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A215199
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Smallest number k such that k and k+1 are both of the form p*q^n where p and q are distinct primes.
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5
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14, 44, 135, 2511, 8991, 29888, 916352, 12393728, 155161088, 2200933376, 6856828928, 689278976, 481758175232, 3684603215871, 35419114668032, 2035980763136, 174123685117952, 9399153082499072, 19047348965998592, 203368956137832447, 24217192574746623, 2503092614937444351
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OFFSET
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1,1
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COMMENTS
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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
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
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LINKS
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EXAMPLE
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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.
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MAPLE
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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:
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
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PROG
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(Python)
from sympy import isprime, nextprime
from sympy.ntheory.modular import crt
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
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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