OFFSET
1,3
COMMENTS
From Robert Dawson, Apr 04 2018: (Start)
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)
REFERENCES
C. A. Pickover, Wonders of Numbers, Chap. 65, Oxford Univ. Press NY 2000; pp. 158-160.
LINKS
Giovanni Resta, Table of n, a(n) for n = 1..9000 (terms > 10^11 generated according to Robert Dawson's comment)
Robert Dawson, On Some Sequences Related to Sums of Powers, J. Int. Seq., Vol. 21 (2018), Article 18.7.6.
MATHEMATICA
l = {0}; s = 0; Do[s = s + n^2; If[ Mod[s, 10^Floor[ Log[10, n] + 1]] == n, AppendTo[l, n]], {n, 10^6}]; l (* Robert G. Wilson v, May 24 2004 *)
PROG
(PARI) isok(n) = frac((n*(n+1)*(2*n+1)/6 - n)/10^#Str(n)) == 0; \\ Michel Marcus, Aug 01 2018
CROSSREFS
KEYWORD
nonn,base
AUTHOR
Lekraj Beedassy, May 14 2004
EXTENSIONS
More terms from Robert G. Wilson v, May 24 2004
Term corrected (6025 -> 6625) by Robert Dawson, Jul 31 2018
0 inserted by David A. Corneth, Aug 02 2018
STATUS
approved