A093534 Square pyramorphic numbers: integers m such that the sum of the first m squares (A000330) ends in m.

0, 1, 5, 25, 40, 65, 80, 160, 225, 385, 400, 560, 625, 785, 800, 960, 1185, 2560, 2625, 4000, 5185, 6560, 6625, 8000, 9185, 9376, 10625, 26560, 37185, 40000, 50625, 66560, 77185, 80000, 90625, 226560, 317185, 400000, 490625, 626560, 717185, 800000

Offset: 1

Author

Lekraj Beedassy, May 14 2004

Keywords

• nonn
• base

Comments

From Robert Dawson, Apr 04 2018: (Start)

This sequence is the union of the following twelve subsequences.

Terms in 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.

Crossrefs

A060204 gives the corresponding sums of squares. Cf. A000330.

Programs

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 *)

PARI

isok(n) = frac((n*(n+1)*(2*n+1)/6 - n)/10^#Str(n)) == 0; \\ Michel Marcus, Aug 01 2018

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