A054841 If n = 2^a * 3^b * 5^c * 7^d * ... then a(n) = a + 10 * b + 100 * c + 1000 * d + ... .

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

Offset: 1

Author

Henry Bottomley, Apr 11 2000

Keywords

• base
• nonn

Comments

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 mM. 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

Examples

			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.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000 (n=1..1023 from Michael De Vlieger)
• Evans A Criswell, A Sequence Puzzle (Posted to rec.puzzles Jan 01 1997)
• Walter Nissen, Exponential Prime Power Representation, sci.math newsgroup, May 23 1995.

Crossrefs

Row 10 of A104244.

Left inverse of A054842.

Cf. A001222, A048675, A090880, A090881, A090882, A276075, A276085 (analogous constructions for other bases), A090883, A090884, A049084, A027748, A124010, A056239.

Programs

Haskell

a054841 1 = 0
a054841 n = sum $ zipWith (*)
                  (map ((10 ^) . subtract 1 . a049084) $ a027748_row n)
                  (map fromIntegral $ a124010_row n)
-- Reinhard Zumkeller, Aug 03 2015

Maple

A:= n -> add(t[2]*10^(numtheory:-pi(t[1])-1),t= ifactors(n)[2]);
seq(A(n), n=1..1000); # Robert Israel, Jul 24 2014

Mathematica

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]] & /@
      Flatten[ConstantArray @@@ FactorInteger[n]]]]]] (* Michael De Vlieger, Jul 24 2014 *)

PARI

A054841(n)=sum(i=1,#n=factor(n)~,10^primepi(n[1,i])*n[2,i])/10 \\ M. F. Hasler, Nov 16 2008

Python

from sympy import factorint, primepi
def a(n): return sum(e*10**(primepi(p)-1) for p, e in factorint(n).items())
print([a(n) for n in range(1, 41)]) # Michael S. Branicky, Mar 17 2024

Formula

a(m*n) = a(m) + a(n) for all m,n > 0. A007953(a(n))=A001222(n) for all n such that A051903(n) < 10. - M. F. Hasler, Nov 16 2008

a(n) = sum(10^(A049084(A027748(k))-1) * A124010(k): k = 1..A001221(n)). - Reinhard Zumkeller, Aug 03 2015

a(A054842(n)) = n for all n >= 0. - Antti Karttunen, Aug 29 2016

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