A284788 -id:A284788 - 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

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)

cp's OEIS Frontend

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

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