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 A269844 Primes equal to the sum of a pair of consecutive integers which are both squarefree. 1

%I

%S 5,11,13,29,43,59,61,67,83,131,139,157,173,211,227,229,277,283,317,

%T 331,347,373,389,419,421,443,461,509,547,563,571,619,643,653,659,661,

%U 691,709,733,787,797,821,853,859,877,907,941,947,997,1019,1021,1069,1091,1093,1109,1123,1163,1181,1213

%N Primes equal to the sum of a pair of consecutive integers which are both squarefree.

%C The associated prime factors will always include 2 and 3.

%C Will every prime number be encountered as a prime factor from the sequence entries?

%C The sequence appears to share many of it terms with A001122.

%C What is the asymptotic behavior?

%C Conjecture: sequence has density A271780/2 = A005597*4/Pi^2 = 0.2675535... in the primes. - _Charles R Greathouse IV_, Jan 24 2018

%H Charles R Greathouse IV, <a href="/A269844/b269844.txt">Table of n, a(n) for n = 1..10000</a>

%H Bill McEachen, <a href="/A269844/a269844.png">A269844_vs_A001122</a>

%e 277 = 138 + 139 = 2*3*23 + 139 is in the sequence since both terms are squarefree.

%e 281 = 140 + 141 = 2^2*5*7 + 3*47 is not in the sequence since the former term is divisible by 2^2.

%t Select[Prime@ Range[3, 200], PrimeOmega@ # == PrimeNu@ # &[# (# + 1)] &@ Floor[#/2] &] (* _Michael De Vlieger_, Mar 07 2016 *)

%o (PARI)

%o genit(maxx)={for(i5=3,maxx,n=prime(i5);a=factor(floor(n/2.));b=factor(ceil(n/2.));clear=1;for(j5=1,omega(floor(n/2.)),if(a[j5,2]<>1,clear=0));

%o for(j7=1,omega(ceil(n/2.)),if(b[j7,2]<>1,clear=0));if(clear>0,print1(n,",")));}

%o (PARI) is(n)=isprime(n) && issquarefree(n\2) && issquarefree(n\2+1) \\ _Charles R Greathouse IV_, Jan 24 2018

%o (PARI) list(lim)=my(v=List(),t=1); forfactored(k=3,(lim+1)\2, if(vecmax(k[2][,2])>1, t=0, ; if(t && isprime(t=2*k[1]-1), listput(v,t)); t=1)); Vec(v) \\ _Charles R Greathouse IV_, Jan 24 2018

%Y Cf. A001122, primes with primitive root 2

%K nonn,easy

%O 1,1

%A _Bill McEachen_, Mar 06 2016

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Last modified November 14 19:12 EST 2018. Contains 317214 sequences. (Running on oeis4.)