A279314 Composite numbers n such that the sum of the prime factors of n, with multiplicity, is congruent to n (mod 9).

4, 22, 27, 58, 85, 94, 105, 114, 121, 150, 166, 202, 204, 222, 224, 265, 274, 315, 319, 342, 346, 355, 378, 382, 391, 438, 445, 450, 454, 483, 517, 526, 535, 540, 560, 562, 576, 588, 612, 627, 634, 636, 640, 645, 648, 654, 663, 666, 690, 697, 706, 728, 729, 762, 778, 825, 840, 841, 852

Offset: 1

Author

Ely Golden, Dec 09 2016

Keywords

• nonn

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.

Examples

			105 is a member as 105 = 3*5*7 with 105 mod 9 = 6 and (3+5+7) mod 9 = 15 mod 9 = 6.
		

Links

• Ely Golden, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A006753.

Programs

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 *)

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

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")