A076770 -id:A076770 - cp's OEIS frontend

A284787 Even numbers representable in at least two ways as the sum of two odd composites.

30, 36, 42, 48, 50, 54, 58, 60, 64, 66, 70, 72, 74, 76, 78, 80, 82, 84, 86, 88, 90, 92, 94, 96, 98, 100, 102, 104, 106, 108, 110, 112, 114, 116, 118, 120, 122, 124, 126, 128, 130, 132, 134, 136, 138, 140, 142, 144, 146, 148, 150, 152, 154, 156, 158, 160, 162

Offset: 1

Bernard Schott, Apr 03 2017

nonn

If n is even and n > 68, at least two of n-15, n-25, n-35, n-45, n-55, n-65, are odd numbers divisible by 3 and greater than 3, with n = (n-55) + 55 for example.

So if n is even and n > 68, then n can be written in at least two ways as the sum of two odd positive composite numbers.

			30 = 9 + 21 = 15 + 15;
66 = 15 + 51 = 21 + 45.

D. Wells, The Penguin Dictionary of Curious and Interesting Numbers, 1997, page 111.

R. E. Ruemmler and Minnich, Problem 1328: Sums of Composite Odd Numbers, Mathematics Magazine, 63 (1990), 276.

Cf. A076770, A284788.

up = 200; oddco = Select[Range[9, up, 2], ! PrimeQ[#] &]; Select[ Range[2, up, 2], Length@ Quiet@ IntegerPartitions[#, {2}, oddco, 2] == 2 &] (* Giovanni Resta, Apr 03 2017 *)

a(42)-a(57) from Giovanni Resta, Apr 03 2017

A366190 Minimal lengths of prime knots formed by orthogonal unit line segments of the cubic lattice.

Original entry on oeis.org

4, 24, 30, 34, 36, 40, 42, 44, 46, 48, 50, 52, 54, 56, 58, 60, 62, 64

Offset: 1

Tamas Sandor Nagy and Thomas Scheuerle, Oct 12 2023

The same term may correspond to more than one knot. For the initial terms, the minimum lengths were found within three layers of the lattice and it is conceivable that the tightest representation of larger knots expand in all three axes.
Length 24: 3_1.
Length 30: 4_1.
Length 34: 5_1.
Length 36: 5_2.
Length 40: 6_1, 6_2, 6_3.
Length 42: 8_19.
Length 44: 8_20.
Length 46: 7_2, 7_5, 7_6, 8_21.
Length 48: 8_3, 8_7, 9_42, 10_124.
Length 50: 8_1, 8_2, 8_4, 8_5, 8_6, 8_8, 8_9, 8_10, 8_11, 8_13, 8_14, 8_16, 9_43, 9_44, 9_46, 9_47, 10_139.
Length 52: 8_12, 8_15, 8_17, 8_18, 9_45, 9_48, 9_49, 10_132.
Length 54: 9_1, 9_3-5, 9_14, 9_19, 9_26, 9_31, 9_40, 9_41, ... .
Conjecture: All even numbers >= 40 will appear in this sequence.

			a(1) = 4 because the unknot is represented by four joined unit line segments, forming a closed loop, in the lattice.
a(2) = 24 because the second simplest knot, the trefoil knot, 3_1, can be described by 24 joined unit line segments, forming a self-avoiding closed loop in the lattice.

Andrew Rechnitzer, List of knot data for different cubic lattices.
Rob Scharein, Kai Ishihara, Javier Arsuaga, Yuanan Diao, Koya Shimokawa and Mariel Vazquez, Bounds for the minimum step number of knots in the simple cubic lattice, J. Phys. A: Math. Theor. 42 475006 (2009).
Thomas Scheuerle, Examples for a(2) - a(6) with up to 6 crossings.
Index entries for sequences related to knots

Cf. A122059, A076770.

A345339 a(n) = 18*n + 20.

Original entry on oeis.org

20, 38, 56, 74, 92, 110, 128, 146, 164, 182, 200, 218, 236, 254, 272, 290, 308, 326, 344, 362, 380, 398, 416, 434, 452, 470, 488, 506, 524, 542, 560, 578, 596, 614, 632, 650, 668, 686, 704, 722, 740, 758, 776, 794, 812, 830, 848, 866, 884, 902, 920, 938, 956, 974, 992, 1010

Offset: 0

Bernard Schott, Jun 14 2021

The largest even integer which cannot be written as the sum of 2n composite odd integers, for n >= 1, is 18*n + 20, proved by the Shippensburg University Mathematical Problem Solving Group (see Links).

			For n = 1, a(1) = A118081(14) = 38.

Ronald E. Ruemmler, Problem 1328, Mathematics Magazine, Vol. 62, No. 4 (October 1989), p. 274; Sums of Composite Odd Numbers, Solution to problem 1328 by Garrett R. Vargas, ibid., Vol. 63, No. 4 (October 1990), pp. 276-277.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A076770, A118081, A284787, A284788.

Table[18*n + 20, {n, 0, 55}] (* Amiram Eldar, Jun 14 2021 *)
LinearRecurrence[{2,-1},{20,38},60] (* Harvey P. Dale, Jan 15 2023 *)

a(n) = 18*n + 20.
G.f.: 2*(10 - x)/(1 - x)^2. - Stefano Spezia, Jun 14 2021
From Elmo R. Oliveira, Dec 08 2024: (Start)
E.g.f.: 2*exp(x)*(10 + 9*x).
a(n) = 2*a(n-1) - a(n-2) for n >= 2. (End)

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A284787 Even numbers representable in at least two ways as the sum of two odd composites.

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A366190 Minimal lengths of prime knots formed by orthogonal unit line segments of the cubic lattice.

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A345339 a(n) = 18*n + 20.

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