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A215199 Smallest number k such that k and k+1 are both of the form p*q^n where p and q are distinct primes. 5
14, 44, 135, 2511, 8991, 29888, 916352, 12393728, 155161088, 2200933376, 6856828928, 689278976, 481758175232, 3684603215871, 35419114668032, 2035980763136, 174123685117952, 9399153082499072, 19047348965998592, 203368956137832447, 24217192574746623, 2503092614937444351 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
a(15) <= 35419114668032. - Donovan Johnson, Aug 22 2012
If k is a term such that k = p*q^n and k+1 = r*s^n, where p,q,r,s are primes, then clearly q != s. Conjecture: q and s are either 2 or 3 for all terms. - Chai Wah Wu, Mar 10 2019
Since q^n and s^n are coprime, the Chinese Remainder Theorem can be used to find candidate terms to test, i.e., numbers k such that k+1 == 0 (mod s^n) and k+1 == 1 (mod q^n) (see Python code). - Chai Wah Wu, Mar 12 2019
From David A. Corneth, Mar 13 2019: (Start)
Conjecture: Let 1 <= D < 2^n be the denominator of N/D of (3/2)^n. Without loss of generality, if the conjecture above holds that (q, s) = (2, 3) then r = D + k*2^n for some n.
Example: for n = 100, we have the continued fraction of (3/2)^100 to be 406561177535215237, 2, 1, 1, 14, 9, 1, 1, 2, 2, 1, 4, 1, 2, 6, 5, 1, 195, 3, 26, 39, 6, 1, 1, 1, 2, 7, 1, 4, 2, 1, 11, 1, 25, 6, 1, 4, 3, 2, 112, 1, 2, 1, 3, 1, 3, 4, 8, 1, 1, 12, 2, 1, 3, 2, 2 from which we compute D = 519502503658624787456021964081. We find r = 1100840223501761745286594404230449 = D + 868 * 2^100 giving a(100) + 1 = r*3^100. - David A. Corneth, Mar 13 2019
LINKS
Chai Wah Wu, Table of n, a(n) for n = 1..1279 (terms 25..32 from David A. Corneth)
EXAMPLE
a(3) = 135 because 135 = 5*3^3 and 136 = 17*2^3;
a(4) = 2511 because 2511 = 31*3^4 and 2512 = 157*2^4.
MAPLE
psig := proc(n)
local s, p ;
s := [] ;
for p in ifactors(n)[2] do
s := [op(s), op(2, p)] ;
end do:
sort(s) ;
end proc:
A215199 := proc(n)
local slim, smi, sma, ca, qi, q, p, k ;
for slim from 0 do
smi := slim*1000 ;
sma := (slim+1)*1000 ;
ca := sma ;
q := 2 ;
for qi from 1 do
p := nextprime(floor(smi/q^n)-1) ;
while p*q^n < sma do
if p <> q then
k := p*q^n ;
if psig(k+1) = [1, n] then
ca := min(ca, k) ;
end if;
end if;
p := nextprime(p) ;
end do:
if q^n >= sma then
break;
end if;
q := nextprime(q) ;
end do:
if ca < sma then
return ca ;
end if;
end do:
end proc:
for n from 1 do
print(A215199(n)) ;
end do; # R. J. Mathar, Aug 07 2012
PROG
(Python)
from sympy import isprime, nextprime
from sympy.ntheory.modular import crt
def A215199(n):
l = len(str(3**n))-1
l10, result = 10**l, 2*10**l
while result >= 2*l10:
l += 1
l102, result = l10, 20*l10
l10 *= 10
q, qn = 2, 2**n
while qn <= l10:
s, sn = 2, 2**n
while sn <= l10:
if s != q:
a, b = crt([qn, sn], [0, 1])
if a <= l102:
a = b*(l102//b) + a
while a < l10:
p, t = a//qn, (a-1)//sn
if p != q and t != s and isprime(p) and isprime(t):
result = min(result, a-1)
a += b
s = nextprime(s)
sn = s**n
q = nextprime(q)
qn = q**n
return result # Chai Wah Wu, Mar 12 2019
CROSSREFS
Sequence in context: A189807 A009942 A031130 * A216258 A064348 A206215
KEYWORD
nonn
AUTHOR
Michel Lagneau, Aug 05 2012
EXTENSIONS
a(10)-a(14) from Donovan Johnson, Aug 22 2012
a(15)-a(17) from Chai Wah Wu, Mar 09 2019
a(18)-a(22) from Chai Wah Wu, Mar 10 2019
STATUS
approved

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Last modified May 25 09:23 EDT 2024. Contains 372786 sequences. (Running on oeis4.)