Having searched Sloane's OEIS for 4937775, I'll answer #2 first: I'd call them Smith numbers or joke numbers; that's what they're called in Sloane's A006753.
Sloane gives the answer to #1:
composite numbers n such that sum of digits of n = sum of digits of prime factors of n (counted with multiplicity)
And for #3:
4, 22, 27, 58, 85, 94, 121, 166, 202, 265, 274, 319, 346, 355, 378, 382, 391, 438, 454, 483, 517, 526, 535, 562, 576, 588, 627, 634, 636, 645, 648, 654, 663, 666, 690, 706, 728, 729, 762, 778, 825, 852, 861, 895, 913, 915, 922, 958, 985, 1086, 1111, 1165
... and some interesting asides:
Of course primes also have this property, trivially.
The current puzzle of course doesn't necessarily imply that the primes are excluded from the class that is required, so possibly the puzzle is looking for a class consisting of primes and Smith numbers; maybe we could call them identiSOD numbers.
From Sloane also:
a(133809) = 4937775 is the first Smith number historically: 4937775 = 3*5*5*65837 and 4+9+3+7+7+7+5 = 3+5+5+(6+5+8+3+7) = 42, Albert Wilansky coined the term Smith number when he noticed the defining property in the phone number of his brother-in-law Harold Smith: 493-7775.
There are 248483 7-digit Smith numbers, corresponding to US phone numbers without area codes (like 4937775). - Charles R Greathouse IV, May 19 2013
I have confirmed via the below program that there are indeed 248483 7-digit Smith numbers (about a 15-minute run time for the program). However there is no correspondence with US phone numbers. 1000165, 1155122, 1454613, to take just a sampling, all begin with the digit 1, which no US phone number has, and 4691245, a valid US phone number, again, to take just one of many, is not a Smith number (it's 1 more than Smith number 4691244).
Since there are 834564 lines in the output file, there are 834564 - 248483 = 586081 7-digit prime numbers in addition to the 248483 7-digit Smith numbers.
The program below creates a file with all the 7-digit prime (so marked) and Smith numbers.
DefDbl A-Z
Dim crlf$, ct, fct(20, 1)
Function mform$(x, t$)
a$ = Format$(x, t$)
If Len(a$) < Len(t$) Then a$ = Space$(Len(t$) - Len(a$)) & a$
mform$ = a$
End Function
Private Sub Form_Load()
Text1.Text = ""
crlf$ = Chr(13) + Chr(10)
Form1.Visible = True
Open "smith numbers.txt" For Output As #2
For n = 1000000 To 9999999
DoEvents
If prmdiv(n) = n Then
Print #2, n; " prime"
ElseIf smith(n) Then
Print #2, n
ct = ct + 1
End If
If n Mod 10000 = 0 Then Text1.Text = Text1.Text & Str(n)
Next
Close 2
Text1.Text = Text1.Text & crlf & ct & " done"
End Sub
Function factor(num)
diffCt = 0: good = 1
n = Abs(num): If n > 0 Then limit = Sqr(n) Else limit = 0
If limit <> Int(limit) Then limit = Int(limit + 1)
dv = 2: GoSub DivideIt
dv = 3: GoSub DivideIt
dv = 5: GoSub DivideIt
dv = 7
Do Until dv > limit
GoSub DivideIt: dv = dv + 4 '11
GoSub DivideIt: dv = dv + 2 '13
GoSub DivideIt: dv = dv + 4 '17
GoSub DivideIt: dv = dv + 2 '19
GoSub DivideIt: dv = dv + 4 '23
GoSub DivideIt: dv = dv + 6 '29
GoSub DivideIt: dv = dv + 2 '31
GoSub DivideIt: dv = dv + 6 '37
If INKEY$ = Chr$(27) Then s$ = Chr$(27): Exit Function
Loop
If n > 1 Then diffCt = diffCt + 1: fct(diffCt, 0) = n: fct(diffCt, 1) = 1
factor = diffCt
Exit Function
DivideIt:
cnt = 0
Do
q = Int(n / dv)
If q * dv = n And n > 0 Then
n = q: cnt = cnt + 1: If n > 0 Then limit = Sqr(n) Else limit = 0
If limit <> Int(limit) Then limit = Int(limit + 1)
Else
Exit Do
End If
Loop
If cnt > 0 Then
diffCt = diffCt + 1
fct(diffCt, 0) = dv
fct(diffCt, 1) = cnt
End If
Return
End Function
Function sod(n)
s$ = LTrim(Str(n))
tot = 0
For i = 1 To Len(s$)
tot = tot + Val(Mid(s$, i, 1))
Next
sod = tot
End Function
Function smith(n)
s1 = sod(n)
f = factor(n)
s2 = 0
For i = 1 To f
s2 = s2 + sod(fct(i, 0)) * fct(i, 1)
Next
If s1 = s2 Then
smith = 1
Else
smith = 0
End If
End Function
Function prmdiv(num)
Dim n, dv, q
If num = 1 Then prmdiv = 1: Exit Function
n = Abs(num): If n > 0 Then limit = Sqr(n) Else limit = 0
If limit <> Int(limit) Then limit = Int(limit + 1)
dv = 2: GoSub DivideIt
dv = 3: GoSub DivideIt
dv = 5: GoSub DivideIt
dv = 7
Do Until dv > limit
GoSub DivideIt: dv = dv + 4 '11
GoSub DivideIt: dv = dv + 2 '13
GoSub DivideIt: dv = dv + 4 '17
GoSub DivideIt: dv = dv + 2 '19
GoSub DivideIt: dv = dv + 4 '23
GoSub DivideIt: dv = dv + 6 '29
GoSub DivideIt: dv = dv + 2 '31
GoSub DivideIt: dv = dv + 6 '37
Loop
If n > 1 Then prmdiv = n
Exit Function
DivideIt:
Do
q = Int(n / dv)
If q * dv = n And n > 0 Then
prmdiv = dv: Exit Function
Else
Exit Do
End If
Loop
Return
End Function
Sample output (the ones not marked prime are Smith):
1000003 prime
1000033 prime
1000037 prime
1000039 prime
1000081 prime
1000099 prime
1000117 prime
1000121 prime
1000133 prime
1000151 prime
1000159 prime
1000165
1000171 prime
1000183 prime
1000187 prime
1000193 prime
1000199 prime
1000211 prime
1000213 prime
1000231 prime
1000249 prime
1000253 prime
1000273 prime
1000289 prime
1000291 prime
1000303 prime
1000313 prime
1000329
1000333 prime
1000357 prime
1000367 prime
...
4916453 prime
4916461
4916467 prime
4916481
4916489 prime
4916557 prime
4916581 prime
4916591 prime
4916603 prime
4916629 prime
4916633 prime
4916637
4916643
4916659
4916669 prime
4916683 prime
4916707 prime
4916709
4916713
4916717
4916719 prime
4916741 prime
4916759 prime
4916771 prime
...
9999433 prime
9999454
9999463 prime
9999469 prime
9999481 prime
9999511 prime
9999520
9999533 prime
9999562
9999593 prime
9999601 prime
9999633
9999637 prime
9999653 prime
9999659 prime
9999667 prime
9999677 prime
9999713 prime
9999739 prime
9999742
9999749 prime
9999761 prime
9999778
9999823 prime
9999863 prime
9999877 prime
9999883 prime
9999889 prime
9999895
9999901 prime
9999907 prime
9999922
9999929 prime
9999931 prime
9999937 prime
9999943 prime
9999971 prime
9999973 prime
9999991 prime
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Posted by Charlie
on 2017-10-29 11:35:41 |
researched solution