
COMMENTS

This sequence is the union of the following twelve subsequences.
Terms in <angle brackets> have fewer than d digits: they are pyramorphic, and always appear elsewhere, as an earlier term in the same sequence or in a related sequence. Dashes replace solutions to the congruences for which the inequalities, or other conditions proving pyramorphicity, are not satisfied; these are not part of the subsequences.
(i) a(d) := 4 * 10^(d1) for d >= 2:
(, 40,400,4000,40000,400000,...)
(ii) 2a(d) for d >= 2:
(, 80,800,8000,80000,800000,...)
(iii) b(d) such that 2^(d+1)b(d), 5^db(d)1, b(d) < 10^d:
(,,,9376,,,7109376,,...)
(iv) c(d) such that 2^(d+1)c(d), 5^(d1)2c(d)+5, c(d) < 4*10^(d1):
(0,<0>,160,2560,26560,226560,<226560>,12226560,...)
(v) c(d) + a(d) for d >= 2:
(,40,560,6560,66560,626560,42265609,41226560,...)
(vi) c(d) + 2a(d) for d >= 2, when this is less than 10^d:
(, 80,960,,,,8226560,81226560,...)
(vii) c'(d) such that 2^(d+1)c'(d)1, 5^(d1)2c'(d)+5, c'(d) < 4*10^(d1):
(1,25,385,1185,37185,317185,1117185,25117185,...)
(viii)c'(d) + a(d) for d >= 2:
(,65,785,5185,77185,717185,5117185,65117185,...)
(ix) c'(d) + 2a(d) for d >= 2, when this is less than 10^d:
(,,,9185,,,9117185,,...)
(x) c"(d) such that 2^(d+1)c"(d)1, 5^(d1)c"(d), c"(d) < 4*10^(d1):
(5,25,225,2625,10625,<90625>,<890625>,12890625,...)
(xi) c"(d) + a(d) for d >= 2:
(,65,625,6625,50625,490625,4890625,52890626,...)
(xii) c"(d) + 2a(d) for d >= 2, when this is less than 10^d:
(,,,,90625,890625,8890625,92890625,...)
For d >= 3 the dth terms of these sequences are always distinct.
For d > 3 there are at least eight and at most eleven square pyramorphic numbers with d digits (not including leading zeros). The minimum is first achieved for d=6; the maximum is first achieved for d=49.
(End)
