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A054841
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If n = 2^a * 3^b * 5^c * 7^d * ... then a(n) = a + 10 * b + 100 * c + 1000 * d + ... .
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29
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0, 1, 10, 2, 100, 11, 1000, 3, 20, 101, 10000, 12, 100000, 1001, 110, 4, 1000000, 21, 10000000, 102, 1010, 10001, 100000000, 13, 200, 100001, 30, 1002, 1000000000, 111, 10000000000, 5, 10010, 1000001, 1100, 22, 100000000000, 10000001, 100010
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OFFSET
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1,3
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COMMENTS
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Are there any other numbers besides n=12 for which n=a(n) ? - Ctibor O. Zizka, Oct 08 2008
The sequence is a morphism from (N*,*) into (N,+), cf. formula. Up to n=1023, the digit sum A007953(a(n)) equals Omega(n) = A001222(n). This holds whenever A051903(n)<10. Restricted to these n, the sequence is also injective. However, when n is a multiple of 2^10, 3^10, 5^10 etc, then a(n) is equal to some a(m) with m<n. - M. F. Hasler, Nov 16 2008
This has been called the "Exponential Prime Power Representation" of n by W. Nissen in a post to the sci.math newsgroup (where probably some more sophisticated convention for representing digits > 10 would be used). - M. F. Hasler, Jul 03 2016
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LINKS
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FORMULA
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a(n) = Sum_{i>0} e_i*10^(i-1) when n = Product_{i>0} prime(i)^e_i. - M. F. Hasler, Mar 14 2018
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EXAMPLE
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a(25) = 200 because 25 = 5^2 * 3^0 * 2^0.
a(1024) = 10 = a(3), because 1024 = 2^10; but this two-digit multiplicity overflows into the 10^1 position, which encodes for powers of three.
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MAPLE
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A:= n -> add(t[2]*10^(numtheory:-pi(t[1])-1), t= ifactors(n)[2]);
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MATHEMATICA
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a054841[n_Integer] := Catch[FromDigits[IntegerDigits[Apply[Plus,
Which[n == 0, Throw["undefined"],
n == 1, 0,
Max[Last /@ FactorInteger @ n] > 9, Throw["overflow"],
True, Power[10, PrimePi[Abs[#]] - 1]] & /@
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PROG
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(PARI) A054841(n)=sum(i=1, #n=factor(n)~, 10^primepi(n[1, i])*n[2, i])/10 \\ M. F. Hasler, Nov 16 2008
(Haskell)
a054841 1 = 0
a054841 n = sum $ zipWith (*)
(map ((10 ^) . subtract 1 . a049084) $ a027748_row n)
(map fromIntegral $ a124010_row n)
(Python)
from sympy import factorint, primepi
def a(n): return sum(e*10**(primepi(p)-1) for p, e in factorint(n).items())
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CROSSREFS
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Cf. A001222, A048675, A090880, A090881, A090882, A276075, A276085 (analogous constructions for other bases), A090883, A090884, A049084, A027748, A124010, A056239.
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KEYWORD
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base,nonn
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AUTHOR
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STATUS
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approved
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