A073904 - OEIS

%I #43 Dec 21 2024 03:11:25

%S 1,2,9,8,625,12,117649,24,36,80,25937424601,60,23298085122481,448,

%T 2025,384,48661191875666868481,180,104127350297911241532841,240,35721,

%U 11264,907846434775996175406740561329,360,10000,53248,26244,1344

%N Smallest multiple k*n of n having n divisors.

%C Smallest refactorable number, m, such that m=k*n has n divisors. - _Robert G. Wilson v_, Oct 31 2005

%H Jon E. Schoenfield, <a href="/A073904/b073904.txt">Table of n, a(n) for n = 1..388</a> (first 115 terms from Carole Dubois)

%H Carole Dubois, <a href="/A073904/a073904.jpg">semi-Logarithmic pin plot of A073904(n)</a>

%F If p is a prime then a(p) = p^(p-1). If n = p^2 then a(n) = 2^(p-1)*p^(p-1).

%F a(p^r) = (2*3*5*...*p_r)^(p-1) for r < p <= p_r. a(p^r) = (2*3*...*p_(r-1))^(p-1)*p^(p-1) for p > p_r. Else a(p^r) = ...? for r >= p. Problem a(2^r) = ...? Cf. A005179(p^n)=(2*3*...*p_n)^(p-1) for p_n < 2^p. - _Thomas Ordowski_, Aug 20 2005

%F a(p^r) = (2*3...*p_(r-1)*p)^(p-1) for p > p_r; else a(p^r) = (2*3...*p...*p_m)^(p-1)*p^(p^k-p) for p <= p_r and p_m < 2^p, where m=r-k+1 for smallest k such that p^k > r, so k=floor(log(r)/log(p))+1 and p > log(p_m)/log(2). Examples: If k=1 then a(p^r) = (2*3*...*p_r)^(p-1) for r < p <= p_r. If p=2 then a(2^r) = (2*3*...*p_m)*2^(2^k-2) for r < 5. For instance, let r=4 so k=3, m=2 and a(2^4)=384. - _Thomas Ordowski_, Aug 22 2005

%F If p is a prime and n=p^r then a(p^r) = (s_1*s_2*...*s_r)^(p-1) where (s_r) is a permutation of the (ascending sequence) numbers of the form q^(p^j) for every prime q and j>=0; permutation such that s_(p^j)=p^(p^j) and shifted remainder. For example, if p=3 then (s_r): 3, 2, 3^3, 5, 7, 2^3, 11, 13, 3^9, 17, 19, ... so a(3^r) = (3*2*27*5*...*s_r)^2. - _Thomas Ordowski_, Aug 29 2005

%F If n=2^r then a(2^r) is the product of the first r members of the A109429 sequence. - _Thomas Ordowski_, Aug 29 2005

%F a(n) = n * A076931(n). - _Thomas Ordowski_, Oct 07 2005

%F a(4) = 8; a(2*prime(n)) = A299795(n), for n>1. - _Bernard Schott_, Nov 06 2022

%e Smallest multiple a(n)=k*n; a(1)=1*1, a(2)=1*2, a(3)=3*3, a(4)=2*4, a(5)=125*5, a(6)=2*6, ... having d(k*n)=n divisors; d(1)=1, d(2)=2, d(3^2)=3, d(2^3)=4, d(5^4)=5, d(2^2*3)=3*2=6, ...

%t f[n_] := Block[{k = 1}, If[ PrimeQ[n], n^(n - 1), While[d = DivisorSigma[0, k*n]; d != n, k++ ]; k*n]]; Table[ f[n], {n, 28}] (* _Robert G. Wilson v_ *)

%Y Cf. A076931, A050376, A005179, A037992, A050376, A111172, A299795.

%Y Cf. A033950 (refactorable numbers, also known as tau numbers).

%Y Cf. A110821 (SuperRefactorable numbers).

%K nice,nonn

%O 1,2

%A _Amarnath Murthy_, Aug 18 2002

%E a(12) corrected by _Thomas Ordowski_, Aug 18 2005

%E Further corrections from _Thomas Ordowski_, Oct 07 2005

%E a(21), a(27) & a(28) from _Robert G. Wilson v_, Oct 31 2005