^n * S(n, 2/sqrt(13)), n >= 0, with Chebyshev's S polynomials (A049310).
a(n) = (Ap^(n+1) - Am^(n+1))/(Ap - Am) with Ap:= 1 + 2*sqrt(3)*i and Am = 1 - 2*sqrt(3)*i, (Binet - de Moivre formula), where i is the imaginary unit.
E.g.f.: (sqrt(3)*sin(2*sqrt(3)*x) + 6*cos(2*sqrt(3)*x))*exp(x)/6. - Ilya Gutkovskiy, Apr 27 2016
Previous Showing 41-43 of 43 results.

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