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A147619 Numbers n = concat(a,b) such that phi(n) = phi(a) * phi(b), where phi = A000010. 7
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Concat(a,b) means decimal concatenation of a and b, i.e., a*10^[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.

Farideh Firoozbakht has proved that if n is in this sequence, then n*10 is again in the sequence. Thus one could call "primitive" elements of this sequence those which aren't a multiple 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.

LINKS

Paolo P. Lava, Table of n, a(n) for n = 1..500

MAPLE

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

PROG

(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", "))

CROSSREFS

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

Sequence in context: A220409 A129238 A297759 * A119093 A128951 A272383

Adjacent sequences:  A147616 A147617 A147618 * A147620 A147621 A147622

KEYWORD

base,easy,nonn

AUTHOR

M. F. Hasler, Nov 08 2008

STATUS

approved

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Last modified April 25 04:14 EDT 2019. Contains 322451 sequences. (Running on oeis4.)