A372631 -id:A372631 - cp's OEIS frontend

A372630 Numbers k with property that there exists an m>k such that the sum of the natural numbers from k^2 to m^2 inclusive is a square number.

1, 3, 8, 11, 12, 14, 17, 23, 30, 33, 35, 37, 41, 48, 59, 60, 65, 68, 77, 79, 82, 84, 89, 93, 94, 99

Offset: 1

Nicolay Avilov, May 07 2024

			The number 3 is a member of the sequence because the sum of all natural numbers from 3^2 to 4^2 inclusive is 9 + 10 + 11 + 12 + 13 + 14 + 15 + 16 = 100, with 100 = 10^2.

Nicolay Avilov, Problem 1876. Segment of a natural series (in Russian).

Cf. A000290, A372631, A373330.

check(k, mm=100) = my(d=2*k^2-1, v=List([]), x, y, z); for(t=d+1, 17*d, if(issquare((t^2-d^2)/2), listput(v, t))); if(v[#v\2] != 3*d, return(-1)); for(i=1, #v\2, x=v[i]; y=v[i+#v\2]; for(j=1, mm, if(issquare((x-1)/2) && x>d+2, return(1)); z=6*y-x; x=y; y=z)); 0; \\ Jinyuan Wang, Jul 06 2024; just for checking

A372782 Least number m for which there exists some positive k < m where the sum of the integers from k + 1 to m inclusive is an n-th power > 1.

Original entry on oeis.org

2, 4, 7, 13, 22, 40, 67, 121, 202, 364, 607, 1093, 1822, 3280, 5467, 9841, 16402, 29524, 49207, 88573, 147622, 265720, 442867, 797161, 1328602, 2391484, 3985807, 7174453, 11957422, 21523360, 35872267, 64570081, 107616802, 193710244, 322850407, 581130733, 968551222, 1743392200

Offset: 1

Jean-Marc Rebert, May 14 2024

With triangular sum T(i) = i*(i+1)/2 = Sum_{j=1..i} i, the aim is T(m) - T(k) = b^n for some b. T(m) - T(k) = (m+k+1)*(m-k)/2 so if m-k is odd then use m-k = b^x to eliminate k in the other factor (m+k+1)/2 = b^(n-x), so that m = b^(n-x) + (b^x - 1)/2. If instead m+k+1 is odd then the resulting expression is the same. This form is an integer and minimized at b=3 and x = ceiling(n/2). - Kevin Ryde, May 18 2024

			a(2) = 4 because the sum of all integers from 3 + 1 to 4 inclusive is 4 = 2^2, a square.
a(3) = 7 as we have 2 + 3 + 4 + 5 + 6 + 7 = 27 = 3^3, i.e., m = 7 and k = 1. - _David A. Corneth_, May 15 2024

Paolo Xausa, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,3,-3).

Cf. A002452, A052993, A087503, A372631.

LinearRecurrence[{1, 3, -3}, {2, 4, 7}, 50] (* Paolo Xausa, Jun 09 2024 *)

isok(m, n) = my(s = m*(m+1)/2); for (k=1, m-1, s -= k; if (ispower(s, n), return(k)););
a(n) = my(m=1); while (! isok(m, n), m++); m; \\ Michel Marcus, May 16 2024

a(n) = {
	my(res, c, l, u);
	res = 2^n; c = 8*3^n;
	l = (sqrt(c) - 1)\2; u = res;
	for(i = l, u,
		if(issquare(4*(i + 1)*i + 1 - c),
			return(i);
		)
	);
	return(2^n)
} \\ David A. Corneth, May 16 2024

a(n) = A087503(n-1) + 1.
a(n) = 3*a(n-2) + 1.
G.f.: x*(2 + 2*x - 3*x^2)/((1 - x)*(1 - 3*x^2)). - Stefano Spezia, May 18 2024

More terms from David A. Corneth, May 16 2024

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A372630 Numbers k with property that there exists an m>k such that the sum of the natural numbers from k^2 to m^2 inclusive is a square number.

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A372782 Least number m for which there exists some positive k < m where the sum of the integers from k + 1 to m inclusive is an n-th power > 1.

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