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.

A322375 a(n) = A318366(A025487(n)).

Original entry on oeis.org

0, 0, 1, 2, 4, 8, 10, 20, 12, 20, 24, 40, 34, 35, 52, 70, 72, 56, 95, 84, 112, 48, 104, 130, 84, 156, 164, 168, 116, 180, 212, 120, 238, 280, 240, 224, 284, 189, 322, 165, 304, 344, 258, 300, 438, 330, 380, 420, 348, 464, 220, 500, 477, 160, 472, 460, 644, 440, 592
Offset: 1

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Author

David A. Corneth, Jan 12 2019

Keywords

Comments

A318366(n) computes a sum over the divisors of n in such a way that A318366(n) only depends on the prime signature of n. A025487 lists least numbers with a given prime signature with the exponents in nonincreasing order. This sequence hence shows this sum over divisors for distinct prime signatures.

Examples

			a(8) = A318366(A025487(8)) = A318366(24) = 20 (See example of finding A318366 at that sequence).

Links

Crossrefs

Cf. A025487, A318366.

A331669 List of distinct numbers that occur in A318366 (the Dirichlet convolution square of bigomega).

Original entry on oeis.org

0, 1, 2, 4, 8, 10, 12, 20, 24, 34, 35, 40, 48, 52, 56, 70, 72, 84, 95, 104, 112, 116, 120, 130, 156, 160, 164, 165, 168, 180, 189, 212, 220, 224, 238, 240, 258, 280, 284, 286, 300, 304, 322, 330, 344, 348, 352, 364, 380, 420, 438, 440, 455, 460, 464, 472, 477, 480
Offset: 1

Views

Author

Torlach Rush, Jan 23 2020

Keywords

Comments

There is a strong correlation between values of this function and values of other arithmetic functions. In other words, a(n) correlates to a single distinct value from one or more of the arithmetic functions.
Terms of this sequence select from the positive integers as follows:
A318366(k) = a(1), 1 followed by the primes (A008578).
A318366(k) = A008836(k) = A001221(k) = a(2), primes squared (A001248).
A318366(k) = A001221(k) = a(3), squarefree semiprimes (A006881).
A318366(k) = A000005(k) = a(4), primes cubed (A030078).
A318366(k) = a(5), a prime squared times a prime (A054753).
A318366(k) = a(6), primes to the fourth power (A030514).
A318366(k) = a(7), sphenic numbers (A007304).
A318366(k) = a(8), union of A050997 and A065036.
A318366(k) = a(9), squarefree semiprimes squared (A085986).
A318366(k) = a(10), product of four primes, three distinct (A085987).
A318366(k) = a(11), primes to the sixth power (A030516).
A318366(k) = a(12), product of prime to fourth power and a different prime (A178739).
A318366(k) = a(13), product of four distinct primes (A046386).
...

Examples

			0 is a term because the only divisors of a prime (p) are 1 and a prime itself and bigomega(1) * bigomega(p) + bigomega(p) * bigomega(1) = 0 * 1 + 1 * 0 = 0.
1 is a term because a prime squared gives bigomega(1) * bigomega(p^2) + bigomega(p) * bigomega(p) + bigomega(p^2) * bigomega(1) = 0 * 2 + 1 * 1 + 2 * 0 = 1.

Crossrefs

Cf. A001222, A318366.
Cf. also A101296.

Extensions

More terms, using A318366 extended b-file, from Michel Marcus, Jan 24 2020

A328486 Dirichlet g.f.: zeta(s)^4 * (1 - 2^(-s))^2.

Original entry on oeis.org

1, 2, 4, 3, 4, 8, 4, 4, 10, 8, 4, 12, 4, 8, 16, 5, 4, 20, 4, 12, 16, 8, 4, 16, 10, 8, 20, 12, 4, 32, 4, 6, 16, 8, 16, 30, 4, 8, 16, 16, 4, 32, 4, 12, 40, 8, 4, 20, 10, 20, 16, 12, 4, 40, 16, 16, 16, 8, 4, 48, 4, 8, 40, 7, 16, 32, 4, 12, 16, 32, 4, 40, 4, 8, 40, 12, 16, 32, 4, 20
Offset: 1

Views

Author

Ilya Gutkovskiy, Oct 16 2019

Keywords

Comments

Dirichlet convolution of A001227 with itself.

