A103152 Smallest odd number which can be written as a sum 2p+q (where p and q are both odd primes, A065091) in exactly n ways, zero if there are no such odd number.

9, 13, 17, 29, 45, 51, 69, 81, 99, 93, 105, 135, 153, 201, 195, 165, 231, 237, 321, 297, 225, 363, 725, 393, 285, 315, 471, 483, 435, 405, 465, 561, 555, 495, 609, 783, 675, 867, 849, 963, 645, 525, 693, 897, 795, 915, 987, 735, 855, 825, 765, 1095, 975, 1467

Offset

1,1

Comments

Conjecture: except for the 2nd, 3rd and 4th terms, all other terms are divisible by 3; See also comments in A103151.

a(23)=725 is also not divisible by 3. [D. S. McNeil, Sep 06 2010]

The only terms a(n) not divisible by 3 for n <= 1450 are a(2),a(3),a(4) and a(23). - Robert Israel, Mar 17 2020

Links

Robert Israel, Table of n, a(n) for n = 1..1450

Example

9 is the smallest odd number with just one such composition: 9 = 3+2*3, thus a(1)=9.
Similarly, 13 is smallest with exactly 2 compositions: 13 = 3+2*5 = 7+2*3, thus a(2)=13.

Maple

N:= 2000: # for terms before the first term > N
P:= select(isprime, [seq(i, i=3..N, 2)]):
nP:= nops(P):
V:= Vector(N):
for i from 1 while 2*P[i]<N do
  for j from 1 to nP do
    k:= 2*P[i]+P[j];
    if k > N then break fi;
    V[k]:= V[k]+1;
od od:
A:= Vector(N):
for i from 1 to N by 2 do if V[i] <> 0 and A[V[i]] = 0 then A[V[i]]:= i fi od:
for i from 1 to N do if A[i] = 0 then break fi od:
seq(A[j], j=1..i-1); # Robert Israel, Mar 17 2020

Mathematica

Array[a, 300]; Do[a[n] = 0, {n, 1, 300}]; n = 9; ct = 0; While[ct < 200, m = 3; ct = 0; While[(m*2) < n, If[PrimeQ[m], cp = n - (2* m); If[PrimeQ[cp], ct = ct + 1]]; m = m + 2]; If[a[ct] == 0, a[ct] = n]; n = n + 2]; Print[a]

Prog

(Scheme) (define (A103152 n) (+ 1 (* 2 (first-n-where-fun_n-is-i1 A103151 n))))
(define (first-n-where-fun_n-is-i1 fun i) (let loop ((n 1)) (cond ((= i (fun n)) n) (else (loop (+ n 1)))))) ;; Antti Karttunen, Jun 19 2007

Crossrefs

Cf. A103151, A001172, A001031.

Sequence in context: A273059 A188220 A211429 * A295113 A263941 A090415

Adjacent sequences: A103149 A103150 A103151 * A103153 A103154 A103155

Keyword

nonn

Author

Lei Zhou, Feb 09 2005

Extensions

Starting offset changed from 0 to 1 by Antti Karttunen, Jun 19 2007

Status

approved