d= m^p-n^q

[ZAKIRS]

Don! Thanks for the great job!
 
Could you say exact number of perfect powers  < 10^19,
and in general what about function pp(n), number of  perfect powers < n?

And to all who can/wish to fill/extend this list:

d=m^p-n^q:

{d,m,p,n,q}=
{{1, 3, 2, 2, 3},
{2, 3, 3, 5, 3},
{3, 2, 2, 1, 2},
{4, 2, 3, 2, 2},
{5, 3, 2, 2, 2},
{6, 6, 1, 0, 1},
{7, 2, 4, 3, 2},
{8, 2, 4, 2, 3},
{9, 6, 2, 3, 3},
{10, 13, 3, 3, 7},
{11,  3,  3, 4, 2},
{12,  4, 2, 2, 2}, 
{13,  7, 2, 6, 2},
{14,  14, 1, 0, 1},
{15, 4, 2, 1, 2},
{16, 5, 2, 3, 2},
{17, 5, 2, 2, 3},
{18, 3, 3, 3, 2},
{19, 3, 3, 2, 3},
{20, 6, 2, 4, 2},
{21, 5, 2, 2, 2},
{22, 7, 2, 3, 3}, 
{23, 3, 3, 2, 2},
{24, 7, 2, 5, 2},
{25, 5, 3, 10, 2},
{26, 3, 3, 1, 2}, 
{27, 6, 2, 3, 2},
{28, 2, 5, 2, 2},
{29, 15, 2, 14, 2},
{30, 30, 1, 0, 1},  
{31, 2, 5, 1, 2}, 
{32, 6, 2, 2, 2},
{33, 7, 2, 4, 2},
{34, 34, 1, 0, 1},
{35, 6, 2, 1, 2}, 
{36, 10, 2, 8, 2},
{37, 8, 2, 3, 3},
{38, 38, 1, 0, 1},
{39, 8, 2, 5, 2},
{40, 7, 2, 3, 2},
{41, 7, 2, 2, 3},
{42, 42, 1, 0, 1},
{43, 43, 1, 0, 1},
{44, 13, 2, 5, 3},
{45, 7, 2, 2, 2},
{46, 17, 2, 3, 5},
{47,  2, 7, 9, 2},
{48, 7, 2, 1, 2},
{49, 9, 2, 2, 5},
{50, 50, 1, 0, 1},
{51, 10, 2,  7, 2},
{52, 14, 2, 12, 2}, 
{53, 53, 1, 0, 1},
{54, 3, 4, 3, 3}, 
{55, 8, 2, 3, 2},
{56, 8, 2, 2, 3},
{57, 11, 2, 8, 2},
{58, 58, 1, 0, 1 },
{59,59, 1, 0, 1},
{60, 2, 6, 2, 2},
{61, 5, 3, 8, 2}, 
{62, 62, 1, 0, 1},
{63, 8, 2, 1, 2},
{64, 10, 2, 6, 2},
{65, 9, 2, 4, 2},
{66,66, 1, 0, 1},
{67,67, 1, 0, 1},
{68,68,2,2,5},
{69, 13, 2, 10, 2},
{70,70, 1, 0, 1},
{71, 14, 2, 5, 3}, 
{72, 9, 2, 3, 2}, 
{73, 9, 2, 2, 3},
{74, 3, 5, 13, 2},
{75, 10, 2, 5, 2},
{76, 5, 3, 7, 2},
{77, 9, 2, 2, 2},
{78,78, 1, 0, 1},
{79, 2, 7, 7, 2}, 
{80, 9, 2, 1, 2}, 
{81, 15, 2, 12, 2}, 
{82,82, 1, 0, 1}, 
{83,83, 1, 0, 1}, 
{84, 10, 2, 4, 2},
{85, 11, 2, 6, 2}, 
{86,86, 1, 0, 1}, 
{87, 16, 2, 13, 2},
{88, 13, 2, 3, 4},
{89, 11, 2, 2, 5},
{90,90, 1, 0, 1},
{91, 10, 2, 3, 2},
{92, 10, 2,  2, 3},
{93,  5, 3, 2, 5},
{94, 11, 2,  3, 3},
{95, 12, 2, 7, 2},
{96,  10, 2,  2, 3},
{97, 15, 2, 2, 7},
{98,  5, 3,  3, 3},
{99, 10, 2, 1, 2},
{100, 5, 3, 5, 2},
{101, 2, 7, 3, 3},
{102,102, 1, 0, 1},
{103, 2, 7, 5, 2},
{104, 15, 2, 11, 2},
{105, 11, 2, 4, 2},
{106,106, 1, 0, 1},
{107,107, 1, 0, 1},
{108, 12, 2, 6, 2},
{109, 5, 3, 4, 2},
{110,110, 1, 0, 1 },
{111,111, 1, 0, 1},
{112, 11, 2, 3, 2},
{113, 11, 2, 2, 3},
{114,114, 1, 0, 1},
{115, 14, 2, 9, 2},
{116, 5, 3, 3, 2},
{117, 5, 3, 2, 3},
{118, 3, 5, 5, 3},
{119, 2, 7, 3, 2},
{120, 2, 7, 2, 3},
{121, 5, 3, 2, 2},
{122, 3, 5, 11, 2},
{123,123, 1, 0, 1}, 
  {124, 5, 3,  1, 2}, 
 {125, 15, 2,  10, 2}, 
{126,126, 1, 0, 1},
{127, 2, 7, 1, 2},
 {128, 12, 2, 4, 2}, 
{129,129, 1, 0, 1},
{130,130, 1, 0, 1},
  {131, 16, 2, 5, 2}, 
{132, 14, 2, 8, 2}, 
{133, 13, 2, 6, 2}, 
{134,134, 1, 0, 1},
{135, 12, 2, 3, 2}, 
 {136, 12, 2, 2, 3}, 
{137, 13, 2, 2, 5},
{138,138, 1, 0, 1},
{139,139, 1, 0, 1}, 
{140, 12, 2, 2, 2}, 
{141,141, 1, 0, 1},
{142, 13, 2, 3, 3}, 
{143, 12, 2, 1, 2}, 
 {144, 13, 2, 5, 2}, 
 {145, 17, 2, 12, 2},
{146, 195, 3, 2723,2},
{147, 14, 2, 7, 2},
{148,148, 1, 0, 1},
{149,149, 1, 0, 1},  
{150,175,3, 2315,2},
{151,151, 1, 0, 1},
{152, 6, 3, 8, 2}, 
{153, 13, 2, 4, 2},
{154,154, 1, 0, 1},
{155, 18, 2, 13, 2}, 
{156, 16, 2, 10, 2},
{157,157, 1, 0, 1}, 
{158,158, 1, 0, 1},
{159,159, 1, 0, 1},
{160, 13, 2, 3, 2}, 
{161, 13, 2, 2, 3}, 
{162, 3, 5, 9, 2}, 
{163,163, 1, 0, 1},
{164, 14, 2, 2, 5}, 
{165, 13, 2, 2, 2}, 
{166,166, 1, 0, 1},
{167, 6, 3, 7, 2}, 
{168, 13, 2, 1, 2}, 
{169, 14, 2, 3, 3}, 
{170,170, 1, 0, 1}, 
{171, 14, 2, 5, 2}, 
{172,172, 1, 0, 1},
{173,173, 1, 0, 1},
{174, 3, 5, 13, 2}, 
{175, 16, 2, 9, 2}, 
{176, 15, 2, 7, 2},
{177,177, 1, 0, 1},
{178,178, 1, 0, 1}, 
{179, 3, 5, 8, 2}, 
{180, 14, 2, 4, 2}, 
{181,181, 1, 0, 1},
{182,182, 1, 0, 1},
{183,183, 1, 0, 1},
{184, 6, 3, 2, 5}, 
{185,185, 1, 0, 1},
{186,186, 1, 0, 1},
{187, 14, 2, 3, 2}, 
{188, 14, 2, 2, 3}, 
{189, 6, 3, 3, 3}, 
{190,190, 1, 0, 1},
{191, 6, 3, 5, 2}, 
{192, 14, 2, 2, 2},
{193,193, 1, 0, 1},
{194, 3, 5, 7, 2},
{195, 14, 2, 1, 2}, 
{196, 18, 2, 2, 7}, 
{197,197, 1, 0, 1},
{198, 15, 2, 3, 3}, 
{199, 18, 2, 5, 3}, 
{200, 6, 3, 4, 2}}

