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 A071139 Numbers k such that the sum of distinct primes dividing k is divisible by the largest prime dividing k. 13
 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, 25, 27, 29, 30, 31, 32, 37, 41, 43, 47, 49, 53, 59, 60, 61, 64, 67, 70, 71, 73, 79, 81, 83, 89, 90, 97, 101, 103, 107, 109, 113, 120, 121, 125, 127, 128, 131, 137, 139, 140, 149, 150, 151, 157, 163, 167, 169, 173, 179, 180 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS All primes and prime powers are terms, as are certain other composites (see Example section). If k is a term then every multiple of k having no prime factors other than those of k are also terms. E.g., since 286 = 2*11*13 is a term, so are 572 = 286*2 and 3146 = 286*11. If k = 2*p*q where p and q are twin primes, then sum = 2+p+q = 2q is divisible by q, the largest prime factor, so 2*A037074 is a subsequence. LINKS Reinhard Zumkeller, Table of n, a(n) for n = 1..10000 FORMULA A008472(k)/A006530(k) is integer. EXAMPLE 30 = 2*3*5; sum of distinct prime factors is 2+3+5 = 10, which is divisible by 5, so 30 is a term; 2181270 = 2*3*5*7*13*17*47; sum of distinct prime factors is 2+3+5+7+13+17+47 = 94, which is divisible by 47, so 2181270 is a term. MATHEMATICA ffi[x_] := Flatten[FactorInteger[x]] lf[x_] := Length[FactorInteger[x]] ba[x_] := Table[Part[ffi[x], 2*w-1], {w, 1, lf[x]}] sb[x_] := Apply[Plus, ba[x]] ma[x_] := Part[Reverse[Flatten[FactorInteger[x]]], 2] Do[s=sb[n]/ma[n]; If[IntegerQ[s], Print[{n, ba[n]}]], {n, 2, 1000000}] PROG (Haskell) a071139 n = a071139_list !! (n-1) a071139_list = filter (\x -> a008472 x `mod` a006530 x == 0) [2..] -- Reinhard Zumkeller, Apr 18 2013 (PARI) isok(n) = if (n != 1, my(f=factor(n)[, 1]); (sum(k=1, #f~, f[k]) % vecmax(f)) == 0); \\ Michel Marcus, Jul 09 2018 CROSSREFS Cf. A008472, A006530, A000961, A025475, A037074. Sequence in context: A030230 A089352 A086486 * A326837 A326847 A298538 Adjacent sequences:  A071136 A071137 A071138 * A071140 A071141 A071142 KEYWORD nonn AUTHOR Labos Elemer, May 13 2002 EXTENSIONS Edited by Jon E. Schoenfield, Jul 08 2018 STATUS approved

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Last modified August 20 18:11 EDT 2019. Contains 326154 sequences. (Running on oeis4.)