A110172 Conjectured numbers j such that phi(j) + phi(k) = phi(j+k) has no solution k, where phi is Euler's totient function.

3, 15, 21, 39, 45, 57, 69, 105, 147, 165, 177, 195, 213, 273, 285, 315, 345, 393, 399, 465, 489, 525, 585, 615, 633, 645, 651, 681, 717, 777, 807, 813, 843, 855, 879, 885, 903, 915, 933, 939, 1005, 1035, 1041, 1065, 1095, 1149, 1263, 1281, 1293, 1317, 1395

Offset

1,1

Comments

All k < 10^8 have been checked. All of these numbers are multiples of 3.

The observation above is true for every term. Substituting k=j into phi(j) + phi(k) = phi(j+k) gives phi(j) + phi(j) = phi(j+j), i.e., 2*phi(j) = phi(2j), which is true for every positive even number j; thus k=j yields a solution for every positive even number j. Substituting k=2j into phi(j) + phi(k) = phi(j+k) gives phi(j) + phi(2j) = phi(j+2j), i.e., phi(j) + phi(2j) = phi(3j); since phi(j) = phi(2j) for every odd number j, this is equivalent (for odd j) to phi(j) + phi(j) = phi(3j), i.e., 2*phi(j) = phi(3j), which holds for every odd j that is not a multiple of 3; thus, k=2j yields a solution for every odd j that is not a multiple of 3. Consequently, every term of the sequence is an odd multiple of 3. - Flávio V. Fernandes, May 10 2022

Links

Table of n, a(n) for n=1..51.

Crossrefs

Cf. A066426 (least k such that phi(n) + phi(k) = phi(n+k)).

Cf. A306771.

Sequence in context: A243128 A306771 A216521 * A261274 A014493 A147025

Adjacent sequences: A110169 A110170 A110171 * A110173 A110174 A110175

Keyword

nonn

Author

T. D. Noe, Jul 15 2005

Status

approved