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.

Previous Showing 11-18 of 18 results.

A325257 a(1) = 1; a(n) = Sum_{d|n, d

Original entry on oeis.org

1, 3, 5, 16, 11, 43, 17, 88, 48, 95, 31, 320, 41, 145, 157, 486, 59, 554, 67, 696, 243, 265, 83, 2204, 218, 347, 458, 1062, 109, 1961, 127, 2668, 447, 493, 523, 5044, 157, 565, 577, 4780, 179, 3021, 191, 1938, 1998, 697, 211, 14590, 516, 2538, 823, 2526, 241, 6622, 939
Offset: 1

Views

Author

Ilya Gutkovskiy, Sep 05 2019

Keywords

Crossrefs

Cf. A000040, A007445, A008966 (parity of a(n)), A034696.

Programs

  • Magma
    sol:=[1]; for n in [2..60] do Append(~sol,&+[NthPrime(Floor(n/d))*sol[d]:d in Set(Divisors(n)) diff {n}]); end for; sol; // Marius A. Burtea, Sep 05 2019
  • Mathematica
    a[n_] := If[n == 1, n, Sum[If[d < n, Prime[n/d] a[d], 0], {d, Divisors[n]}]]; Table[a[n], {n, 1, 55}]
nmax = 55; A[] = 0; Do[A[x] = x + Sum[Prime[k] A[x^k], {k, 2, nmax}] + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x] // Rest
  • PARI
    seq(n)={my(v=vector(n)); v[1]=1; for(n=2, #v, v[n] = sumdiv(n, d, v[d]*prime(n/d))); v} \\ Andrew Howroyd, Sep 05 2019

Formula

G.f. A(x) satisfies: A(x) = x + Sum_{k>=2} prime(k) * A(x^k).

A333178 a(n) = Sum_{d|n, gcd(d, n/d) = 1} prime(d).

Original entry on oeis.org

2, 5, 7, 9, 13, 23, 19, 21, 25, 45, 33, 51, 43, 65, 65, 55, 61, 89, 69, 91, 97, 115, 85, 115, 99, 147, 105, 133, 111, 223, 129, 133, 175, 203, 179, 183, 159, 235, 215, 205, 181, 337, 193, 233, 233, 287, 213, 283, 229, 331, 299, 289, 243, 359, 301, 301, 343, 385, 279, 461
Offset: 1

Views

Author

Ilya Gutkovskiy, Mar 10 2020

Keywords

Crossrefs

Cf. A000040, A001221, A007445, A076479, A333177.

Programs

  • Mathematica
    a[n_] := Sum[If[GCD[n/d, d] == 1, Prime[d], 0], {d, Divisors[n]}]; Table[a[n], {n, 1, 60}]
  • PARI
    a(n) = sumdiv(n, d, if (gcd(d, n/d) ==1, prime(d))); \\ Michel Marcus, Mar 10 2020

Formula

Sum_{d|n, gcd(d, n/d) = 1} (-1)^omega(n/d) * a(d) = prime(n).

A333449 a(n) = Sum_{k=1..n} prime(floor(n/k)).

Original entry on oeis.org

2, 5, 9, 14, 20, 27, 33, 40, 48, 61, 65, 80, 86, 95, 107, 120, 128, 141, 149, 168, 178, 189, 195, 218, 232, 243, 253, 268, 272, 297, 313, 330, 342, 353, 373, 396, 404, 419, 431, 458, 466, 483, 495, 510, 530, 539, 553, 594, 604, 627, 641, 660, 664, 689, 703, 726, 742, 749, 757, 798
Offset: 1

Views

Author

Ilya Gutkovskiy, Mar 21 2020

Keywords

Crossrefs

Cf. A000040, A001223, A007445, A007504, A008683, A333450.

Programs

  • Mathematica
    Table[Sum[Prime[Floor[n/k]], {k, 1, n}], {n, 1, 60}]
g[1] = 2; g[n_] := Prime[n] - Prime[n - 1]; a[n_] := Sum[Sum[g[d], {d, Divisors[k]}], {k, 1, n}]; Table[a[n], {n, 1, 60}]
  • PARI
    a(n) = sum(k=1, n, prime(n\k)); \\ Michel Marcus, Mar 22 2020

Formula

G.f.: (1/(1 - x)) * (2*x/(1 - x) + Sum_{k>=2} (prime(k) - prime(k-1))*x^k/(1 - x^k)).
Sum_{k=1..n} mu(k) * a(floor(n/k)) = prime(n).

A333558 a(n) = Sum_{d|n} phi(d) * prime(d).

Original entry on oeis.org

2, 5, 12, 19, 46, 41, 104, 95, 150, 165, 312, 203, 494, 365, 432, 519, 946, 545, 1208, 747, 990, 1105, 1828, 991, 1986, 1709, 2004, 1663, 3054, 1481, 3812, 2615, 3062, 3173, 3724, 2519, 5654, 4145, 4512, 3591, 7162, 3449, 8024, 4979, 5298, 6209, 9708, 4983, 9638, 6685
Offset: 1

Views

Author

Ilya Gutkovskiy, Mar 26 2020

Keywords

Crossrefs

Cf. A000010, A000040, A007445, A057660, A061150, A130029.

Programs

  • Mathematica
    Table[Sum[EulerPhi[d] Prime[d], {d, Divisors[n]}], {n, 1, 50}]
Table[Sum[Prime[n/GCD[n, k]], {k, 1, n}], {n, 1, 50}]
  • PARI
    a(n) = sumdiv(n, d, prime(d)*eulerphi(d)); \\ Michel Marcus, Mar 27 2020

Formula

G.f.: Sum_{k>=1} phi(k) * prime(k) * x^k / (1 - x^k).
a(n) = Sum_{k=1..n} prime(n/gcd(n,k)).
a(n) = Sum_{k=1..n} prime(gcd(n,k))*phi(gcd(n,k))/phi(n/gcd(n,k)). - Richard L. Ollerton, May 09 2021

A353078 Inverse Moebius transform of odd primes.

