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A147619
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Numbers n = concat(a,b) such that phi(n) = phi(a) * phi(b), where phi = A000010.
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7
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78, 780, 897, 918, 1179, 1365, 1776, 2574, 2598, 2967, 3168, 3762, 4758, 5775, 5796, 7800, 7875, 7917, 8217, 8970, 9180, 9576, 11790, 13650, 13662, 13875, 13896, 14391, 17760, 18564, 18858, 19812, 20097, 25740, 25935, 25974, 25980, 27573, 28776
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OFFSET
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1,1
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COMMENTS
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Concat(a,b) means decimal concatenation of a and b, i.e., a*10^floor(log_10(b)+1) + b, since we do not allow leading zeros in b. However, allowing for leading zeros in b would not give any additional term below 10^6.
This sequence was suggested by Farideh Firoozbakht and David Wilson on the SeqFan mailing list, Oct 27 and Nov 06 2008 (see Links).
Farideh Firoozbakht has proved that if n is in this sequence, then n*10 is again in the sequence. Thus one could call "primitive" terms of this sequence those which aren't multiples of 10.
A possible variant would be to allow decomposition of n into an arbitrary number of substrings. If one requires decomposition of n into each of its digits, i.e., the analog of A098771 with sigma replaced by phi, then 78 appears to be the only number having this property.
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LINKS
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MAPLE
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with(numtheory): P:=proc(q) local s, t, k, n; for n from 1 to q do
for k from 1 to ilog10(n) do s:=n mod 10^k; t:=trunc(n/10^k);
if s*t>0 then if phi(s)*phi(t)=phi(n)
then print(n); break; fi; fi; od; od; end: P(10^5); # Paolo P. Lava, Jan 27 2015
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PROG
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(PARI) is_A147619(n)={ local(p=1, s=eulerphi(n)); while( n>p*=10, n%p*10<p & next; s==eulerphi( n\p )*eulerphi( n%p ) & return(1))}
for( n=1, 10^5, is_147619(n) & print1(n", "))
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CROSSREFS
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KEYWORD
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base,easy,nonn
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AUTHOR
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STATUS
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approved
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