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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 (n = 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) 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 Cf. A074172, A215173, A215197, A215198. Sequence in context: A189807 A009942 A031130 * A216258 A064348 A206215 Adjacent sequences:  A215196 A215197 A215198 * A215200 A215201 A215202 KEYWORD nonn,hard 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 September 20 14:44 EDT 2021. Contains 347586 sequences. (Running on oeis4.)