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A147619 Numbers n = concat(a,b) such that phi(n) = phi(a) * phi(b), where phi = A000010. 7

%I #16 Jan 02 2023 12:30:47

%S 78,780,897,918,1179,1365,1776,2574,2598,2967,3168,3762,4758,5775,

%T 5796,7800,7875,7917,8217,8970,9180,9576,11790,13650,13662,13875,

%U 13896,14391,17760,18564,18858,19812,20097,25740,25935,25974,25980,27573,28776

%N Numbers n = concat(a,b) such that phi(n) = phi(a) * phi(b), where phi = A000010.

%C 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.

%C This sequence was suggested by _Farideh Firoozbakht_ and David Wilson on the SeqFan mailing list, Oct 27 and Nov 06 2008 (see Links).

%C _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.

%C 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.

%H Paolo P. Lava, <a href="/A147619/b147619.txt">Table of n, a(n) for n = 1..500</a>

%H David Wilson, <a href="http://list.seqfan.eu/oldermail/seqfan/2008-November/000064.html">A nice (decimal) property of 78</a>, Seqfan mailing list (Nov 2008).

%p with(numtheory): P:=proc(q) local s, t, k, n; for n from 1 to q do

%p for k from 1 to ilog10(n) do s:=n mod 10^k; t:=trunc(n/10^k);

%p if s*t>0 then if phi(s)*phi(t)=phi(n)

%p then print(n); break; fi; fi; od; od; end: P(10^5); # _Paolo P. Lava_, Jan 27 2015

%o (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))}

%o for( n=1,10^5, is_147619(n) & print1(n","))

%Y Cf. A000010, A147616 (analog for sigma), A147624 (analog for omega), A147627 (analog for bigomega).

%K base,easy,nonn

%O 1,1

%A _M. F. Hasler_, Nov 08 2008

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