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A272403
Numbers n = concat(x,y) such that (x - phi(x)) + (y - phi(y)) = n - phi(n), where n - phi(n) is the Euler cototient function of n.
2
13, 17, 31, 71, 103, 107, 113, 131, 137, 167, 173, 179, 191, 197, 311, 431, 701, 971, 1013, 1019, 1031, 1061, 1091, 1097, 1103, 1109, 1151, 1163, 1181, 1193, 1219, 1223, 1229, 1277, 1283, 1301, 1307, 1339, 1367, 1373, 1409, 1433, 1439, 1487, 1499, 1511, 1523
OFFSET
1,1
COMMENTS
Essentially primes. Only 58 squarefree composite in the first 10000 terms: 1219, 1339, 2869, 3743, 4427, 9707, 11569, 14269, 16105, 17125, 18733, 19375, 22927, 74069, 106159, 107629, 115069, 134959, 137533, 137843, 142417, 146207, 147943, 150421, 156857, 158899, 165625, 170033, 183595, 184375, 194627, 220417, 226417, 243293, 280873, 284371, 325067, 345827, 425261, 740821, 765403, 794837, 857257, 908647, 914231, 1005673, 1007509, 1037749, 1043527, 1188211, 1188919, 1296497, 1416019, 1428773, 1527167, 1528913, 1587227, 15906225.
LINKS
EXAMPLE
137 - phi(137) = (1 - phi(1)) + (37 - phi(37)) = 1;
1219 - phi(1219) = (1- phi(1)) + (219 - phi(219)) = 75;
4427- phi(4427) = (442 - phi(442)) + (7- phi(7)) = 251.
MAPLE
with(numtheory): P:=proc(q) local x, y, k, n; for n from 1 to q do
for k from 1 to ilog10(n) do x:=n mod 10^k; y:=trunc(n/10^k);
if (x-phi(x))+(y-phi(y))=n-phi(n) then print(n); break; fi;
od; od; end: P(10^6);
MATHEMATICA
Select[Range@ 1600, Function[m, AnyTrue[Map[FromDigits /@ TakeDrop[IntegerDigits@ m, #] &, Range[IntegerLength@ m - 1]], Total@ Map[# - EulerPhi@ # &, #] == m - EulerPhi@ m &]]] (* Michael De Vlieger, Apr 29 2016, Version 10.2 *)
CROSSREFS
KEYWORD
nonn,base,easy
AUTHOR
Paolo P. Lava, Apr 29 2016
STATUS
approved