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 A279314 Composite numbers n such that the sum of the prime factors of n, with multiplicity, is congruent to n (mod 9). 1

%I

%S 4,22,27,58,85,94,105,114,121,150,166,202,204,222,224,265,274,315,319,

%T 342,346,355,378,382,391,438,445,450,454,483,517,526,535,540,560,562,

%U 576,588,612,627,634,636,640,645,648,654,663,666,690,697,706,728,729,762,778,825,840,841,852

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

%C Supersequence of A006753 (Smith numbers).

%C Sequence is proven infinite due to the infinitude of the Smith numbers.

%C 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.

%H Ely Golden, <a href="/A279314/b279314.txt">Table of n, a(n) for n = 1..10000</a>

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

%t 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 *)

%t cnnQ[n_]:=CompositeQ[n]&&Mod[Total[Flatten[Table[#[],#[]]&/@ FactorInteger[ n]]],9]==Mod[n,9]; Select[Range,cnnQ] (* Requires Mathematica version 10 or later *) (* _Harvey P. Dale_, Jul 17 2017 *)

%o (SageMath)

%o def factorSum(f):

%o s=0

%o for c in range(len(f)):

%o s+=(f[c]*f[c])

%o return s

%o #this variable affects the modulus

%o modulus=9

%o c=2

%o index=1

%o while(index<=10000):

%o f=list(factor(c))

%o if(((len(f)>1)|(f>1))&(factorSum(f)%modulus==c%modulus)):

%o print(str(index)+" "+str(c))

%o index+=1

%o c+=1

%o print("complete")

%o (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

%Y Cf. A006753.

%K nonn

%O 1,1

%A _Ely Golden_, Dec 09 2016

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Last modified May 29 04:33 EDT 2020. Contains 334697 sequences. (Running on oeis4.)