Links

Crossrefs

Cf. A000005, A001227, A007426, A070288, A318366, A328487.

Programs

  • Maple
    with(numtheory):
b:= proc(n) option remember; tau(2*n)-tau(n) end:
a:= n-> add(b(d)*b(n/d), d=divisors(n)):
seq(a(n), n=1..100);  # Alois P. Heinz, Oct 16 2019
  • Mathematica
    nmax = 80; A001227 = Table[DivisorSum[n, Mod[#, 2] &], {n, 1, nmax}]; Table[DivisorSum[n, A001227[[#]] A001227[[n/#]] &], {n, 1, nmax}]
f[2, e_] := e + 1; f[p_, e_] := (e + 1)*(e + 2)*(e + 3)/6; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Nov 30 2020 *)

Formula

a(n) = Sum_{d|n} A001227(d) * A001227(n/d).
Sum_{k=1..n} a(k) ~ n * (log(n)^3/24 + (g/2 + log(2)/4 - 1/8)* log(n)^2 + (1/4 - g + 3*g^2/2 - log(2)/2 + 2*g*log(2) - sg1)* log(n) - 1/4 + (1 - 2*log(2))*g + (3*log(2) - 3/2)*g^2 + g^3 + log(2)/2 - log(2)^3/6 + (1 - 3*g - 2*log(2))* sg1 + sg2/2), where g is the Euler-Mascheroni constant A001620 and sg1, sg2 are the Stieltjes constants, see A082633 and A086279. - Vaclav Kotesovec, Oct 17 2019
Multiplicative with a(2^e) = e + 1, and a(p^e) = (e + 1)*(e + 2)*(e + 3)/6 for odd primes p. - Amiram Eldar, Nov 30 2020

A349711 a(n) = Sum_{d|n} sopfr(d) * sopfr(n/d).

Original entry on oeis.org

0, 0, 0, 4, 0, 12, 0, 16, 9, 20, 0, 44, 0, 28, 30, 40, 0, 54, 0, 68, 42, 44, 0, 104, 25, 52, 36, 92, 0, 124, 0, 80, 66, 68, 70, 147, 0, 76, 78, 152, 0, 164, 0, 140, 108, 92, 0, 200, 49, 110, 102, 164, 0, 144, 110, 200, 114, 116, 0, 298, 0, 124, 144, 140, 130, 244, 0, 212, 138, 236, 0, 300, 0, 148, 140
Offset: 1

Views

Author

Ilya Gutkovskiy, Nov 26 2021

Keywords

Comments

Dirichlet convolution of A001414 with itself.

Links

Crossrefs

Cf. A000292, A001414, A034761, A318366, A349712.

Programs

  • Maple
    b:= proc(n) option remember; add(i[1]*i[2], i=ifactors(n)[2]) end:
a:= n-> add(b(d)*b(n/d), d=numtheory[divisors](n)):
seq(a(n), n=1..75);  # Alois P. Heinz, Nov 26 2021
  • Mathematica
    sopfr[1] = 0; sopfr[n_] := Plus @@ Times @@@ FactorInteger@n; a[n_] := Sum[sopfr[d] sopfr[n/d], {d, Divisors[n]}]; Table[a[n], {n, 1, 75}]
  • PARI
    sopfr(n) = (n=factor(n))[, 1]~*n[, 2]; \\ A001414
a(n) = sumdiv(n, d, sopfr(d)*sopfr(n/d)); \\ Michel Marcus, Nov 26 2021
  • Python
    from itertools import product
from sympy import factorint
def A349711(n):
    f = factorint(n)
    plist, m = list(f.keys()), sum(f[p]*p for p in f)
    return sum((lambda x: x*(m-x))(sum(d[i]*p for i, p in enumerate(plist))) for d in product(*(list(range(f[p]+1)) for p in plist))) # Chai Wah Wu, Nov 27 2021

Formula

Dirichlet g.f.: ( zeta(s) * Sum_{p prime} p/(p^s-1) )^2.
a(p^k) = (k^3-k)*p^2/6 = A000292(k-1)*p^2 for p prime. - Chai Wah Wu, Nov 28 2021
Showing 1-4 of 4 results.

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