 All about flooble | fun stuff | Get a free chatterbox | Free JavaScript | Avatars  perplexus dot info  A common feature (Posted on 2017-10-29) 22 & 4,937,775 share a certain very distinctive feature.

1. What is it?
2. How would you call such numbers?
3. Please list some samples with the above feature.

 No Solution Yet Submitted by Ady TZIDON No Rating Comments: ( Back to comment list | You must be logged in to post comments.) researched solution Comment 1 of 1
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

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

 Posted by Charlie on 2017-10-29 11:35:41 Please log in:

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