Order 13 Hexamagic Series Impossibility

Theorem. There are no order 13 hexamagic series.
Proof.
S1 = 1105           ≡ 6 (mod 7)
S2 = 124865         ≡ 6 (mod 7)
S3 = 15873325       ≡ 6 (mod 7)
S4 = 2152397897     ≡ 6 (mod 7)
S5 = 304021793725   ≡ 6 (mod 7)
S6 = 44169254755145 ≡ 2 (mod 7)

From the Modulo 7 Hexamagic Series Lemma,
this hexamagic series must have at least
  3 entries of 1 (mod 7) and
  4 entries each of 2 ... 6 (mod 7).

An order 13 hexamagic series requires
at least 23 entries.