A185347
Semiprimes that are the sum of 10 consecutive primes.
Original entry on oeis.org
129, 158, 382, 1114, 1546, 2374, 2582, 3446, 3578, 6218, 6826, 7978, 8266, 9298, 9382, 10202, 12946, 14002, 15178, 15406, 15766, 16382, 16466, 17282, 17362, 18374, 18838, 19226, 19606, 23878, 24074, 25154, 25642, 26206, 29782, 30034, 30638, 32902, 33526, 34862, 34934, 35678, 35978, 36602
Offset: 1
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(* First run the program for A109611 to define semiPrimeQ *) Select[Table[Plus@@Prime[Range[n, n + 9]], {n, 500}], semiPrimeQ] (* Alonso del Arte, Feb 15 2011 *)
Select[Total/@Partition[Prime[Range[600]],10,1],PrimeOmega[#]==2&] (* Harvey P. Dale, Sep 06 2014 *)
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{s=129;for(n=1,2000,if(2==bigomega(s), print1(s", ")); s=s-prime(n)+prime(n+10))}
A217985
Primes that are arithmetical average of 100 consecutive primes.
Original entry on oeis.org
5009, 6047, 8039, 9311, 9497, 11171, 15137, 17029, 18869, 20983, 26339, 28627, 31699, 35023, 38833, 43579, 49199, 49727, 50549, 60089, 68447, 72469, 76603, 87931, 89659, 98809, 99733, 102547, 111217, 143281, 143831, 150431, 185401, 191341, 195137, 195161
Offset: 1
a(1) = 5009 because prime(620)+...+prime(719) = 4583+...+5441 = 500900.
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h = 100; t = {}; s = Prime[Range[h]]; pCnt = h; Do[sm = Total[s]; If[Mod[sm, h] == 0 && PrimeQ[sm/h], AppendTo[t, sm/h]]; pCnt++; s = Append[Rest[s], Prime[pCnt]], {20000}]; t (* T. D. Noe, Oct 22 2012 *)
A284102
Numbers that are the sum of 10 consecutive primes and also the sum of 10 consecutive semiprimes.
Original entry on oeis.org
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
Offset: 1
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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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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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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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
A125270
Coefficient of x^2 in polynomial whose zeros are 5 consecutive primes starting with the n-th prime.
Original entry on oeis.org
1358, 3954, 10478, 22210, 43490, 78014, 129530, 206650, 324350, 466270, 621466, 853742, 1132130, 1436690, 1870850, 2388050, 2886370, 3440410, 4133410, 4904906, 5926654, 7195670, 8425430, 9792950, 11040910, 12098990, 13917898, 16097810
Offset: 1
Cf.
A001043,
A034961,
A034963,
A034964,
A127333,
A127334,
A127335,
A127336,
A127337,
A127338,
A127339,
A127340,
A127341,
A127342,
A127343,
A127345,
A127346,
A127347,
A127348,
A127349,
A127351,
A037171,
A034962,
A034965,
A082246,
A082251,
A070934,
A006094,
A046301,
A046302,
A046303,
A046324,
A046325,
A046326,
A046327,
A127489.
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a = {}; Do[AppendTo[a, (Prime[x] Prime[x + 1] Prime[x + 2] + Prime[x] Prime[x + 1] Prime[x + 3] + Prime[x] Prime[x + 1] Prime[x + 4] + Prime[x] Prime[x + 2] Prime[x + 3] + Prime[x] Prime[x + 2] Prime[x + 4] + Prime[x] Prime[x + 3] Prime[x + 4] + Prime[x + 1] Prime[x + 2] Prime[x + 3] + Prime[x + 1] Prime[x + 2] Prime[x + 4] + Prime[x + 1] Prime[x + 3] Prime[x + 4] + Prime[x + 2] Prime[x + 3] Prime[x + 4])], {x, 1, 100}]; a
fcp[{p_,q_,r_,s_,t_}]:=p*q(r+s+t)+(p+q)r(s+t)+(p+q+r)s*t; fcp/@Partition[ Prime[ Range[40]],5,1] (* Harvey P. Dale, Sep 05 2014 *)
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