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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A059975 For n > 1, a(n) is the least number of prime factors (counted with multiplicity) of any integer with n divisors; fully additive with a(p) = p-1.

Original entry on oeis.org

0, 1, 2, 2, 4, 3, 6, 3, 4, 5, 10, 4, 12, 7, 6, 4, 16, 5, 18, 6, 8, 11, 22, 5, 8, 13, 6, 8, 28, 7, 30, 5, 12, 17, 10, 6, 36, 19, 14, 7, 40, 9, 42, 12, 8, 23, 46, 6, 12, 9, 18, 14, 52, 7, 14, 9, 20, 29, 58, 8, 60, 31, 10, 6, 16, 13, 66, 18, 24, 11, 70, 7, 72, 37, 10, 20, 16, 15, 78, 8, 8, 41
Offset: 1

Views

Author

Yong Kong (ykong(AT)curagen.com), Mar 05 2001

Keywords

Comments

n*a(n) is the number of complex multiplications needed for the fast Fourier transform of n numbers, writing n = r1 * r2 where r1 is a prime.
This sequence with offset 1 and a(1) = 0 is completely additive with a(p^e) = e*(p-1) for prime p and e >= 0. - Werner Schulte, Feb 23 2019

Examples

			a(18) = 5 since 18 = 2*3^2, a(18) = 1*(2-1) + 2*(3-1) = 5.

References

  • Herbert S. Wilf, Algorithms and complexity, Internet Edition, Summer, 1994, p. 56.

Links

Crossrefs

Essentially same as A087656 apart from offset.
Cf. A000005, A138618, A309155, A309239, A327250, A341865, A373368 [= gcd(n, a(n))], A373369 [= gcd(A001414(n), a(n))].
Cf. A003159 (positions of even terms), A096268 (with offset 1, parity of terms), A373385 (positions of multiples of 3).
Leftmost column of irregular table A355029.
Other completely additive sequences with primes p mapped to a function of p include: A001222 (with a(p)=1), A001414 (with a(p)=p), A276085 (with a(p)=p#/p), A341885 (with a(p)=p*(p+1)/2), A373149 (with a(p)=prevprime(p)), A373158 (with a(p)=p#).

Programs

  • Maple
    A059975 := proc(n)
        local a,pf,p,e ;
        a := 0 ;
        for pf in ifactors(n)[2] do
                p := op(1,pf) ;
                e := op(2,pf) ;
                a := a+e*(p-1) ;
        end do:
        a ;
end proc: # R. J. Mathar, Oct 17 2011
  • Mathematica
    Table[Total[(First /@ FactorInteger[n] - 1) Last /@ FactorInteger[n]], {n, 1, 100}] (* Danny Marmer, Nov 13 2014 *)
f[p_, e_] := e*(p - 1); a[n_] := Plus @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Mar 27 2023 *)
  • PARI
    a(n) = {my(f = factor(n)); sum(i = 1, #f~, f[i, 2]*(f[i, 1] - 1));} \\ Amiram Eldar, Mar 27 2023

Formula

a(n) = Sum ( e_i * (p_i - 1) ) where n = Product ( p_i^e_i ) is the canonical factorization of n.
a(n) = min(A001222(x) : A000005(x)=n).
a(n) = row sums of A138618 - row products of A138618. - Mats Granvik, May 23 2013
a(n) = A001414(n) - A001222(n). - David James Sycamore, Jul 17 2019
a(n) = n - A341865(n). - Antti Karttunen, Jun 05 2024

Extensions

Definition revised by Hugo van der Sanden, May 21 2010
Term a(1)=0 prepended and Werner Schulte's comment adopted as an alternative definition - Antti Karttunen, Jun 05 2024

A355030 a(n) is the number of possible values of the number of prime divisors (counted with multiplicity) of numbers with n divisors.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 4, 1, 2, 2, 5, 1, 4, 1, 4, 2, 2, 1, 7, 2, 2, 3, 4, 1, 5, 1, 7, 2, 2, 2, 8, 1, 2, 2, 6, 1, 5, 1, 4, 4, 2, 1, 11, 2, 4, 2, 4, 1, 7, 2, 7, 2, 2, 1, 11, 1, 2, 4, 11, 2, 5, 1, 4, 2, 5, 1, 14, 1, 2, 4, 4, 2, 5, 1, 11, 5, 2, 1, 11, 2
Offset: 1

Views

Author

Amiram Eldar, Jun 16 2022

Keywords

Comments

First differs from A305254 at n = 40, from A001055 and A252665 at n = 36, from A218320 at n = 32 and from A317791, A318559 and A326334 at n = 30.

Examples

			a(2) = 1 since numbers with 2 divisors are primes, i.e., numbers k with the single value Omega(k) = 1.
a(4) = 2 since numbers with 4 divisors are either of the following 2 forms: p1 * p2 with p1 and p2 being distinct primes, or of the form p^3 with p prime.
a(8) = 3 since numbers with 8 divisors are either of the following 3 forms: p1 * p2 * p3 with p1, p2 and p3 being distinct primes, p1 * p2^3, or p1^7.

Links

Crossrefs

Row lengths of A355029.
Cf. A000005, A001222, A118914, A162247, A355027.
Cf. A001055, A218320, A305254, A317791, A318559, A326334.

Programs

  • Mathematica
    Table[Length[Union[Total[#-1]& /@ f[n]]], {n, 1, 100}] (* using the function f by T. D. Noe at A162247 *)

Formula

a(n) <= A001055(n).
a(p) = 1 for p prime.
a(A355031(n)) = n.

A355031 a(n) is the least number k such that A355030(k) = n, or -1 if no such k exists.

Original entry on oeis.org

1, 4, 8, 12, 16, 40, 24, 36, 90, 126, 48, 112, 546, 72, 108, 96, 160, 352, 168, 120, 256, 2475, 144, 588, 300, 320, 216, 448, 1216, 240, 810, 420, 288, 1040, 384, 660, 360, 640, 432, 1408, 540, 504, 480, 600, 648, 1176, 792, 672, 1500, 576, 2000, 900, 1824, 1248
Offset: 1

Views

Author

Amiram Eldar, Jun 16 2022

Keywords

Links

Crossrefs

Cf. A000005, A001222, A162247, A355029, A355030.

Programs

  • Mathematica
    m[n_] := Length[Union[Total[#-1]& /@ f[n]]]; seq[len_, max_] := Module[{s = Table[0, {len}], c = 0, n = 1, k}, While[c < len && n < max, k = m[n]; If[k <= len && s[[k]] == 0, c++; s[[k]] = n]; n++]; s]; seq[60, 10^4] (* using the function f by T. D. Noe at A162247 *)
Showing 1-3 of 3 results.

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