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A279314
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Composite numbers n such that the sum of the prime factors of n, with multiplicity, is congruent to n (mod 9).
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1
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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
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OFFSET
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1,1
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COMMENTS
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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.
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LINKS
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EXAMPLE
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105 is a member as 105 = 3*5*7 with 105 mod 9 = 6 and (3+5+7) mod 9 = 15 mod 9 = 6.
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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