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.

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A178147 Sum of squares d^2 of distinct divisors of n, d in {2, 3, 5}.

Original entry on oeis.org

0, 4, 9, 4, 25, 13, 0, 4, 9, 29, 0, 13, 0, 4, 34, 4, 0, 13, 0, 29, 9, 4, 0, 13, 25, 4, 9, 4, 0, 38, 0, 4, 9, 4, 25, 13, 0, 4, 9, 29, 0, 13, 0, 4, 34, 4, 0, 13, 0, 29, 9, 4, 0, 13, 25, 4, 9, 4, 0, 38, 0, 4, 9, 4, 25, 13, 0, 4, 9, 29, 0, 13, 0, 4, 34, 4, 0, 13, 0
Offset: 1

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Author

Vladimir Shevelev, May 21 2010, May 23 2010

Keywords

Comments

The sequence is periodic with period {0 4 9 4 25 13 0 4 9 29 0 13 0 4 34 4 0 13 0 29 9 4 0 13 25 4 9 4 0 38} of length 30.
A generalization: let B={b_1,...,b_t} be a set of t positive (not necessarily distinct) integers and m>=0 an integer.
For m>=0, let A(n)=Sum d^m over divisors d of n which are elements of B (with the multiplicities as in B). Calculating directly values of
A(b_i),A(b_i+b_j),A(b_i+b_j+b_k),...,
A(b_1+...+b_t), for the other values of A(n) we have the recursion:
A(n)=Sum{1<=i<=t}A(n-b_i)- Sum{1<=i

Links

Crossrefs

Cf. A005063, A098002, A178142-A178144, A178146.

Formula

a(n)= a(n-2) +a(n-3) -a(n-7)- a(n-8) +a(n-10), n>10.
By the comment, up to 10 it is sufficient to
calculate directly only values a(2)=4, a(3)=9, a(5)=25, a(7)=0, a(8)=4, a(10)=29.
For other n's we can use the recursion, accepting formally a(n)=0 for n<0. So a(1)=0; a(4)=a(2)+a(1)=4;a(6)=a(4)+a(3)=4+9=13,
a(9)=a(7)+a(6)-a(2)-a(1)=0+13-4+0=9.
a(n) = -2*a(n-1) -2*a(n-2) -a(n-3) +a(n-5) +2*a(n-6) +2*a(n-7) +a(n-8). - R. J. Mathar, Jul 13 2010
G.f. -x^2*(4+17*x+30*x^2+55*x^3+80*x^4+38*x^6+76*x^5) / ( (x-1)*(1+x)*(1+x+x^2)*(x^4+x^3+x^2+x+1) ). - R. J. Mathar, Dec 17 2012

A228182 a(n) is the smallest k such that the sum of squares of prime divisors of k is equal to the sum of prime divisors of n+k.

Original entry on oeis.org

12, 810, 35152, 18, 9, 67881, 6, 36, 20, 12, 3, 7203, 14688, 162, 350, 6, 81, 75, 9, 24, 25, 3648, 37905, 2125, 3, 18, 455, 225, 27, 3800, 81, 12, 343, 54, 26730, 1540, 180, 6, 14, 48, 5, 10010, 96348, 798, 49, 360, 9, 45, 3430, 192, 126, 36, 3, 225, 729, 648
Offset: 1

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Author

Michel Lagneau, Aug 15 2013

Keywords

Comments

Smallest k such that A005063(k) = A008472(n+k), where A008472(n) is the sum of the distinct primes dividing n and A005063(n) is the sum of squares of primes dividing n.

Examples

			a(2) = 810 because the prime divisors of 810 are {2, 3, 5}, the prime divisors of 810 + 2 = 812 are {2, 7, 29} and 2^2 + 3^2 + 5^2 = 2 + 7 + 29 = 38, hence 810 is in the sequence.

Crossrefs

Cf. A005063, A008472, A228181.

Programs

  • Mathematica
    sk[n_]:=Module[{k=1},While[Plus@@(First@#&/@FactorInteger[k]^2)!=Plus@@(First@#&/@FactorInteger[n+k]),k++];k];Array[sk,65,1]

A280385 a(n) = Sum_{k=1..n} prime(k)^2*floor(n/prime(k)) .

Original entry on oeis.org

0, 4, 13, 17, 42, 55, 104, 108, 117, 146, 267, 280, 449, 502, 536, 540, 829, 842, 1203, 1232, 1290, 1415, 1944, 1957, 1982, 2155, 2164, 2217, 3058, 3096, 4057, 4061, 4191, 4484, 4558, 4571, 5940, 6305, 6483, 6512, 8193, 8255, 10104, 10229, 10263, 10796, 13005, 13018, 13067, 13096, 13394, 13567, 16376, 16389, 16535
Offset: 1

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Author

Ilya Gutkovskiy, Jan 01 2017

Keywords

Comments

Sum of all squares of prime divisors of all positive integers <= n.
Partial sums of A005063.

