login
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.
2
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
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
KEYWORD
nonn,base,easy
AUTHOR
Paolo P. Lava, Apr 29 2016
STATUS
approved