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A054842 If n = a + 10 * b + 100 * c + 1000 * d + ... then a(n) = (2^a) * (3^b) * (5^c) * (7^d) * ...

%I A054842 #49 Nov 29 2024 08:08:58
%S A054842 1,2,4,8,16,32,64,128,256,512,3,6,12,24,48,96,192,384,768,1536,9,18,
%T A054842 36,72,144,288,576,1152,2304,4608,27,54,108,216,432,864,1728,3456,
%U A054842 6912,13824,81,162,324,648,1296,2592,5184,10368,20736,41472,243,486,972,1944
%N A054842 If n = a + 10 * b + 100 * c + 1000 * d + ... then a(n) = (2^a) * (3^b) * (5^c) * (7^d) * ...
%C A054842 a((10^k-1)/9) = Primorial(k)= A061509((10^k-1)/9). This is a rearrangement of whole numbers. a(m) = a(n) iff m = n. (Unlike A061509, in which a(n) = a(n*10^k).) - _Amarnath Murthy_ and Meenakshi Srikanth (menakan_s(AT)yahoo.com), Jul 14 2003
%C A054842 Part of the previous comment is incorrect: as a set, this sequence consists of numbers n such that the largest exponent appearing in the prime factorization of n is 9. So this cannot be a rearrangement (or permutation) of the natural numbers. - _Tom Edgar_, Oct 20 2015
%H A054842 Reinhard Zumkeller, <a href="/A054842/b054842.txt">Table of n, a(n) for n = 0..9999</a>
%F A054842 a(n) = f(n, 1, 1) with f(x, y, z) = if x > 0 then f(floor(x/10), y*prime(z)^(x mod 10), z+1) else y. - _Reinhard Zumkeller_, Mar 13 2010
%e A054842 a(15)=96 because 3^1 * 2^5 = 3*32 = 96.
%t A054842 A054842[n_] := Times @@ (Prime[Range[Length[#], 1, -1]]^#) & [IntegerDigits[n]];
%t A054842 Array[A054842, 100, 0] (* _Paolo Xausa_, Nov 25 2024 *)
%o A054842 (Haskell)
%o A054842 a054842 = f a000040_list 1 where
%o A054842    f _      y 0 = y
%o A054842    f (p:ps) y x = f ps (y * p ^ d) x'  where (x', d) = divMod x 10
%o A054842 -- _Reinhard Zumkeller_, Aug 03 2015
%o A054842 (PARI) a(n)= my(d=Vecrev(digits(n))); factorback(primes(#d), d); \\ _Ruud H.G. van Tol_, Nov 28 2024
%Y A054842 Cf. A054841, A069877, A085840.
%Y A054842 Cf. analogous sequences for other bases: A019565 (base 2), A101278 (base 3), A101942 (base 4), A101943 (base 5), A276076 (factorial base), A276086 (primorial base).
%K A054842 base,nonn,easy,look
%O A054842 0,2
%A A054842 _Henry Bottomley_, Apr 11 2000