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 1-4 of 4 results.

A332967 Sum of all integers m satisfying Omega(m) = n and pi(p) <= n for all prime factors p of m.

Original entry on oeis.org

1, 2, 19, 410, 14343, 1139166, 89131918, 10861230692, 1271028562739, 203393524967230, 52274418436233714, 11160490802017899420, 3415612116240107778630, 1088775430914588654276060, 311608007930071575510930780, 99738699420765496000734958440
Offset: 0

Views

Author

Alois P. Heinz, Mar 04 2020

Keywords

Examples

			a(2) = 4 + 6 + 9 = 2*2 + 2*3 + 3*3 = 19.

Links

Crossrefs

Row sums of A330394.
Main diagonal of A343751.
Cf. A000040, A000720, A001222, A027748, A088218, A124960.

Programs

  • Maple
    b:= proc(n, i) option remember; `if`(n=0, 1,
      add(ithprime(j)*b(n-1, j), j=1..i))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..17);

Formula

a(n) = [x^n] Product_{i=1..n} 1/(1-prime(i)*x).
a(n) = A124960(2n,n).
a(n) = Sum_{k=1..A088218(n)} A330394(n,k).
a(n) = A343751(n,n).

A357251 a(n) = Sum_{1<=i<=j<=n} prime(i)*prime(j).

Original entry on oeis.org

4, 19, 69, 188, 496, 1029, 2015, 3478, 5778, 9519, 14479, 21768, 31526, 43609, 59025, 79218, 105178, 135739, 173795, 219164, 271140, 333629, 406171, 491878, 594698, 711959, 842151, 988848, 1150168, 1330177, 1548617, 1791098, 2063454, 2359107, 2698231, 3064708, 3470396, 3918157, 4404795, 4938846
Offset: 1

Views

Author

J. M. Bergot and Robert Israel, Sep 20 2022

Keywords

Comments

a(n) is the sum of products of unordered pairs of (not necessarily distinct) elements from the first n primes.
It appears that 4 is the only square in the sequence.

Examples

			a(3) = 2*2 + 2*3 + 2*5 + 3*3 + 3*5 + 5*5 = 69.

Links

Crossrefs

Cf. A007504, A024447, A024450, A065762, A357252.
Partial sums of A143215.
Row n=2 of A343751.

Programs

  • Maple
    P:= [seq(ithprime(i),i=1..100)]:
S:= ListTools:-PartialSums(P):
ListTools:-PartialSums(zip(`*`,P,S));
  • Mathematica
    Accumulate[(p = Prime[Range[40]]) * Accumulate[p]] (* Amiram Eldar, Sep 20 2022 *)
  • Python
    from itertools import accumulate
from sympy import prime, primerange
def aupton(nn):
    p = list(primerange(2, prime(nn)+1))
    return list(accumulate(c*d for c, d in zip(p, accumulate(p))))
print(aupton(40)) # Michael S. Branicky, Sep 24 2022 after Amiram Eldar

Formula

a(n) = (A007504(n)^2 + A024450(n))/2.
a(n) = A024447(n) + A024450(n).
a(n) = A065762(n)/2. - Hugo Pfoertner, Sep 24 2022

A016273 Expansion of g.f. 1/((1-2x)(1-3x)(1-5x)).

Original entry on oeis.org

1, 10, 69, 410, 2261, 11970, 61909, 315850, 1598421, 8050130, 40425749, 202656090, 1014866581, 5079099490, 25409813589, 127092049130, 635589254741, 3178333432050, 15892828897429, 79467630222970, 397348609370901, 1986774423719810, 9933966253389269
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Column k=3 of A343751.

Programs

  • Mathematica
    CoefficientList[ Series[ 1/((1 - 2x)(1 - 3x)(1 - 5x)), {x, 0, 20} ], x ]
LinearRecurrence[{10,-31,30},{1,10,69},20] (* Harvey P. Dale, Oct 05 2014 *)
  • PARI
    Vec(1/((1-2*x)*(1-3*x)*(1-5*x))+O(x^99)) \\ Charles R Greathouse IV, Sep 26 2012
  • Sage
    [((5^n - 2^n)/3-(3^n - 2^n))/2 for n in range(2,22)] # Zerinvary Lajos, Jun 05 2009

Formula

From Hieronymus Fischer, Jun 25 2007: (Start)
a(n) = Sum_{0<=i,j,k,<=n, i+j+k=n} 2^i*3^j*5^k.
a(n) = (2^(n+3) + 5^(n+2) - 3^(n+3))/6. (End)
a(n) = ((5^n - 2^n)/3 - (3^n - 2^n))/2 , n >= 2. - Zerinvary Lajos, Jun 05 2009
From Vincenzo Librandi, Mar 15 2011: (Start)
a(n) = 10*a(n-1) - 31*a(n-2) + 30*a(n-3), n >= 3.
a(n) = 8*a(n-1) - 15*a(n-2) + 2^n, a(0)=1, a(1)=10. (End)
E.g.f.: exp(2*x)*(8 - 27*exp(x) + 25*exp(3*x))/6. - Stefano Spezia, Sep 27 2023

A025931 Expansion of 1/((1-2x)(1-3x)(1-5x)(1-7x)).

Original entry on oeis.org

1, 17, 188, 1726, 14343, 112371, 848506, 6255392, 45386165, 325753285, 2320698744, 16447547298, 116147697667, 818112983159, 5752200695702, 40392496919044, 283383067688049, 1986859807248393, 13923911479636180, 97546847987676230, 683225284523104511
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Column k=4 of A343751.

Programs

  • Maple
    b:= proc(n, k) option remember; `if`(n=0, 1,
     `if`(k=0, 0, ithprime(k)*b(n-1, k)+b(n, k-1)))
    end:
a:= n-> b(n, 4):
seq(a(n), n=0..28);  # Alois P. Heinz, Jul 14 2021
  • Mathematica
    CoefficientList[Series[1/((1-2x)(1-3x)(1-5x)(1-7x)),{x,0,20}],x] (* or *) LinearRecurrence[{17,-101,247,-210},{1,17,188,1726},20] (* Harvey P. Dale, Aug 05 2013 *)

Formula

a(n) = 12*a(n-1) - 35*a(n-2) + 3^(n+1) - 2^(n+1), n >= 2. - Vincenzo Librandi, Mar 19 2011
a(n) = 7^(n+3)/40 - 2^(n+3)/15 + 3^(n+3)/8 - 5^(n+3)/12. - R. J. Mathar, Mar 19 2011
Showing 1-4 of 4 results.

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

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