A118081 -id:A118081 - cp's OEIS frontend

A141100 Number of unordered pairs of odd composite numbers that sum to 2n.

0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 2, 0, 1, 2, 0, 1, 3, 1, 1, 3, 2, 1, 4, 1, 2, 5, 1, 3, 5, 1, 4, 5, 3, 3, 6, 3, 3, 7, 3, 3, 9, 3, 4, 7, 4, 6, 9, 5, 5, 8, 6, 6, 10, 5, 5, 12, 4, 6, 12, 5, 9, 11, 7, 7, 11, 9, 9, 13, 8, 8, 16, 7, 11, 14, 8, 11, 14, 9, 9, 17, 13, 10, 16, 11, 11, 19, 11, 12, 18, 10

Offset: 1

T. D. Noe, Jun 02 2008, Jun 05 2008

nonn

See A141099 for pairs of odd nonprime numbers. We have a(n) > 0 except for the 14 values of 2n given in A118081.

			a(18)=2 because 36 = 9+27 = 15+21.

T. D. Noe, Table of n, a(n) for n=1..1000

Cf. A005171, A118081, A141095, A141097, A141099.

Table[cnt=0; Do[If[ !PrimeQ[i] && !PrimeQ[2n-i], cnt++ ], {i,3,n,2}]; cnt, {n,100}]

a(n) = 1 - floor(n/2) + Sum_{i=3..n} c(i) * c(2n-i), n>1, where c = A005171. - Wesley Ivan Hurt, Dec 27 2013

A076770 Even numbers representable as the sum of two odd composites.

Original entry on oeis.org

18, 24, 30, 34, 36, 40, 42, 44, 46, 48, 50, 52, 54, 56, 58, 60, 62, 64, 66, 68, 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

Offset: 1

Jon Perry, Nov 14 2002

nonn

If n is even and n > 38, at least one of (n-15), (n-25), (n-35) is an odd number divisible by 3 and greater than 3. So every even number > 38 is a term. [Jack Brennen]

			40 can be represented as 15+25 where 15 and 25 are both composite numbers, so 40 is a term.

Cf. A118081.

v=vector(5000); vc=1; forstep (n=9,300,2, if (isprime(n),continue, forstep (j=9,300,2,if (isprime(j),continue,x=n+j; fl=true; for (i=1,vc,if (v[i]==x,fl=false; break)); if (fl==true,v[vc]=x; vc++))))); print(vc); v=vecsort(vecextract(v,concat("1..",vc-1)))

Corrected by Don Reble, Nov 20 2006

A046458 Positive even integers that are not the sum of two nonprime odd integers.

Original entry on oeis.org

4, 6, 8, 12, 14, 20, 32, 38

Offset: 1

Patrick De Geest, Aug 15 1998

See A118081 for numbers not representable as the sum of two odd composite numbers. Roberts shows that every even number greater than 38 is the sum of an odd composite number and one of the five composite numbers 9, 15, 21, 27, 33. - T. D. Noe, Jun 01 2008, Jun 05 2008

Joe Roberts, "Lure of the Integers", The Mathematical Association of America, 1992, p. 189.

A152483 Even numbers which are not the sum of 2 odd semiprimes.

Original entry on oeis.org

2, 4, 6, 8, 10, 12, 14, 16, 20, 22, 26, 28, 32, 38, 52, 62

Offset: 1

Donovan Johnson, Dec 06 2008

No more terms < 10^9. Conjectured to be complete.

			62 is in this sequence because no 2 odd semiprimes sum to 62. 64 is not in this sequence because the sum of 9 (odd semiprime) and 55 (odd semiprime) is 64.

Cf. A118081, A046315, A152482, A152484.

A284788 Even numbers that cannot be represented in at least two ways as the sum of two odd composites.

Original entry on oeis.org

2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 32, 34, 38, 40, 44, 46, 52, 56, 62, 68

Offset: 1

Bernard Schott, Apr 03 2017

If n is even and n > 68, then n can be written as at least two distinct sums of two composite odd integers.

			34 is in the sequence because 34 = 9 + 25 but cannot be represented in a second way as the sum of two odd composites with 9, 15, 21, 25, 27, 33.

