OFFSET
1,1
COMMENTS
Supersequence of A006753 (Smith numbers).
Sequence is proven infinite due to the infinitude of the Smith numbers.
Can be generalized for other moduli. Setting the modulus to 1 yields the composite numbers. Setting the modulus to m (m>=2) yields the supersequence which includes the Smith numbers in base (m+1). Of course, m=1 includes all Smith numbers for any base.
LINKS
Ely Golden, Table of n, a(n) for n = 1..10000
EXAMPLE
105 is a member as 105 = 3*5*7 with 105 mod 9 = 6 and (3+5+7) mod 9 = 15 mod 9 = 6.
MATHEMATICA
Select[Range[4, 860], Function[n, And[CompositeQ@ n, Mod[#, 9] == Mod[n, 9] &@ Total@ Flatten@ Map[ConstantArray[#1, #2] & @@ # &, FactorInteger@ n]]]] (* Michael De Vlieger, Dec 10 2016 *)
cnnQ[n_]:=CompositeQ[n]&&Mod[Total[Flatten[Table[#[[1]], #[[2]]]&/@ FactorInteger[ n]]], 9]==Mod[n, 9]; Select[Range[900], cnnQ] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jul 17 2017 *)
PROG
(SageMath)
def factorSum(f):
s=0
for c in range(len(f)):
s+=(f[c][0]*f[c][1])
return s
#this variable affects the modulus
modulus=9
c=2
index=1
while(index<=10000):
f=list(factor(c))
if(((len(f)>1)|(f[0][1]>1))&(factorSum(f)%modulus==c%modulus)):
print(str(index)+" "+str(c))
index+=1
c+=1
print("complete")
(PARI) isok(n) = !isprime(n) && (f=factor(n)) && ((n % 9) == (sum(k=1, #f~, f[k, 1]*f[k, 2]) % 9)); \\ Michel Marcus, Dec 10 2016
CROSSREFS
KEYWORD
nonn
AUTHOR
Ely Golden, Dec 09 2016
STATUS
approved