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A270142 a(n) = product of first k composites, with the i-th composite raised to the d-th power, where k = A055642(n) and d is the i-th digit of n.

%I A270142 #20 Dec 21 2022 12:01:09
%S A270142 4,16,64,256,1024,4096,16384,65536,262144,4,24,144,864,5184,31104,
%T A270142 186624,1119744,6718464,40310784,16,96,576,3456,20736,124416,746496,
%U A270142 4478976,26873856,161243136,64,384,2304,13824,82944,497664,2985984,17915904,107495424
%N A270142 a(n) = product of first k composites, with the i-th composite raised to the d-th power, where k = A055642(n) and d is the i-th digit of n.
%C A270142 All terms are multiples of 4, since A002808(1) = 4 and the most significant digit of n is always nonzero.
%C A270142 Does a term exist such that a(n) = n? Such a number would be the analog of a Meertens number when raising composites to the powers of the digits of n instead of raising primes to the powers of the digits.
%C A270142 From _Chai Wah Wu_, Dec 15 2022: (Start)
%C A270142 If a(n) is defined using digits of n in base b, then there are bases b and numbers n such that a(n) = n. For instance:
%C A270142 base b             n
%C A270142 ------------------------------------------------
%C A270142 2                  4, 24, 36, 24192000, 85155840
%C A270142 3                  2592
%C A270142 4                  4, 103680
%C A270142 6                  20736
%C A270142 8                  16, 256, 13824
%C A270142 12                 1327104
%C A270142 16                 21233664
%C A270142 23                 24
%C A270142 24                 746496
%C A270142 (End)
%H A270142 Felix Fröhlich, <a href="/A270142/b270142.txt">Table of n, a(n) for n = 1..10000</a>
%e A270142 a(12) = 144, since A002808(1) = 4, A002808(2) = 6 and 4^1 * 6^2 = 144.
%o A270142 (PARI) composite(n) = my(i=0, c=2); while(1, if(!ispseudoprime(c), i++); if(i==n, return(c)); c++)
%o A270142 compopowerprod(n) = my(d=digits(n)); for(k=1, #d, p=prod(i=1, #d, composite(i)^d[i])); p
%o A270142 a(n) = compopowerprod(n)
%o A270142 (Python)
%o A270142 from math import prod
%o A270142 from sympy import composite
%o A270142 def A270142(n): return prod(composite(i)**int(d) for i, d in enumerate(str(n),1)) # _Chai Wah Wu_, Dec 09 2022
%Y A270142 Cf. A189398, A246532.
%K A270142 nonn,base,easy
%O A270142 1,1
%A A270142 _Felix Fröhlich_, Mar 12 2016