A384947 Positive integers m for which A183136(m) != f(m), where f(m) = floor( (m*(m+1)/2)/phi - m/2 + 1/(2*phi) ) and phi = (1+sqrt(5))/2 is the golden ratio.

15, 18, 36, 39, 41, 47, 49, 52, 91, 94, 96, 102, 103, 104, 107, 109, 123, 125, 128, 130, 136, 138, 141, 235, 238, 240, 246, 247, 248, 251, 252, 253, 267, 268, 269, 272, 273, 274, 277, 280, 281, 282, 285, 287, 303, 306, 322, 324, 327, 328

Offset: 1

Author

Hoang Xuan Thanh, Jun 13 2025

Keywords

• nonn

Comments

f(m) is an approximation to A183136(m) = Sum_{k=1..m} floor(k/phi) based on assuming the floor in each term decreases it by 1/2 from what is otherwise a triangular sum; and further offset + 1/(2*phi) in f(m) chosen to improve the accuracy of this approximation.

The actual values of frac(k/phi) can differ from 1/2 each by a net amount which is enough to make m a term of this sequence.

Examples

			41 is term, because A183136(41) = 512 != 511 = floor(((41^2+1)*phi - 41) / (2*phi^2)).
		

Links

• Paolo Xausa, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Hoang Xuan Thanh)

Crossrefs

Cf. A183136, A001622, A060143.

Programs

Mathematica

PositionIndex[MapIndexed[# != Floor[PolygonalNumber[#2[[1]]]/GoldenRatio - #2[[1]]/2 + 1/(2*GoldenRatio)] &, Accumulate[Floor[Range[500]/GoldenRatio]]]][True] (* Paolo Xausa, Jun 20 2025 *)