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Search: id:A137502
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| A137502 |
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Reverse sequence of powers in prime decomposition of n. |
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+0
1
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| 2, 2, 4, 2, 6, 2, 8, 4, 10, 2, 18, 2, 14, 6, 16, 2, 12, 2, 50, 10, 22, 2, 54, 4, 26, 8, 98, 2, 30, 2, 32, 14, 34, 6, 36, 2, 38, 22, 250, 2, 70, 2, 242, 18, 46, 2, 162, 4, 4, 26, 338, 2, 24, 10, 686, 2, 58, 2, 150, 2, 150, 50, 64, 14, 154, 2, 578, 38, 42, 2, 108, 2, 74, 12, 1058, 6, 506
(list; graph; listen)
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OFFSET
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2,1
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COMMENT
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In the prime decomposition of n we use all the primes up to the highest prime divisor, exponents of zero being allowed except for the largest prime.
If n=(p(1)^e1)*(p(2)^e2)*.......*(p(k)^ek) (ek>0, other ei>=0 and p(n) = n-th prime) then A137502 we reverse the sequence e1,e2,... ek to build a(n): a(n) = (p(1)^ek)*(p(2)^e(k-1))* . . . . *(p(k)^e1)
As p(1)=2 and ek =/=0, a(n) is always even
If n is prime then a(n) =2 and if n is a power of prime, a(n) is the same power of 2
If the sequence e1,e2,. . . . ek is palindromic, a(n)=n
For any given even number Q, we can by reversing the sequence of its powers define not only one but an infinity (by adding as many zeros as we want on the left end) of n such that a(n)=Q. Hence the sequence is a permutation of even integers where each even integer is infinitely repeated.
For example as Q=1224=(2^3)*(3^2)*(5^0)*(7^0)*(11^0)*(13^0)*(17^1),
Q =a((2^1)*(3^0)*(5^0)*(7^0)*(11^0)*(13^2)*(17^3))=a(1660594) but also of an infinity of other ones, the first one being a((2^0)*(3^1)*(5^0)*(7^0)*(11^0)*(13^0)*(17^2)*(19^1))=a(5946753)
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EXAMPLE
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n=9=(2^0)*(3^2), hence a(9) = (2^2)*(3^0)=4
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CROSSREFS
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Sequence in context: A067824 A107067 A046801 this_sequence A143112 A167272 A090624
Adjacent sequences: A137499 A137500 A137501 this_sequence A137503 A137504 A137505
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KEYWORD
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base,easy,nonn
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AUTHOR
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Philippe Lallouet (philip.lallouet(AT)orange.fr), Apr 22 2008
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EXTENSIONS
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Edited by N. J. A. Sloane (njas(AT)research.att.com), Jan 16 2009
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