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 A113029 a(1) = 2, a(2) = 3; for n > 2, a(n) = least prime equal to the sum of two or more previous terms. 0
 2, 3, 5, 7, 17, 19, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS A heuristic argument suggests that all primes except 11, 13 and 23 are included in this sequence (tested on first million primes). - Ryan Murphy (murphy(AT)minegoboom.com), Jan 24 2006 Except for 17 which uses all 4 of the previous terms, all the other terms so far use only two or three of the previous terms. This is a more restrictive application of the Goldbach conjecture. - Robert G. Wilson v, Apr 08 2007, May 05 2007 Up to 10^4, all a(n) requiring 4 terms are of the form a(n)=2+7+m+p with m=5 or m=19, i.e., of the form 14+p or 28+p; no a(n)<10^6 requires more than 4 terms. - M. F. Hasler, May 04 2007 Ryan Murphy's heuristic is correct: a(n) = prime(n+3) for n > 6. It suffices to check that all n in 36..72 are the sum of one or more members of this sequence. Thus, all n > 72 are the sum of two or more distinct members of this sequence, since by Bertrand's postulate there is a prime n/2 < p < n. - Charles R Greathouse IV, Aug 22 2011 LINKS Table of n, a(n) for n=1..58. EXAMPLE 5 = 2+3 follows 3, 7 = 5+2 follows 5, 17 = 2+3+5+7 follows 7. MATHEMATICA (* first do *) Needs["DiscreteMath`Combinatorica`"] (* then *) f[lst_List] := Block[{k = Length@ lst, p = Infinity, q}, lmt = If[k > 5, Sum[Binomial[k, i], {i, 2, 4}], 2^k + 1]; k++; While[k < lmt, q = Plus @@ NthSubset[k, lst]; If[ ! MemberQ[lst, q] && PrimeQ@q && q < p, p = q]; k++ ]; Append[lst, p]]; Nest[f, {2, 3}, 58] (* Robert G. Wilson v, Apr 08 2007 *) PROG (PARI) prevprime(p)={ if( nextprime(p-1)

2))); p=bitor(p-3, 1); while( nextprime(p) > p, p-=2 ); p } \ decomp(n, p)={local(d); if(!p, if(n==2 || n==3, return([n]), p=n), p=min(n, p)); while( p=prevprime(p), if( bittest(disallowed, p), next); if( (n<2*p && isprime(n-p) && !bittest(disallowed, n-p) && d=[n-p]) || d=decomp( n-p, p ), return(concat(d, p)) ))} \ disallowed=0; forprime(p=1, 10^4, if(decomp(p), print1(p", "), disallowed+=1<6, prime(n+3), [2, 3, 5, 7, 17, 19][n]) \\ Charles R Greathouse IV, Aug 22 2011 (Python) from sympy import isprime; P = [2, 3]; B = {2, 3, 5} while P[-1] < 300: S = set() for i in B: if isprime(i) and i > P[-1]: P.append(i); break for j in B: S.add(j+P[-1]) B = B|S print(*P, sep = ', ') # Ya-Ping Lu, Dec 27 2023 CROSSREFS Sequence in context: A089968 A164060 A356405 * A090432 A301918 A127042 Adjacent sequences: A113026 A113027 A113028 * A113030 A113031 A113032 KEYWORD easy,nonn AUTHOR Amarnath Murthy, Jan 03 2006 EXTENSIONS More terms from Ryan Murphy (murphy(AT)minegoboom.com), Jan 24 2006 Edited by N. J. A. Sloane at the suggestion of Andrew S. Plewe, May 14 2007 STATUS approved

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Last modified August 11 11:53 EDT 2024. Contains 375069 sequences. (Running on oeis4.)