A071139 Numbers k such that the sum of distinct primes dividing k is divisible by the largest prime dividing k.

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

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: A366914 A089352 A086486 * A327473 A326837 A326847

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