A147616 -id:A147616 - 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

A147624 Numbers n = concat(a,b) such that omega(n) = omega(a) * omega(b), where omega = A001221.

Original entry on oeis.org

23, 25, 26, 27, 29, 32, 36, 37, 43, 46, 47, 49, 53, 56, 59, 62, 63, 65, 68, 69, 73, 76, 79, 83, 86, 89, 96, 97, 104, 108, 113, 116, 122, 123, 124, 129, 136, 137, 139, 142, 143, 144, 145, 147, 148, 152, 153, 155, 158, 159, 163, 166, 167, 169, 173, 176, 179, 183, 184

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.

			a(1)=23 is in the sequence since omega(23) = 1 = 1*1 = omega(2)*omega(3).
307 is not in this sequence although omega(307) = 1 = 1*1 = omega(3)*omega(07), since we don't allow leading zeros in the second part b.

Cf. A001221, A147616 (analog for sigma), A147619 (analog for eulerphi), A147627 (analog for bigomega).

is_A147624(n)={ local(p=1, o=omega(n)); while( n>p*=10, n%p*10A147624(n) & print1(n","))

A147627 Numbers n = concat(a,b) such that bigomega(n) = bigomega(a) * bigomega(b), where bigomega = A001222.

Original entry on oeis.org

23, 26, 28, 34, 37, 39, 53, 62, 65, 73, 74, 78, 93, 95, 104, 113, 119, 125, 134, 137, 138, 142, 143, 145, 155, 156, 173, 182, 193, 194, 197, 207, 211, 212, 213, 214, 215, 217, 221, 223, 226, 229, 230, 233, 235, 238, 241, 242, 244, 245, 249, 253, 260, 262, 265

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 do not allow leading zeros in b.

			a(4)=34 is in the sequence since bigomega(34) = 2 = 1*2 = bigomega(3)*bigomega(4).
206 is not in this sequence although bigomega(206) = 2 = 1*2 = bigomega(2)*bigomega(06), we do not allow leading zeros in the second part b.

Cf. A001222, A147616 (analog for sigma), A147619 (analog for eulerphi), A147624 (analog for omega).

is_A147627(n)={ local(p=1, o=bigomega(n)); while( n>p*=10, n%p*10A147627(n) & print1(n","))

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

PARI

A147624 Numbers n = concat(a,b) such that omega(n) = omega(a) * omega(b), where omega = A001221.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI

A147627 Numbers n = concat(a,b) such that bigomega(n) = bigomega(a) * bigomega(b), where bigomega = A001222.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI