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.

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

Paolo P. Lava, Table of n, a(n) for n = 1..10000

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

Cf. A000040, A051953, A272157.

Sequence in context: A143863 A158571 A176882 * A159614 A158087 A210546

Adjacent sequences: A272400 A272401 A272402 * A272404 A272405 A272406

Keyword

nonn,base,easy

Author

Paolo P. Lava, Apr 29 2016

Status

approved