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

1,1

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

Links

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

Bill McEachen, A269844_vs_A001122

Example

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.

Mathematica

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

Prog

(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

Crossrefs

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

Cf. A005597, A271780.

Sequence in context: A225754 A098973 A125742 * A116440 A098720 A115091

Adjacent sequences: A269841 A269842 A269843 * A269845 A269846 A269847

Keyword

nonn,easy

Author

Bill McEachen, Mar 06 2016

Status

approved