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COMMENTS
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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^(d-1) 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^d|b(d)-1, b(d) < 10^d:
(-,-,-,9376,-,-,7109376,-,...)
(iv) c(d) such that 2^(d+1)|c(d), 5^(d-1)|2c(d)+5, c(d) < 4*10^(d-1):
(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^(d-1)|2c'(d)+5, c'(d) < 4*10^(d-1):
(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^(d-1)|c"(d), c"(d) < 4*10^(d-1):
(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 d-th 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)
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