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 A212863 Numbers k such that the sum of prime factors of k (counted with multiplicity) equals five times the largest prime divisor of k. 1
 32, 192, 216, 243, 3125, 3750, 4000, 4500, 4800, 5120, 5400, 5760, 6075, 6480, 7290, 16807, 24010, 28812, 34300, 41160, 43904, 46305, 49000, 49392, 55125, 55566, 58800, 62720, 65625, 66150, 70000, 70560, 75264, 78750, 79380, 84000, 84672, 89600, 94500, 95256 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS The numbers prime(n)^5 are in the sequence. Also contains 4^p p and 2^p p^3 for any prime p>2, and 3^p p^2 for any prime > 3. - Robert Israel, Jun 20 2017 From David A. Corneth, Apr 24 2020: (Start) Suppose we look for terms below (inclusive) u. Let maxp be the largest prime factor that is the multiple of at least one of those terms. Then maxp is the largest prime below (inclusive) u^(1/5). Proof: The sum of prime factors counted with multiplicity of a term t divisible by maxp is 5*maxp. The smallest product of primes summing to 5*maxp where the largest prime factor of t is maxp is maxp^5 which must be <= u. Solving this gives maxp is the largest prime below (inclusive) u^(1/5). This enables us to search through the positive integers via a tree starting at 1. (End) LINKS David A. Corneth, Table of n, a(n) for n = 1..10000 (first 200 terms from Robert Israel. Following 4800 terms from Amiram Eldar) David A. Corneth, PARI program David A. Corneth, Terms <= 10^12 EXAMPLE 192 is in the sequence because 192 = 2^6 * 3 => sum of prime divisors = 2*6 + 3 = 15 = 5*3 where 3 is the greatest prime divisor. MAPLE with(numtheory):A:= proc(n) local e, j; e := ifactors(n)[2]: add (e[j][1]*e[j][2], j=1..nops(e)) end: for m from 2 to 500000 do: x:=factorset(m):n1:=nops(x):if A(m)=5*x[n1] then printf(`%d, `, m):else fi:od: MATHEMATICA Select[Range[2, 10^5], Plus @@ Times @@@ (f = FactorInteger[#]) == 5 * f[[-1, 1]] &] (* Amiram Eldar, Apr 24 2020 *) PROG (PARI) See Corneth link \\ David A. Corneth, Apr 24 2020 CROSSREFS Cf. A001414, A163836, A212861, A212862. Sequence in context: A233387 A200840 A208925 * A019560 A130811 A350740 Adjacent sequences: A212860 A212861 A212862 * A212864 A212865 A212866 KEYWORD nonn,easy AUTHOR Michel Lagneau, May 29 2012 STATUS approved

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Last modified August 15 03:10 EDT 2024. Contains 375172 sequences. (Running on oeis4.)