A098771 -id:A098771 - cp's OEIS frontend

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

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

Offset: 1

M. F. Hasler, Nov 08 2008

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.

Paolo P. Lava, Table of n, a(n) for n = 1..500
David Wilson, A nice (decimal) property of 78, Seqfan mailing list (Nov 2008).

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

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

is_A147619(n)={ local(p=1, s=eulerphi(n)); while( n>p*=10, n%p*10

A147616 Numbers k = concat(a,b) such that sigma(k) = sigma(a) * sigma(b), where sigma = A000203.

Original entry on oeis.org

38, 58, 66, 87, 118, 178, 205, 217, 275, 295, 298, 395, 451, 478, 492, 517, 538, 575, 660, 718, 766, 775, 838, 839, 870, 898, 1018, 1138, 1175, 1195, 1318, 1671, 1678, 1775, 1795, 1975, 2050, 2163, 2170, 2295, 2395, 2518, 2578, 2638, 2665, 2750, 2818

Offset: 1

M. F. Hasler, Nov 08 2008

Concat(a,b) means decimal concatenation of a and b, i.e., a*10^[log_10(b)+1] + b, since we don't allow leading zeros in b. (However, allowing leading zeros in b would not give any additional term up to at least 10^6.)
This sequence has been suggested by David Wilson on the SeqFan mailing list, Nov 08 2008.
A possible variant would be to allow decomposition of k into an arbitrary number (>1) of substrings. If one requires decomposition of k into each of its digits, this yields A098771.

			a(1)=38 is in the sequence since sigma(38) = 60 = 4*15 = sigma(3)*sigma(8).

Paolo P. Lava, Table of n, a(n) for n = 1..1000

Cf. A000203, A098771, A147619 (analog for phi), A147624 (analog for omega), A147627 (analog for bigomega).

is_A147616(n)={ local(p=1, s=sigma(n)); while( n>p*=10, n%p*10

Precision about disallowed leading zeros, fix in PARI code, more cross references. - M. F. Hasler, Nov 09 2008

cp's OEIS Frontend

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

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