A127313 - OEIS

%I #11 Sep 07 2025 09:00:22

%S 7,37,53,59,61,67,71,83,97,149,173,181,233,271,277,283,307,421,521,

%T 541,569,613,617,673,691,719,809,859,877,971,983,1031,1039,1181,1237,

%U 1297,1319,1381,1423,1453,1459,1471,1483,1543,1579,1607,1609,1787,1861,1931

%N Primes p such that p+q+r, where p, q, r are consecutive primes, equals s+t+1, where s and t are consecutive primes.

%C The smallest of three consecutive primes whose sum is equal to 1 plus the sum of two consecutive primes.

%e 37, 41, 43 are consecutive primes, 37+41+43 = 121; 59 and 61 are consecutive primes, 59+61+1 = 121. Hence 37 is a term.

%e 19, 23, 29 are consecutive primes, 19+23+29 = 71; 31 and 37 are consecutive primes, 31+37+1 = 69 < 71; 37 and 41 are the next pair of consecutive primes, 37+41+1 = 79 > 71. Hence there are no consecutive primes s and t with s+t+1 = 71 and 19 is not in the sequence.

%p B:={seq(1+ithprime(k)+ithprime(k+1),k=1..500)}: a:=proc(n) if member(ithprime(n)+ithprime(n+1)+ithprime(n+2),B)=true then ithprime(n) else fi end: seq(a(n),n=1..900); # _Emeric Deutsch_, Apr 01 2007

%o (Magma) [ p: p in PrimesInInterval(3, 1950) | not IsPrime(k) and PreviousPrime(k)+NextPrime(k) eq 2*k where k is (p+NextPrime(p)+NextPrime(NextPrime(p))-1) div 2 ]; /* _Klaus Brockhaus_, Apr 03 2007 */

%K nonn

%O 1,1

%A _J. M. Bergot_, Mar 28 2007

%E Edited, corrected and extended by _Emeric Deutsch_ and _Klaus Brockhaus_, Apr 01 2007