A279513 Multiplicative with a(p^k) = p*a(k) for any prime p and k>0.

1, 2, 3, 4, 5, 6, 7, 6, 6, 10, 11, 12, 13, 14, 15, 8, 17, 12, 19, 20, 21, 22, 23, 18, 10, 26, 9, 28, 29, 30, 31, 10, 33, 34, 35, 24, 37, 38, 39, 30, 41, 42, 43, 44, 30, 46, 47, 24, 14, 20, 51, 52, 53, 18, 55, 42, 57, 58, 59, 60, 61, 62, 42, 12, 65, 66, 67, 68

Offset: 1

Author

Rémy Sigrist, Dec 13 2016

Keywords

• nonn
• mult

Comments

To compute a(n): multiply (with multiplicity) each prime number appearing in the prime tower factorization of n (see A182318 for definition).

a(n) = n if n is squarefree.

a(n) <= A000026(n) for any n>0.

First differs from A000026 at n=256: a(256)=12 and A000026(256)=16.

If n = p_1 * p_2 * ... * p_k with p_1, p_2, ..., p_k primes, then a(p_1 ^ p_2 ^ ... ^ p_k) = n.

Examples

			a(6!) = a(2^(2^2)*3^2*5) = 2*2*2*3*2*5 = 240.
		

References

• Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.33 Hall-Montgomery Constant, p. 207.

Links

• Rémy Sigrist, Table of n, a(n) for n = 1..10000
• Richard Alan Gilman and Rainer Tschiersch, Problem 6660, The American Mathematical Monthly, Vol. 100, No. 3 (1993), pp. 296-298.

Crossrefs

Cf. A000026, A182318, A279510.

Programs

Maple

a:= proc(n) option remember; `if`(n=1, 1,
      mul(i[1]*a(i[2]), i=ifactors(n)[2]))
    end:
seq(a(n), n=1..100);  # Alois P. Heinz, Aug 22 2020

Mathematica

a[n_] := a[n] = If[n==1, 1, Times @@ (#[[1]] a[#[[2]]]& /@ FactorInteger[n] )]; Array[a, 256] (* Jean-François Alcover, Mar 31 2017 *)

PARI

a(n) =  my (f=factor(n)); return (prod(i=1, #f~, f[i,1]*a(f[i,2])))

Formula

Sum_{k=1..n} a(k) ~ (1/2) * c * n^2, where c = Product_{p prime} (1 - 1/p^2 + (p-1)*Sum_{k>=2} a(k)/p^(2*k)) = 0.8351076361... (Gilman and Tschiersch, 1993; Finch, 2003; the constant was calculated by Kevin Ford). - Amiram Eldar, Nov 04 2022