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<DIV>Seqfans,</DIV>
<DIV> To clarify, I should have stated my
conjecture as the </DIV>
<DIV>UNION of (i) and (ii) (by using "or", not "and"), like so: </DIV>
<DIV> </DIV>
<DIV>
<DIV>Conjecture: A111075(n) is odd whenever: </DIV>
<DIV>(i) n = m^2 for all m>=1 such that 3 does not
divide m, or<BR>(ii) n = 3*A028982(m) for all m>=1.</DIV>
<DIV> </DIV>
<DIV>Also, A028982 gives "Union of nonzero squares and twice
squares."<BR> </DIV>
<DIV>Underneath it all, this may be trivial, yet for now it seems
mysterious </DIV>
<DIV>why the number 3 and the squares would determine when
A111075 is odd ...</DIV>
<DIV> </DIV>
<DIV>If someone can support the conjecture, they are welcome to submit </DIV>
<DIV>the new sequence: "n for which A111075 is odd." </DIV>
<DIV> </DIV>
<DIV>Thanks,</DIV>
<DIV> Paul</DIV></DIV>
<DIV> </DIV>
<DIV>On Wed, 12 Oct 2005 00:44:40 -0400 "Paul D. Hanna" <<A
href="mailto:pauldhanna@juno.com">pauldhanna@juno.com</A>> writes:</DIV>
<BLOCKQUOTE dir=ltr
style="PADDING-LEFT: 10px; MARGIN-LEFT: 10px; BORDER-LEFT: #000000 2px solid">
<DIV>The question that came to mind was: "when is A111075(n) odd?"</DIV>
<DIV>This gives a new sequence:</DIV>
<DIV>"n for which A111075(n) is odd"</DIV>
<DIV>1,3,4,6,12,16,24,25,27,48,49,54,64,75,96,100,108,121,147,<BR>150,169,192,196,216,243,256,289,294,300,361,363,384,400,432,<BR>484,486,507,529,588,600,625,675,676,726,768,784,841,864,867,<BR>961,972,1014,1024,1083,1156,1176,1200,1225,1323,1350,1369,</DIV>
<DIV> </DIV>
<DIV>Conjecture: A111075(n) is odd whenever: </DIV>
<DIV>(i) n = m^2 for all m>=1 such that 3 does not
divide m, and <BR>(ii) n = 3*A028982(m) for all m>=1.</DIV>
<DIV>Note that A028982 lists positive integers having an odd sum of
divisors.</DIV>
<DIV> </DIV></BLOCKQUOTE></BODY></HTML>