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A093534 Square pyramorphic numbers: integers m such that the sum of the first m squares (A000330) ends in m. 3
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 (list; graph; refs; listen; history; text; internal format)
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

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

Sequence in context: A070389 A098993 A099799 * A070388 A250314 A293571

Adjacent sequences:  A093531 A093532 A093533 * A093535 A093536 A093537

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

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Last modified May 16 19:46 EDT 2021. Contains 343951 sequences. (Running on oeis4.)