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A003963 Fully multiplicative with a(p) = k if p is the k-th prime. 70
1, 1, 2, 1, 3, 2, 4, 1, 4, 3, 5, 2, 6, 4, 6, 1, 7, 4, 8, 3, 8, 5, 9, 2, 9, 6, 8, 4, 10, 6, 11, 1, 10, 7, 12, 4, 12, 8, 12, 3, 13, 8, 14, 5, 12, 9, 15, 2, 16, 9, 14, 6, 16, 8, 15, 4, 16, 10, 17, 6, 18, 11, 16, 1, 18, 10, 19, 7, 18, 12, 20, 4, 21, 12, 18, 8, 20, 12, 22, 3, 16, 13, 23, 8, 21, 14, 20, 5 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

a(n) is the Matula number of the rooted tree obtained from the rooted tree T having Matula number n, by contracting its edges that emanate from the root. Example: a(49) = 16. Indeed, the rooted tree with Matula number 49 is the tree obtained by merging two copies of the tree Y at their roots. Contracting the two edges that emanate from the root, we obtain the star tree with 4 edges having Matula number 16. - Emeric Deutsch, May 01 2015

The Matula (or Matula-Goebel) number of a rooted tree can be defined in the following recursive manner: to the one-vertex tree there corresponds the number 1; to a tree T with root degree 1 there corresponds the t-th prime number, where t is the Matula-Goebel number of the tree obtained from T by deleting the edge emanating from the root; to a tree T with root degree m>=2 there corresponds the product of the Matula-Goebel numbers of the m branches of T. - Emeric Deutsch, May 01 2015

a(n) = the product of the parts of the partition having Heinz number n. We define the Heinz number of a partition p = [p_1, p_2, ..., p_r] as Product(p_j-th prime, j=1...r) (concept used by Alois P. Heinz in A215366 as an "encoding" of a partition). For example, for the partition [1, 1, 2, 4, 10] we get 2*2*3*7*29 = 2436. Example: a(75) = 18; indeed, the partition having Heinz number 75 = 3*5*5 is [2,3,3] and 2*3*3 = 18. - Emeric Deutsch, Jun 03 2015

LINKS

T. D. Noe, Table of n, a(n) for n = 1..10000

E. Deutsch, Rooted tree statistics from Matula numbers, Discrete Appl. Math., 160, 2012, 2314-2322.

F. Göbel, On a 1-1-correspondence between rooted trees and natural numbers, J. Combin. Theory, B 29 (1980), 141-143.

I. Gutman and A. Ivic, On Matula numbers, Discrete Math., 150, 1996, 131-142.

I. Gutman and Yeong-Nan Yeh, Deducing properties of trees from their Matula numbers, Publ. Inst. Math., 53 (67), 1993, 17-22.

D. W. Matula, A natural rooted tree enumeration by prime factorization, SIAM Rev. 10 (1968) 273.

Index entries for sequences related to Matula-Göbel numbers

Index entries for sequences computed from indices in prime factorization

FORMULA

If n = product prime(k)^e(k) then a(n) = product k^e(k).

Multiplicative with a(p^e) = A000720(p)^e. - David W. Wilson, Aug 01 2001

a(n) = Product(A049084(A027748(n,k))^A124010(n,k): k=1..A001221(n)). - Reinhard Zumkeller, Jun 30 2012

Rec. eq.: a(1)=1, a(k-th prime) = a(k), a(rs)=a(r)a(s). The Maple program is based on this. - Emeric Deutsch, May 01 2015

a(n) = A243504(A241909(n)) = A243499(A156552(n)) = A227184(A243354(n)) - Antti Karttunen, Mar 07 2017

MAPLE

with(numtheory): a := proc (n) local r, s: r := proc (n) options operator, arrow: op(1, factorset(n)) end proc: s := proc (n) options operator, arrow: n/r(n) end proc: if n = 1 then 1 elif bigomega(n) = 1 then pi(n) else a(r(n))*a(s(n)) end if end proc: seq(a(n), n = 1 .. 88);

# Alternative:

seq(mul(numtheory:-pi(t[1])^t[2], t=ifactors(n)[2]), n=1..100); # Robert Israel, May 01 2015

MATHEMATICA

a[n_] := Times @@ (PrimePi[ #[[1]] ]^#[[2]]& /@ FactorInteger[n]); a[1] = 1; Table[a[n], {n, 1, 88}]

PROG

(PARI) a(n)=f=factor(n); prod(i=1, #f[, 1], primepi(f[i, 1])*f[i, 2]) \\ Charles R Greathouse IV, Apr 26 2012

(PARI) a(n) = {f = factor(n); for (i=1, #f~, f[i, 1] = primepi(f[i, 1]); ); factorback(f); } \\ Michel Marcus, Feb 08 2015

(PARI) A003963(n)={n=factor(n); n[, 1]=apply(primepi, n[, 1]); factorback(n)} \\ M. F. Hasler, May 03 2018

(Haskell)

a003963 n = product $

   zipWith (^) (map a049084 $ a027748_row n) (a124010_row n)

-- Reinhard Zumkeller, Jun 30 2012

CROSSREFS

Cf. A000720, A001221, A001222, A027748, A049084, A056239, A064553, A124010, A156552, A215366, A227184, A241909, A243354, A243499, A243504.

Product of entries on row n of A112798.

Sequence in context: A136523 A228731 A163507 * A003960 A243499 A124223

Adjacent sequences:  A003960 A003961 A003962 * A003964 A003965 A003966

KEYWORD

nonn,nice,easy,mult

AUTHOR

Marc LeBrun

STATUS

approved

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Last modified August 18 04:26 EDT 2018. Contains 313821 sequences. (Running on oeis4.)