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A071139
Numbers k such that the sum of distinct primes dividing k is divisible by the largest prime dividing k.
14
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
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
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
KEYWORD
nonn
AUTHOR
Labos Elemer, May 13 2002
EXTENSIONS
Edited by Jon E. Schoenfield, Jul 08 2018
STATUS
approved