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A284102
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Numbers that are the sum of 10 consecutive primes and also the sum of 10 consecutive semiprimes.
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1
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6504, 12946, 12990, 19052, 19764, 21490, 31638, 35604, 41300, 42364, 45212, 52528, 58104, 60034, 63400, 66662, 67858, 69880, 74090, 74824, 78542, 88844, 96256, 96346, 97818, 104584, 106970, 111122, 113120, 117540, 125384
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OFFSET
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1,1
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LINKS
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EXAMPLE
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a(1)=6504 because 6504 is the sum of 10 consecutive primes A000040(114..114+9)={619,631,641,643,647,653,659,661,673,677} and also
6504 is the sum of 10 consecutive semiprimes A001358(192..192+9)={629,633,634,635,649,655,662,667,669,671}.
Note that a(1) = 6504 = A283873(10).
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MAPLE
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N:= 10^6:
P:= select(isprime, [$1..N]):
S:= select(t -> numtheory:-bigomega(t)=2, [$1..N]):
P10:= {seq(add(P[i], i=m..m+9), m=1..nops(P)-9)}:
S10:= {seq(add(S[i], i=m..m+9), m=1..nops(S)-9)}:
sort(convert(P10 intersect S10, list)); # Robert Israel, Mar 20 2017
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MATHEMATICA
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With[{nn = 12600}, Intersection[Total /@ Partition[Prime@ Range@ PrimePi@ nn, 10, 1], Total /@ Partition[Select[Range@ nn, PrimeOmega@ # == 2 &], 10, 1]]] (* Michael De Vlieger, Mar 20 2017 *)
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PROG
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(PARI) list(lim)=if(lim<6504, return([])); my(v=List(), u=v, P=primes(9), x=(lim+10*log(lim))\1, t); forprime(p=2, x\2, forprime(q=2, min(x\p, p), listput(u, p*q))); u=Set(u); while(u[#u]+1+(t=sum(i=0, 8, u[#u-i]))<=lim, for(n=x+1, lim-t, if(issemi(n), u=concat(u, n); next(2))); break); for(i=1, #u-9, u[i]+=sum(j=1, 9, u[i+j])); t=vecsum(P); forprime(p=P[#P]+1, , t+=p; if(t>lim, break); if(setsearch(u, t), listput(v, t)); t-=P[1]; P=concat(P[2..9], p)); Vec(v) \\ Charles R Greathouse IV, Mar 20 2017
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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