As per this page the minimum number of chants is 3 & the other permisible numbers are 11,21,27 & 54.So,in particular, 60 is not a valid japa count number .
For mantra reciting, a lot depends on the type of mantra you intend to
recite and the purpose. Some people recite some mantras daily. It is
up to their convenience. The normal number of times a mantra is
recited is 3, 11, 21, 27, 54 and 108 or more.
Although the above info is generally valid but in some specific cases it does not apply.For example:
For specific Graha shanti, the number of times the mantra to be
recited is prescribed. The prescribed time limit during which it
should be completed is one mandala or 40 days. You are given a grace
period of 8 days, taking into account any emergency or illness. So
maximum 48 days.
In Mahanirvana Tantra Lord Sada Shiva says
The number of daily recitations can be 10,108 or 1008.
From the 7th Chapter, verses 76-81:
Sandhya, whether Vaidika or Tantrika should be performed thrice daily,
and according as the worship changes so does its service (76). The
worshippers of the Brahma-Mantra have performed their Sandhya when
they have made japa of the Gayatri, realizing within themselves the
identity of the Gayatri and Brahman (77). In the case of those who are
not Brahma-worshippers, Vaidika Sandhya consists of the worship of and
offering of oblations to the Sun and the recitation of the Gayatri
(78). O Gentle One! In all daily prayers recitation should be done one
thousand and eight or a hundred and eight or ten times (79). O Devi!
the Shudras and Samanyas may observe any of the rites proclaimed by
the Agamas, and by these they attain that which they desire (80). The
three times of performance (of Sandhya) are at sunrise, at noon, and
at sunset (81).