A272157 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.

23, 37, 53, 73, 113, 137, 173, 193, 197, 211, 223, 229, 233, 241, 271, 283, 293, 307, 311, 313, 317, 331, 337, 347, 353, 359, 367, 373, 379, 383, 389, 397, 433, 503, 523, 541, 547, 571, 593, 613, 617, 673, 677, 719, 733, 743, 761, 773, 797, 977, 1013, 1033, 1093

Offset

1,1

Comments

Essentially primes. Only 36 semiprimes in the first 10000 terms: 3337, 3959, 7519, 13939, 16151, 19879, 28999, 33221, 34669, 35579, 37391, 38579, 43733, 46339, 50299, 59177, 68773, 74743, 77593, 79927, 130733, 140653, 175507, 192649, 197737, 212197, 268561, 269989, 318293, 339629, 345911, 352829, 360379, 362771, 363191, 365399.

For any prime p, cototient(p) = p - phi(p) = 1. All the primes of the sequence are the concatenation of two other primes, x and y, and therefore cototient(x) * cototient(y) = 1 * 1 = 1 = cototient(p).

Links

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

Example

271 - phi(271) = (2 - phi(2)) * (71 - phi(71)) = 1;
3337 - phi(3337) = (333 - phi(333)) * (7 - phi(7)) = 117;
3959 - phi(3959) = (3 - phi(3)) * (959 - phi(959)) = 143.

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);

Crossrefs

Cf. A000040, A051953, A272403.

Sequence in context: A034302 A057878 A019549 * A129800 A105184 A238056

Adjacent sequences: A272154 A272155 A272156 * A272158 A272159 A272160

Keyword

nonn,base,easy

Author

Paolo P. Lava, Apr 29 2016

Status

approved