OFFSET
2,1
COMMENTS
a(n) is the smallest number of the form described above whose symmetric representation of sigma, SRS(a(n)), consists of 2 parts that have a unimodal width pattern of type 121 and that meet at the diagonal. Since floor( (sqrt(8*a(n) + 1) - 1)/2 ) = 2^(k+1) * p, the central 0 width extent of SRS(a(n)) equals 0.
Conjecture: The sequence is infinite.
LINKS
Chai Wah Wu, Table of n, a(n) for n = 2..75 [Notice that a(15) has 355 digits, so the graph cuts off after 14 terms. - N. J. A. Sloane, Feb 18 2024]
FORMULA
a(n) = min( 2^k * p * (2^(k+1) * p + 1) : p = prime(n), 2 < p < 2^(k+1), 2^(k+1) * p + 1 is prime ), n>=2.
EXAMPLE
a(2) = 78 = 2 * 3 * 13 = A262259(3) and SRS(78) consists of 2 unimodal parts 121 that meet at diagonal position (54, 54).
a(4) = 6328 = 8 * 7 * 113 = A262259(11) which demonstrates that 2^k < p < 2^(k+1) need not be true.
a(15) with k = 582 and p = 47, its second prime factor 2^(k+1) * p + 1 has 178 digits so that a(15) has 355 digits.
a(16) = 24129129742336 = 2^16 * 53 * 6946817.
Table of records of number of digits a(2) through a(500):
sequence index 2 3 4 5 6 7 15 76 419 438
number of digits 2 3 4 5 7 12 355 3854 5856 20049
MATHEMATICA
minExp[p_] := Module[{k=Floor[Log[2, p]]}, NestWhile[#+1&, k+1, !PrimeQ[2^# p+1]&]-1]/; PrimeQ[p]
a370209[p_] := Module[{k=minExp[p]}, 2^k p(2^(k+1)p+1)]/; PrimeQ[p]
Map[a370209[Prime[#]]&, Range[2, 14]] (* a(15) is too large to list *)
PROG
(Python)
from itertools import count
from sympy import prime, isprime
def A370209(n):
p = prime(n)
return next((p<<k)*m for k in count(p.bit_length()-1) if isprime(m:=(p<<k+1)+1)) # Chai Wah Wu, Feb 17 2024
CROSSREFS
KEYWORD
nonn
AUTHOR
Hartmut F. W. Hoft, Feb 11 2024
STATUS
approved