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 11-17 of 17 results.

A167342 Totally multiplicative sequence with a(p) = p*(p+3) = p^2+3p for prime p.

Original entry on oeis.org

1, 10, 18, 100, 40, 180, 70, 1000, 324, 400, 154, 1800, 208, 700, 720, 10000, 340, 3240, 418, 4000, 1260, 1540, 598, 18000, 1600, 2080, 5832, 7000, 928, 7200, 1054, 100000, 2772, 3400, 2800, 32400, 1480, 4180, 3744, 40000, 1804, 12600, 1978, 15400, 12960
Offset: 1

Views

Author

Jaroslav Krizek, Nov 01 2009

Keywords

Links

Programs

  • Mathematica
    a[1] = 1; a[n_] := (fi = FactorInteger[n]; Times @@ ((fi[[All, 1]] + 3)^fi[[All, 2]])); Table[a[n]*n, {n, 1, 100}] (* G. C. Greubel, Jun 10 2016 *)

Formula

Multiplicative with a(p^e) = (p*(p+3))^e. If n = Product p(k)^e(k) then a(n) = Product (p(k)*(p(k)+3))^e(k). a(n) = n * A166591(n).
Sum_{k>=1} 1/a(k) = Product_{primes p} (1 + 1/(p^2 + 3*p - 1)) = 1.256741057020447773244230946716370792268447699628630376844295183469512964116... - Vaclav Kotesovec, Sep 20 2020

A167353 Totally multiplicative sequence with a(p) = (p+1)*(p+3) = p^2+4p+3 for prime p.

Original entry on oeis.org

1, 15, 24, 225, 48, 360, 80, 3375, 576, 720, 168, 5400, 224, 1200, 1152, 50625, 360, 8640, 440, 10800, 1920, 2520, 624, 81000, 2304, 3360, 13824, 18000, 960, 17280, 1088, 759375, 4032, 5400, 3840, 129600, 1520, 6600, 5376, 162000, 1848, 28800
Offset: 1

Views

Author

Jaroslav Krizek, Nov 01 2009

Keywords

Links

Crossrefs

Cf. A003959, A166591.

Programs

  • Mathematica
    a[1] = 1; a[n_] := (fi = FactorInteger[n]; Times @@ ((fi[[All, 1]] + 3)^fi[[All, 2]])); b[1] = 1; b[n_] := (fi = FactorInteger[n]; Times @@ ((fi[[All, 1]] + 1)^fi[[All, 2]])); Table[a[n]*b[n], {n, 1, 100}] (* G. C. Greubel, Jun 11 2016 *)

Formula

Multiplicative with a(p^e) = ((p+1)*(p+3))^e. If n = Product p(k)^e(k) then a(n) = Product ((p(k)+1)*(p(k)+3))^e(k). a(n) = A003959(n) * A166591(n).
Sum_{k>=1} 1/a(k) = Product_{primes p} (1 + 1/(p^2 + 4*p + 2)) = 1.1854020769112984236586594287311260820805752130814044791625914047437286210... - Vaclav Kotesovec, Sep 20 2020

A167357 Totally multiplicative sequence with a(p) = (p-2)*(p+3) = p^2+p-6 for prime p.

Original entry on oeis.org

1, 0, 6, 0, 24, 0, 50, 0, 36, 0, 126, 0, 176, 0, 144, 0, 300, 0, 374, 0, 300, 0, 546, 0, 576, 0, 216, 0, 864, 0, 986, 0, 756, 0, 1200, 0, 1400, 0, 1056, 0, 1716, 0, 1886, 0, 864, 0, 2250, 0, 2500, 0, 1800, 0, 2856, 0, 3024, 0, 2244, 0, 3534, 0, 3776, 0, 1800
Offset: 1

Views

Author

Jaroslav Krizek, Nov 01 2009

Keywords

Links

Crossrefs

Cf. A166586, A166591.

