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.

Showing 1-3 of 3 results.

A030633 Numbers with 15 divisors.

Original entry on oeis.org

144, 324, 400, 784, 1936, 2025, 2500, 2704, 3969, 4624, 5625, 5776, 8464, 9604, 9801, 13456, 13689, 15376, 16384, 21609, 21904, 23409, 26896, 29241, 29584, 30625, 35344, 42849, 44944, 55696, 58564, 59536, 60025, 68121, 71824, 75625
Offset: 1

Views

Author

Jeff Burch

Keywords

Comments

Numbers of the form p^14 (subset of A010802) or p^2*q^4 (A189988) where p and q are distinct primes. - R. J. Mathar, Mar 01 2010

Links

Crossrefs

Cf. A000005, A010802, A030630, A030631, A030632, A189988.

Programs

  • Mathematica
    Select[Range[300000],DivisorSigma[0,#]==15&] (* Vladimir Joseph Stephan Orlovsky, May 05 2011 *)
  • PARI
    is(n)=numdiv(n)==15 \\ Charles R Greathouse IV, Jun 19 2016
  • Python
    from math import isqrt
from sympy import primepi, primerange, integer_nthroot
def A030633(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        kmin = kmax >> 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x): return n+x-sum(primepi(isqrt(x//p**4)) for p in primerange(integer_nthroot(x,4)[0]+1))+primepi(integer_nthroot(x,6)[0])-primepi(integer_nthroot(x,14)[0])
    return bisection(f,n,n) # Chai Wah Wu, Feb 22 2025

Formula

From Amiram Eldar, Jul 03 2022: (Start)
A000005(a(n)) = 15.
Sum_{n>=1} 1/a(n) = P(2)*P(4) - P(6) + P(14) = 0.0178111..., where P is the prime zeta function. (End)

A168664 a(n) = n^7*(n^7 + 1)/2.

Original entry on oeis.org

0, 1, 8256, 2392578, 134225920, 3051796875, 39182222016, 339111948196, 2199024304128, 11438398618965, 50000005000000, 189874926535206, 641959250190336, 1968688224223903, 5556003465485760, 14596463098125000, 36028797153181696, 84188913484869801
Offset: 0

Views

Author

N. J. A. Sloane, Dec 11 2009

Keywords

Comments

Number of unoriented rows of length 14 using up to n colors. For a(0)=0, there are no rows using no colors. For a(1)=1, there is one row using that one color for all positions. For a(2)=8256, there are 2^14=16384 oriented arrangements of two colors. Of these, 2^7=128 are achiral. That leaves (16384-128)/2=8128 chiral pairs. Adding achiral and chiral, we get 8256. - Robert A. Russell, Nov 13 2018

Links

Crossrefs

Cf. A001015 (Seventh Powers: n^7), A000217 (Triangular Numbers).
Row 14 of A277504.
Cf. A010802 (oriented), A001015 (achiral).

Programs

Formula

From Wesley Ivan Hurt, Oct 30 2014: (Start)
G.f.: (x + 8241*x^2 + 2268843*x^3 + 99203675*x^4 + 1285873650*x^5 + 6421633938*x^6 + 13985577438*x^7 + 13985598654*x^8 + 6421628925*x^9 + 1285868525*x^10 + 99207111*x^11 + 2268471*x^12 + 8128*x^13)/(1 - x)^15.
a(n) = 15*a(n-1) - 105*a(n-2) + 455*a(n-3) - 1365*a(n-4) + 3003*a(n-5) - 5005*a(n-6) + 6435*a(n-7) - 6435*a(n-8) + 5005*a(n-9) - 3003*a(n-10) + 1365*a(n-11) - 455*a(n-12) + 105*a(n-13) - 15*a(n-14) + a(n-15).
a(n) = n^7*(n^7 + 1)/2 = A000217(A001015(n)). (End)
From Robert A. Russell, Nov 13 2018: (Start)
a(n) = (A010802(n) + A001015(n)) / 2 = (n^14 + n^7) / 2.
G.f.: (Sum_{j=1..14} S2(14,j)*j!*x^j/(1-x)^(j+1) + Sum_{j=1..7} S2(7,j)*j!*x^j/(1-x)^(j+1)) / 2, where S2 is the Stirling subset number A008277.
G.f.: x*Sum_{k=0..13} A145882(14,k) * x^k / (1-x)^15.
E.g.f.: (Sum_{k=1..14} S2(14,k)*x^k + Sum_{k=1..7} S2(7,k)*x^k) * exp(x) / 2, where S2 is the Stirling subset number A008277.
For n>14, a(n) = Sum_{j=1..15} -binomial(j-16,j) * a(n-j). (End)
E.g.f.: x*(2+8254*x +789271*x^2 +10392095*x^3 +40075175*x^4 +63436394*x^5 +49329281*x^6 +20912320*x^7 +5135130*x^8 +752752*x^9 + 66066*x^10 +3367*x^11 +91*x^12 +x^13)*exp(x)/2. - G. C. Greubel, Nov 15 2018

A022530 Nexus numbers (n+1)^14 - n^14.

Original entry on oeis.org

1, 16383, 4766585, 263652487, 5835080169, 72260648471, 599858908753, 3719823438255, 18478745943857, 77123207545039, 279749833583241, 904168630965623, 2653457921150425, 7174630439858727, 18080919199832609, 42864668012537311, 96320232521472993, 206435541022680095
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

References

  • J. H. Conway and R. K. Guy, The Book of Numbers, Copernicus Press, NY, 1996, p. 54.

Links

Crossrefs

Column k=13 of A047969.
Cf. A010802 (n^14).

Programs

  • Magma
    [(n+1)^14 - n^14: n in [0..20]]; // G. C. Greubel, Feb 27 2018
  • Maple
    b:=14: a:=n->(n+1)^b-n^b: seq(a(n),n=0..18); # Muniru A Asiru, Feb 28 2018
  • Mathematica
    Last[#]-First[#]&/@Partition[Range[0,20]^14,2,1] (* Harvey P. Dale, Oct 04 2011 *)
Table[(n+1)^14 - n^14, {n,0,20}] (* G. C. Greubel, Feb 27 2018 *)
  • PARI
    for(n=0,20, print1((n+1)^14 - n^14, ", ")) \\ G. C. Greubel, Feb 27 2018

Formula

G.f.: (1+x) *(x^12 +16368*x^11 +4520946*x^10 +193889840*x^9 +2377852335*x^8 +10465410528*x^7 +17505765564*x^6 +10465410528*x^5 +2377852335*x^4 +193889840*x^3 +4520946*x^2 +16368*x+1) / (x-1)^14 . - R. J. Mathar, Sep 02 2016
a(n) = A010802(n+1) - A010802(n). - Michel Marcus, Feb 28 2018
Showing 1-3 of 3 results.

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

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