

A279237


Let k_i be the multiplicity of prime(i) in the prime factorization of the nth composite number C_n, and let k_i=0 if prime(i) is not a factor of C_n. Then a(n)=1*k_1+10*k_2+100*k_3+...+10^N*k_N, where N is the index of the largest prime factor in C_n.


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2, 11, 3, 20, 101, 12, 1001, 110, 4, 21, 102, 1010, 10001, 13, 200, 100001, 30, 1002, 111, 5, 10010, 1000001, 1100, 22, 10000001, 100010, 103, 1011, 10002, 120, 100000001, 14, 2000, 201, 1000010, 100002, 31, 10100, 1003, 10000010, 1000000001, 112, 10000000001, 1020, 6, 100100, 10011, 1000002
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OFFSET

1,1


LINKS



FORMULA

a(n) = 1*k_1+10*k_2+100*k_3+...+10^N*k_N, where k_i is the exponent of prime(i) in the factorization of the nth composite number C_n, k_i=0 if prime(i) is not a factor in C_n. Also, N is the index of the largest prime factor of C_n, so that C_n = Product_{i=1..N} prime(i)^k_i.


EXAMPLE

The 1st composite number is 4 = 2^2, so a(1)=2.
The 2nd composite number is 6 = 3^1*2^1, so a(2)=11.
The 3rd composite number is 8 = 2^3, so a(3)=3.
The 4th composite number is 9 = 3^2*2^0, so a(4)=20.
The 5th composite number is 10 = 5^1*3^0*2^1, so a(5)=101.
The 6th composite number is 12 = 3^1*2^2, so a(6)=12.
The 7th composite number is 14 = 7^1*5^0*3^0*2^1, so a(7)=1001.
The 8th composite number is 15 = 5^1*3^1*2^0, so a(8)=110.
The 9th composite number is 16 = 2^4, so a(9)=4.
The 10th composite number is 18 = 3^2*2^1, so a(10)=21.


MATHEMATICA

Map[FromDigits@ Reverse@ Function[w, ReplacePart[#, Flatten@ Map[{PrimePi@ #1 > #2} & @@ # &, w]] &@ ConstantArray[0, PrimePi@ Max@ w[[All, 1]]]]@ FactorInteger@ # &, Select[Range[4, 120], CompositeQ]] (* Michael De Vlieger, Dec 10 2016 *)


CROSSREFS



KEYWORD

nonn,base


AUTHOR



STATUS

approved