Programs

  • Mathematica
    a[1] = 1; a[n_] := (fi = FactorInteger[n]; Times @@ ((fi[[All, 1]] - 2)^fi[[All, 2]])); b[1] = 1; b[n_] := (fi = FactorInteger[n]; Times @@ ((fi[[All, 1]] + 3)^fi[[All, 2]])); Table[a[n]*b[n], {n, 1, 100}] (* G. C. Greubel, Jun 11 2016 *)

Formula

Multiplicative with a(p^e) = ((p-2)*(p+3))^e. If n = Product p(k)^e(k) then a(n) = Product ((p(k)-2)*(p(k)+3))^e(k).
a(2k) = 0 for k >= 1.
a(n) = A166586(n) * A166591(n).
Sum_{k=1..n} a(k) ~ c * n^3, where c = (2/Pi^2) / Product_{p prime} (1 - 2/p^2 + 5/p^3 + 6/p^4) = 0.1449357432... . - Amiram Eldar, Dec 15 2022

A167360 Totally multiplicative sequence with a(p) = (p+2)*(p+3) = p^2+5p+6 for prime p.

Original entry on oeis.org

1, 20, 30, 400, 56, 600, 90, 8000, 900, 1120, 182, 12000, 240, 1800, 1680, 160000, 380, 18000, 462, 22400, 2700, 3640, 650, 240000, 3136, 4800, 27000, 36000, 992, 33600, 1122, 3200000, 5460, 7600, 5040, 360000, 1560, 9240, 7200, 448000, 1892, 54000, 2070
Offset: 1

Views

Author

Jaroslav Krizek, Nov 01 2009

Keywords

Links

Crossrefs

Cf. A166590, A166591.

Programs

  • Mathematica
    a[1] = 1; a[n_] := (fi = FactorInteger[n]; Times @@ ((fi[[All, 1]] + 2)^fi[[All, 2]])); b[1] = 1; b[n_] := (fi = FactorInteger[n]; Times @@ ((fi[[All, 1]] + 3)^fi[[All, 2]])); Table[a[n]*b[n], {n, 1, 100}] (* G. C. Greubel, Jun 11 2016 *)

Formula

Multiplicative with a(p^e) = ((p+2)*(p+3))^e. If n = Product p(k)^e(k) then a(n) = Product ((p(k)+2)*(p(k)+3))^e(k). a(n) = A166590(n) * A166591(n).
Sum_{k>=1} 1/a(k) = Product_{primes p} (1 + 1/(p^2 + 5*p + 5)) = 1.1480407951783735490090642594369977652983537687209929674246821640934042061... - Vaclav Kotesovec, Sep 20 2020

A167362 Totally multiplicative sequence with a(p) = (p-3)*(p+3) = p^2-9 for prime p.

Original entry on oeis.org

1, -5, 0, 25, 16, 0, 40, -125, 0, -80, 112, 0, 160, -200, 0, 625, 280, 0, 352, 400, 0, -560, 520, 0, 256, -800, 0, 1000, 832, 0, 952, -3125, 0, -1400, 640, 0, 1360, -1760, 0, -2000, 1672, 0, 1840, 2800, 0, -2600, 2200, 0, 1600, -1280, 0, 4000, 2800, 0, 1792
Offset: 1

Views

Author

Jaroslav Krizek, Nov 01 2009

Keywords

Links

Crossrefs

Cf. A166589, A166591.