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

Cf. A118081, A284787 (complement).

oddco = Select[Range[9, 100, 2], ! PrimeQ[#] &]; Select[Range[2, 100, 2],  Length@ Quiet@ IntegerPartitions[#, {2}, oddco, 2] < 2 &] (* Giovanni Resta, Apr 03 2017 *)

A130702 Possible sides in the Euler V=E-F+2 as roots in a cubic polynomial of the form: P(x)=(x-V)(x-F)(x+E) =x^3+(E-V-F)x^2+(VF-E(V+F))x=EF*V Solves here for F ( Face, Edge, Vertex).

Original entry on oeis.org

4, 8, 12, 16, 18, 20, 22, 24, 28, 30, 32, 36, 40, 42, 44, 48, 52, 54, 56, 60, 64, 66, 72

Offset: 1

Roger L. Bagula, Jul 06 2007

Polynomial cubic of Euler's V,F,E: V=E-F+2 P(x)=(x-V)*(x-F)*(x+E) =x^3+(E-V-F)*x^2+(V*F-E(V+F))*x=E*F*V letting (E-V-F)=-2 V+F=E+2 and product: p=V*F I got P(x)=x3-2*x2+(p-E*(E+2))*x+E*p Setting that polynomial equal to zero gives roots that agree with Euler's equation. In the exceptional groups: ( down to two integer variables) p=16*m ; m-> {1,3,15} E=6*n ; n->{1,2,5} The program works to produce the right roots for {-E,V,F}

			Program to get roots for tetrahedron, (cube, octahedron),
(dodecahedron,
icosahedron):
a = {1, 2, 5}
b = {1, 3, 15}
g[n_, m_] := x /. Solve[e [a[[m]]]*p[b[[m]]] - e [a[[m]]]*(e[a[[
m]]] + 2)*x + p[b[[m]]]* x - 2* x^2 + x^3 == 0, x][[n]]
Table[g[n, m], {n, 1, 3}, {m, 1, 3}]
{{-6, -12, -30}, {4, 6, 12}, {4, 8, 20}}

Cf. edge: A008458; vertex: A118081.

ExpandAll[(x - v)*(x - f)*(x + e)]; e[n_] := 6*n; p[m_] := 16*m; a0 = Table[If[IntegerQ[x /. Solve[e [m]*p[p0] - e [m]*(e[m] + 2)*x +p[p0]* x - 2* x^2 + x^3 == 0, x][[1]]] && IntegerQ[x /. Solve[e [m]*p[p0] - e [m]*(e[m] + 2)*x + p[p0]* x - 2* x^2 + x^3 == 0,x][[2]]] && IntegerQ[x /. Solve[e [m]*p[p0] - e [m]*(e[m] + 2)*x +p[p0]* x - 2* x^2 + x^3 == 0, x][[3]]], {Abs[x] /. Solve[e [m]*p[p0] - e [m]*(e[m] + 2)*x + p[p0]* x - 2* x^2 + x^3 == 0, x][[3]]}, {}], {m, 1, 12}, {p0, 1, 33}]

F roots such that:x^3+(E-V-F)*x^2+(V*F-E(V+F))*x=E*F*V and that are exceptional like ( tetrahedron, cube, octahedron, dodecahedron, icosahedron)

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

A141100 Number of unordered pairs of odd composite numbers that sum to 2n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A076770 Even numbers representable as the sum of two odd composites.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI

Extensions

A046458 Positive even integers that are not the sum of two nonprime odd integers.

Views

Author

Keywords

Comments

References

A152483 Even numbers which are not the sum of 2 odd semiprimes.

Views

Author

Keywords

Comments

Examples

Crossrefs

A284788 Even numbers that cannot be represented in at least two ways as the sum of two odd composites.

Views

Author

Keywords

Comments

Examples

References

Crossrefs

Programs

Mathematica

A130702 Possible sides in the Euler V=E-F+2 as roots in a cubic polynomial of the form: P(x)=(x-V)*(x-F)*(x+E) =x^3+(E-V-F)*x^2+(V*F-E(V+F))*x=E*F*V Solves here for F ( Face, Edge, Vertex).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A130702 Possible sides in the Euler V=E-F+2 as roots in a cubic polynomial of the form: P(x)=(x-V)(x-F)(x+E) =x^3+(E-V-F)x^2+(VF-E(V+F))x=EF*V Solves here for F ( Face, Edge, Vertex).