

A191837


Least even number m which can be written as sum of 2n primes p(1) < ... < p(2n) < m/2 such that mp(i) is also prime for i=1,...,2n.


2



48, 108, 204, 324, 624, 630, 1050, 1320, 1590, 2100, 2400, 2730, 3570, 3960, 4830, 5460, 5880, 6930, 7770, 9240, 9450, 11970, 12810, 13020, 14910, 14910, 17430, 18480, 20160, 21630, 23100, 24150, 28770, 28770, 31290, 32760, 32760, 36960, 36960, 39270, 39270, 50190, 51870, 51870
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OFFSET

2,1


COMMENTS

Original definition: In the Goldbach partitions of 2n, find the first 2n with four prime elements to sum to it; find the first 2n with six elements summing to 2n; and so for 2k elements.
Whenever there is more than one decomposition of m as sum of primes, it must be odd+odd=even. Then, only an even number of (odd prime) summands can yield m. Moreover, we restrict these summands to be the lesser one of the decompositions p+q=m, therefore we need more than 2 such summands to yield m, and a(1) is undefined.
The integers in this sequence are all congruent to 0 mod 6.
There can be more than one composition of m. E.g., for m=48, 48=5+7+17+19 and 48=7+11+13+17.
Conjecture: For all a(n), a(n)1 can be found in A014092 (numbers not the sum of two primes), and a(n)+1 can be found in A007921. (numbers not the difference of two primes).  J. Stauduhar, Aug 28 2012
From J. Stauduhar, Aug 22 2011: (Start)
All a(n) are congruent to 0 mod 6=2*3.
All a(n) >= a(7)=630 are congruent to 0 mod 30=2*3*5.
All a(n) >= a(16)=4830 are congruent to 0 mod 210=2*3*5*7.
All a(n) >= a(279)=3513510 are congruent to 0 mod 2310=2*3*5*7*11.
All a(n) >= a(1440)=137507370 are congruent to 0 mod 30030=2*3*5*7*11*13. (End)


LINKS

J. Stauduhar, Table of n, a(n) for n = 2..2500
J. Stauduhar, C program to generate sequence A191837


EXAMPLE

For 48, we have 48=5+43=7+41=17+31=19+29 (ignoring 11+37), and use 5+7+17+19 to give the first even number having four such primes summing to itself.
Similarly, 108 is the least even number with six prime elements summing to itself: 5+103=7+101=11+97=19+89=29+79=37+71 and taking 5+7+11+19+29+37=108.
a(2) = 48 = 5+7+17+19 = 7+11+13+17
a(3) = 108 = 5+7+11+19+29+37
a(9) = 1320 = 13+17+19+23+29+31+37+41+43+61+71+83+89+97+103+107+149+307


MATHEMATICA

nCk[a_, b_]:=Block[{ndx=ns= a, i=rs=b, ct=t=0}, If[(d[[1]]1)==(nsrs), For[ct=1, ct<=rs, ct++, t+=s[[d[[ct]]]]]; If[t==m, Print[sm/2, " ", t]; sm+=2; m=6; Return[False], Return[False]]]; While[d[[i]]==ndx && i>1, i; ndx]; d[[i]]+=1; i++; While[i<=rs, d[[i]]=d[[i1]]+1; ++i; ]; For[ct=1, ct<=rs, ct++, t+=s[[d[[ct]]]]; If[t>m, Break[]]]; If[t==m, Print[sm/2, " ", t]; sm+=2; m=6; Return[False]]; Return[True]]; For[sm=4; m=6, sm<=60, m+=6, s={}; sum=smndct=pct=0; For[p=5, p<m/2, p+=2, If[PrimeQ[p] && PrimeQ[mp], If[sum+p>m, Break[]]; If[smndct++<sm1, sum+=p]; AppendTo[s, p]; pct++]]; If[pct >= sm, d=Range[sm]; While[nCk[Length[s], sm]]]]; (* J. Stauduhar, Sep 07 2012*)


PROG

(PARI) a(n)=forstep(m=2, 1e9, 2, L=[]; forprime(p=1, m\21, isprime(mp)next; L=concat(L, p)); #L<2*n&next; sum(i=#L2*n+1, #L, L[i])<m&next; forvec(v=vector(2*n, i, [1, #L]), sum(i=1, 2*n, L[v[i]])==m & return(m), 2))


CROSSREFS

Sequence in context: A260294 A260287 A187931 * A044235 A044616 A182679
Adjacent sequences: A191834 A191835 A191836 * A191838 A191839 A191840


KEYWORD

nonn


AUTHOR

J. M. Bergot, Jun 17 2011


EXTENSIONS

a(4)a(5) from M. F. Hasler, Jun 21 2011
a(2) to a(5) verified; a(6) to a(10) added by S Kolman, Jul 03 2011
a(11) to a(13) added by S Kolman, Jul 04 2011
a(14) to a(14) added by S Kolman, Jul 05 2011
Confirmed a(7). a(6) corrected by J. Stauduhar, Jul 08 2011
Corrected a(8)a(14) and extended to a(2500).  J. Stauduhar, Jul 12 2011
Edited by J. Stauduhar, Aug 28 2012


STATUS

approved



