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A269844 Primes equal to the sum of a pair of consecutive integers which are both squarefree. 1
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 (list; graph; refs; listen; history; text; internal format)
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

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

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

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Last modified October 22 16:41 EDT 2018. Contains 316490 sequences. (Running on oeis4.)