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

5, 11, 13, 29, 43, 59, 61, 67, 83, 131, 139, 157, 173, 211, 227, 229, 277, 283, 317, 331, 347, 373, 389, 419, 421, 443, 461, 509, 547, 563, 571, 619, 643, 653, 659, 661, 691, 709, 733, 787, 797, 821, 853, 859, 877, 907, 941, 947, 997, 1019, 1021, 1069, 1091, 1093, 1109, 1123, 1163, 1181, 1213

Offset: 1

Author

Bill McEachen, Mar 06 2016

Keywords

• nonn
• easy

Comments

The associated prime factors will always include 2 and 3.

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

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

What is the asymptotic behavior?

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

The prime terms of A179017 (except 3). - Bill McEachen, Oct 21 2021

Examples

			277 = 138 + 139 = 2*3*23 + 139 is in the sequence since both terms are squarefree.
281 = 140 + 141 = 2^2*5*7 + 3*47 is not in the sequence since the former term is divisible by 2^2.
		

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
• Bill McEachen, A269844_vs_A001122

Crossrefs

Cf. A001122 (primes with primitive root 2), A179017.

Cf. A005597, A271780.

Programs

Mathematica

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

PARI

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

PARI

is(n)=isprime(n) && issquarefree(n\2) && issquarefree(n\2+1) \\ Charles R Greathouse IV, Jan 24 2018

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