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

1,1

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

Cf. A006753.

Sequence in context: A022385 A244411 A213240 * A006753 A098836 A204341

Adjacent sequences: A279311 A279312 A279313 * A279315 A279316 A279317

Keyword

nonn

Author

Ely Golden, Dec 09 2016

Status

approved