Examples

			For n = 6 the prime divisors of the first six positive integers are {0}, {2}, {3}, {2}, {5}, {2, 3} so a(6) = 0^2 + 2^2 + 3^2 + 2^2 + 5^2 + 2^2 + 3^2 = 55.

Links

Crossrefs

Cf. A001157, A005063, A008472, A024924, A064602, A279847, A279910.

Programs

  • Mathematica
    Table[Sum[Prime[k]^2 Floor[n/Prime[k]], {k, 1, n}], {n, 55}]
Table[Sum[DivisorSum[k, #1^2 &, PrimeQ[#1] &], {k, 1, n}], {n, 55}]
nmax = 55; Rest[CoefficientList[Series[(1/(1 - x)) Sum[Prime[k]^2 x^Prime[k]/(1 - x^Prime[k]), {k, 1, nmax}], {x, 0, nmax}], x]]
  • PARI
    a(n) = sum(k=1, n, prime(k)^2 * (n\prime(k))); \\ Indranil Ghosh, Apr 03 2017
  • Python
    from sympy import prime
print([sum([prime(k)**2 * (n//prime(k)) for k in range(1, n + 1)]) for n in range(1, 21)]) # Indranil Ghosh, Apr 03 2017

Formula

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

A300521 Expansion of Product_{k>=1} (1 - x^prime(k))^prime(k).

Original entry on oeis.org

1, 0, -2, -3, 1, 1, 3, 0, 9, 8, 4, -31, -12, -13, 20, -13, 48, -17, 74, -87, 8, -143, 175, -174, 349, -164, 369, -651, 520, -1004, 1142, -1218, 1652, -1739, 3291, -3933, 3546, -5743, 6170, -8022, 11435, -13230, 17196, -18706, 22958, -31884, 38420, -49802, 58916
Offset: 0

Views

Author

Ilya Gutkovskiy, Mar 08 2018

Keywords

Crossrefs

Cf. A000040, A000607, A005063, A007441, A030009, A046675, A073592, A219224, A291647, A291696.

Programs

  • Mathematica
    nmax = 48; CoefficientList[Series[Product[(1 - x^Prime[k])^Prime[k], {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 48; CoefficientList[Series[Exp[-Sum[DivisorSum[k, Boole[PrimeQ[#]] #^2 &] x^k/k, {k, 1, nmax}]], {x, 0, nmax}], x]

Formula

G.f.: Product_{k>=1} (1 - x^A000040(k))^A000040(k).
G.f.: exp(-Sum_{k>=1} A005063(k)*x^k/k).

A372620 Expansion of Sum_{k>=1} k * prime(k) * x^prime(k) / (1 - x^prime(k)).

Original entry on oeis.org

0, 2, 6, 2, 15, 8, 28, 2, 6, 17, 55, 8, 78, 30, 21, 2, 119, 8, 152, 17, 34, 57, 207, 8, 15, 80, 6, 30, 290, 23, 341, 2, 61, 121, 43, 8, 444, 154, 84, 17, 533, 36, 602, 57, 21, 209, 705, 8, 28, 17, 125, 80, 848, 8, 70, 30, 158, 292, 1003, 23, 1098, 343, 34, 2, 93
Offset: 1

Views

Author

Ilya Gutkovskiy, May 07 2024

Keywords

Examples

			a(60) = a(2^2 * 3 * 5) = a(prime(1)^2 * prime(2) * prime(3)) = 1 * 2 + 2 * 3 + 3 * 5 = 23.

Links

Crossrefs

Cf. A000720, A001414, A005063, A008472, A056239, A061150, A066328, A328639.

Programs

  • Mathematica
    nmax = 65; CoefficientList[Series[Sum[k Prime[k] x^Prime[k]/(1 - x^Prime[k]), {k, 1, nmax}], {x, 0, nmax}], x] // Rest
a[n_] := Plus @@ (PrimePi[#[[1]]] #[[1]] & /@ FactorInteger[n]); Table[a[n], {n, 1, 65}]

Formula

L.g.f.: -log( Product_{k>=1} (1 - x^prime(k))^k ).
If n = Product (p_j^k_j) then a(n) = Sum (pi(p_j) * p_j), where pi = A000720.
Previous Showing 31-35 of 35 results.

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