Original entry on oeis.org

3, 8, 10, 19, 16, 32, 22, 42, 39, 52, 40, 84, 46, 74, 76, 101, 64, 128, 74, 136, 108, 128, 92, 204, 117, 154, 146, 194, 116, 256, 134, 238, 186, 218, 186, 337, 166, 246, 226, 338, 184, 368, 196, 336, 304, 308, 226, 490, 251, 386, 310, 406, 254, 492, 316, 486, 352, 398, 284, 664
Offset: 1

Views

Author

Ilya Gutkovskiy, Apr 22 2022

Keywords

Crossrefs

Cf. A007445, A030014, A065091.

Programs

  • Mathematica
    nmax = 60; CoefficientList[Series[Sum[Prime[k + 1] x^k/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x] // Rest
Table[DivisorSum[n, Prime[# + 1] &], {n, 1, 60}]
  • PARI
    a(n) =sumdiv(n, d, prime(d+1)); \\ Michel Marcus, Apr 22 2022

Formula

G.f.: Sum_{k>=1} prime(k+1) * x^k / (1 - x^k).
a(n) = Sum_{d|n} prime(d+1).

A309368 a(n) = Sum_{d|n} prime(n/d)^d.

Original entry on oeis.org

2, 7, 13, 32, 43, 129, 145, 405, 660, 1417, 2079, 5999, 8233, 18903, 37271, 74912, 131131, 300239, 524355, 1139985, 2180263, 4372491, 8388691, 17853809, 33715580, 68704969, 136183123, 274127445, 536871021, 1100025921, 2147483775, 4343912079, 8638792645, 17309012967, 34380645545
Offset: 1

Views

Author

Ilya Gutkovskiy, Jul 25 2019

Keywords

Crossrefs

Cf. A000040, A007445, A055225.

Programs

  • Mathematica
    Table[Sum[Prime[n/d]^d, {d, Divisors[n]}], {n, 1, 35}]
nmax = 35; CoefficientList[Series[Sum[Prime[k] x^k/(1 - Prime[k] x^k), {k, 1, nmax}], {x, 0, nmax}], x] // Rest
nmax = 35; CoefficientList[Series[-Log[Product[(1 - Prime[k] x^k)^(1/k), {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax] // Rest

Formula

G.f.: Sum_{k>=1} prime(k)*x^k/(1 - prime(k)*x^k).
L.g.f.: -log(Product_{k>=1} (1 - prime(k)*x^k)^(1/k)).
a(n) ~ 2^n.

A325891 a(1) = 1; a(n) = -Sum_{d|n, d

Original entry on oeis.org

1, -3, -5, 2, -11, 17, -17, -4, 2, 37, -31, -24, -41, 59, 63, 2, -59, -18, -67, -40, 97, 107, -83, 64, 24, 145, 2, -70, -109, -245, -127, 12, 173, 215, 225, 110, -157, 239, 243, 96, -179, -401, -191, -122, -46, 299, -211, -70, 62, -98, 357, -166, -241, 30, 425
Offset: 1

Views

Author

Ilya Gutkovskiy, Sep 07 2019

Keywords

Crossrefs

Cf. A000040, A007445, A034696, A325257.

Programs

  • Magma
    sol:=[1]; for n in [2..55] do Append(~sol,-&+[NthPrime(Floor(n/d))*sol[d]:d in Set(Divisors(n)) diff {n}]); end for; sol; // Marius A. Burtea, Sep 08 2019
  • Mathematica
    a[n_] := If[n == 1, n, -Sum[If[d < n, Prime[n/d] a[d], 0], {d, Divisors[n]}]]; Table[a[n], {n, 1, 55}]
nmax = 55; A[] = 0; Do[A[x] = x - Sum[Prime[k] A[x^k], {k, 2, nmax}] + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x] // Rest
  • PARI
    a(n) = if (n==1, 1, -sumdiv(n, d, if (d Michel Marcus, Sep 08 2019

Formula

G.f. A(x) satisfies: A(x) = x - Sum_{k>=2} prime(k) * A(x^k).

A333644 a(n) = Sum_{k=1..n} floor(n/k) * prime(k).

Original entry on oeis.org

2, 7, 14, 26, 39, 62, 81, 112, 142, 187, 220, 287, 330, 395, 460, 544, 605, 712, 781, 904, 1001, 1116, 1201, 1376, 1486, 1633, 1766, 1945, 2056, 2279, 2408, 2623, 2798, 3001, 3180, 3482, 3641, 3876, 4091, 4406, 4587, 4924, 5117, 5432, 5717, 6004, 6217, 6668, 6914, 7285
Offset: 1

Views

Author

Ilya Gutkovskiy, Mar 31 2020

Keywords

Comments

Partial sums of A007445.

Crossrefs

Cf. A000040, A007445, A024916, A333471.

Programs

  • Mathematica
    Table[Sum[Floor[n/k] Prime[k], {k, n}], {n, 50}]
  • PARI
    a(n) = sum(k=1, n, (n\k)*prime(k)); \\ Michel Marcus, Mar 31 2020

Formula

G.f.: (1/(1 - x)) * Sum_{k>=1} prime(k) * x^k / (1 - x^k).
Previous Showing 11-18 of 18 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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