OFFSET
1,1
COMMENTS
Subsequence of A014574. The terms represent quintuples of twin primes. As there are only 31 terms < 10^11, the accordance with the k-tuple conjecture is not very good, see links "k-tuple conjecture" and A350541, "Test of the k-tuple conjecture". Moreover, the formalism of the conjecture allows the evaluation of the expected frequencies of eight types of quintuples relative to the frequency of all quintuples. The differences are considerable:
relative frequencies
Example observed expected
(1) 11/31=35.5% 23.7%
(2) 5/31=16.1% 15.0%
(3) 3/31= 9.7% 7.5%
(4) 6/31=19.4% 23.7%
(5) 3/31= 9.7% 15.0%
(6) 2/31= 6.5% 3.8%
(7) 1/31= 3.2% 7.5%
(8) 0 3.8%
Generalization:
Twin primes x such that x is the first and x+d the last of m successive twins.
m d
1 0 A014574(n) twin primes
2 6 A173037(n)-3
3 12 Only one quadruple: (6,12,18,30)
3 18 A350541
4 24 Only one quintuple: (6,12,18,30,42)
4 30 A350542
5 36 6, 39713433660, 66419473020, 71525244600*
5 42 18873492, 180929682, 1170073332, 2550576612, 5807487204, 27523454232, 33497368554, 50062053714, 63167632254, 86883508944, 99939276954*
5 48 Current sequence
Annotations:
*The number of terms < 10^11 is too small for submitting a new sequence.
(m=5,d=30) is empty for divisibility reasons.
LINKS
Eric Weisstein's World of Mathematics, k-Tuple Conjecture.
EXAMPLE
The quintuples of twins have the form (x,x+a,x+b,x+c,x+d)
(a,b,c,d) least example
(1) 6,18,30,48 x= 12
(2) 6,30,36,48 x= 123919212
(3) 6,18,36,48 x= 123240097962
(4) 18,30,42,48 x= 124046054430
(5) 12,18,42,48 x=1217444777880
(6) 12,18,30,48 x=1220905115040
(7) 12,30,42,48 x=1227350615220
(8) 18,30,42,48 x>10^11
PROG
(Maxima)
block(twin:[6], n:1, p1:11, j2:1, nmax: 3,
/*returns nmax terms*/
m:5, d:48, w: makelist(-d, i, 1, m),
while n<nmax do(
p2: next_prime(p1), if p2-p1=2 then(
k:p1+1, j1:j2, j2:1+ mod(j2, m), w[j1]:k,
if w[j1]-w[j2]=d then(n:n+1, twin: append(twin, [k-d]))),
p1:p2), twin);
CROSSREFS
KEYWORD
nonn
AUTHOR
Gerhard Kirchner, Jan 07 2022
STATUS
approved