Programs

  • Mathematica
    a[1] = 1; a[n_] := (fi = FactorInteger[n]; Times @@ ((fi[[All, 1]] - 3)^fi[[All, 2]])); b[1] = 1; b[n_] := (fi = FactorInteger[n]; Times @@ ((fi[[All, 1]] + 3)^fi[[All, 2]])); Table[a[n]*b[n], {n, 1, 100}] (* G. C. Greubel, Jun 11 2016 *)

Formula

Multiplicative with a(p^e) = ((p-3)*(p+3))^e. If n = Product p(k)^e(k) then a(n) = Product ((p(k)-3)*(p(k)+3))^e(k).
a(n) = A166589(n) * A166591(n).
Sum_{k=1..n} a(k) ~ c * n^3, where c = (2/Pi^2) / Product_{p prime} (1 - 1/p^2 + 9/p^3 + 9/p^4) = 0.05980933853... . - Amiram Eldar, Dec 15 2022

A167363 Totally multiplicative sequence with a(p) = (p+3)^2 = p^2+6p+9 for prime p.

Original entry on oeis.org

1, 25, 36, 625, 64, 900, 100, 15625, 1296, 1600, 196, 22500, 256, 2500, 2304, 390625, 400, 32400, 484, 40000, 3600, 4900, 676, 562500, 4096, 6400, 46656, 62500, 1024, 57600, 1156, 9765625, 7056, 10000, 6400, 810000, 1600, 12100, 9216, 1000000, 1936, 90000
Offset: 1

Views

Author

Jaroslav Krizek, Nov 01 2009

Keywords

Links

Crossrefs

Cf. A166591.

Programs

  • Mathematica
    b[1] = 1; b[n_] := (fi = FactorInteger[n]; Times @@ ((fi[[All, 1]] + 3)^fi[[All, 2]])); Table[b[n]^2, {n, 1, 100}] (* G. C. Greubel, Jun 11 2016 *)

Formula

Multiplicative with a(p^e) = ((p+3)^2)^e. If n = Product p(k)^e(k) then a(n) = Product ((p(k)+3)^2)^e(k). a(n) = A166591(n)^2.
Sum_{k>=1} 1/a(k) = Product_{primes p} (1 + 1/(p^2 + 6*p + 8)) = 1.1245403934230393267573507658470307221064356442604979888687782305037985824... - Vaclav Kotesovec, Sep 20 2020

A359530 Multiplicative with a(p^e) = (p + 4)^e.

Original entry on oeis.org

1, 6, 7, 36, 9, 42, 11, 216, 49, 54, 15, 252, 17, 66, 63, 1296, 21, 294, 23, 324, 77, 90, 27, 1512, 81, 102, 343, 396, 33, 378, 35, 7776, 105, 126, 99, 1764, 41, 138, 119, 1944, 45, 462, 47, 540, 441, 162, 51, 9072, 121, 486, 147, 612, 57, 2058, 135, 2376, 161
Offset: 1

Views

Author

Vaclav Kotesovec, Feb 26 2023

Keywords

Links

Crossrefs

Cf. A166589 (multiplicative with a(p^e) = (p-3)^e), A166586 (p-2), A003958 (p-1), A000027 (p), A003959 (p+1), A166590 (p+2), A166591 (p+3).

Programs

  • Mathematica
    g[p_, e_] := (p + 4)^e; a[1] = 1; a[n_] := Times @@ g @@@ FactorInteger[n]; Array[a, 100]
  • PARI
    for(n=1, 100, print1(direuler(p=2, n, 1/(1-p*X-4*X))[n], ", "))
  • Python
    from math import prod
from sympy import factorint
def A359530(n): return prod((p+4)**e for p, e in factorint(n).items()) # Chai Wah Wu, Feb 26 2023

Formula

Dirichlet g.f.: Product_{primes p} 1 / (1 - p^(1-s) - 4*p^(-s)).
Dirichlet g.f.: zeta(s-1) * (1 + 4/(2^s - 6)) * Product_{primes p, p>2} (1 + 4/(p^s - p - 4)).
Sum_{k=1..n} a(k) has an average value 2*c*zeta(r-1) * n^r / (3*log(6)), where r = 1 + log(3)/log(2) = 2.5849625007211561814537389439478165... and c = Product_{primes p, p>2} (1 + 4/(p^r - p - 4)) = 1.5747380964592139...
Previous Showing 11-17 of 17 results.

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

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