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A101278 Write n in base 3 as n = b_0 + b_1*3 + b_2*3^2 + b_3*3^3 + ...; then a(n) = Product_{i >= 0} prime(i+1)^b_i.

%I A101278 #28 Sep 04 2025 12:56:55
%S A101278 1,2,4,3,6,12,9,18,36,5,10,20,15,30,60,45,90,180,25,50,100,75,150,300,
%T A101278 225,450,900,7,14,28,21,42,84,63,126,252,35,70,140,105,210,420,315,
%U A101278 630,1260,175,350,700,525,1050,2100,1575,3150,6300,49,98,196,147,294,588
%N A101278 Write n in base 3 as n = b_0 + b_1*3 + b_2*3^2 + b_3*3^3 + ...; then a(n) = Product_{i >= 0} prime(i+1)^b_i.
%C A101278 A permutation of the cubefree numbers (A004709). - _Rémy Sigrist_, Jul 18 2022
%C A101278 These are cubefree numbers organized by the highest factor. By converting to a different base, we avoid the row-by-row triangular entry used in the analogous squarefree A339195. - _Gordon Hamilton_, Aug 13 2025
%H A101278 Reinhard Zumkeller, <a href="/A101278/b101278.txt">Table of n, a(n) for n = 0..6560</a>
%F A101278 If a(bn)=x then a(bn+1)=2x, a(bn+2)=4x, ... a(bn+b-1)=2^b*x. - _Robert G. Wilson v_, Dec 24 2004
%F A101278 G.f.: (1+2x+4x^2)(1+3x^3+9x^6)(1+5x^9+25x^18)... - _Paul Boddington_, Jul 21 2005
%F A101278 a(n) = f(n, 1, 1) with f(x, y, z) = if x > 0 then f(floor(x/3), y*prime(z)^(x mod 3), z+1) else y. - _Reinhard Zumkeller_, Mar 13 2010
%e A101278 The first few terms are computed as follows:
%e A101278   n b2 b1 b0 a(n)
%e A101278   0, 0, 0, 0,  1
%e A101278   1, 0, 0, 1,  2
%e A101278   2, 0, 0, 2,  4
%e A101278   3, 0, 1, 0,  3
%e A101278   4, 0, 1, 1,  6
%e A101278   5, 0, 1, 2, 12
%e A101278 a(11) = a(102_3) and so we get prime(3)^1 * prime(2)^0 * prime(1)^2 = 5^1 * 3^0 * 2^2 = 5 * 1 * 4 = 20. - _Gordon Hamilton_, Aug 13 2025
%t A101278 primeBase[n_Integer?Positive, base_Integer]/;base>1 := Times @@ (Table[Prime[i], {i, Floor[Log[base, n] + 1], 1, -1}]^IntegerDigits[n, base]); Table[primeBase[n, 3], {n, 59}] (* _Robert G. Wilson v_, Dec 24 2004 *)
%o A101278 (PARI) a(n) = {my(d = digits(n, 3), pr = primes(#d)); prod(i = 1, #d, pr[#d + 1 - i]^d[i])} \\ David A. Corneth, Aug 13 2025
%Y A101278 Cf. A019565 (base 2), A101942 (base 4), A101943 (base 5), A054842 (base 10).
%Y A101278 Cf. A004709, A339195.
%K A101278 nonn,easy,base,changed
%O A101278 0,2
%A A101278 Orges Leka (oleka(AT)students.uni-mainz.de), Dec 20 2004
%E A101278 More terms from _Robert G. Wilson v_, Dec 24 2004