notes: 1.i prefer to write e.g. 4^2 rather than 2^4 (that is a lower degree
of power not a lower base)
          2. only one soln with minimal p,q is included for each d
          3.not to annoy Mathematica i include artificially m=d, p=1, n=0,
q=1 
            when solution for given d isn't known to me.
          4. when list will reach say d=1000, then each next entry may be
ascribed by author's name!

Neil! is it possible to include such a list into the database of OEIS?
thanks to all  (Don, Jud and all!) again, zak
 

-----Original Message-----
From: N. J. A. Sloane [mailto:njas at research.att.com]
Sent: Thursday, October 10, 2002 12:11 AM
To: seqfan at ext.jussieu.fr
Cc: davidwwilson at attbi.com; djr at nk.ca; zfseidov at ycariel.yosh.ac.il;
seidovzf at yahoo.com; njas at research.att.com
Subject: Re: not of form m^p-n^q



There has been a lot of discussion about this "sequence",
so I have put a version of it in the database:

%I A074981
%S A074981
6,14,34,42,50,58,62,66,70,78,82,86,90,102,110,114,130,158,178,182,202,
%T A074981
210,226,230,238,246,254,258,266,274,278,290,302,306,310,314,322,326,
%U A074981
330,342,358,374,378,390,394,398,402,410,418,422,426,430,434,438,442
%N A074981 Conjectured list of numbers which are not of form m^p-n^q, where
m,n,p,q are integers with m>0, n>0, p>1, q>1.
%C A074981 These are not the difference of two powers below 10^19. - Don
Reble.
%e A074981 Examples for numbers not in the sequence: 10 = 13^3-3^7, 22 = 7^2
- 3^3, 29 = 15^2 - 14^2, 31 = 2^5 - 1, 52 = 14^2 - 12^2, 54 = 3^4 - 3^3, 60
= 2^6 - 2^2, 68 = 10^2 - 2^5, 72 = 3^4 - 3^2, 76 = 5^3 - 7^2, 84 = 10^2 -
2^4, ...
%Y A074981 Cf. A074980.
%K A074981 nonn,new
%O A074981 1,1
%A A074981 Zakir F. Seidov (seidovzf at yahoo.com), Oct 07 2002
%E A074981 Corrected by Don Reble (djr at nk.ca), Oct 08 2002

The terms may change, and the %E (Extended) line
will probably change too, when i process all the messages
and postings dealin with this sequence.  But at least
it now has an A-number.

There is a closely related sequence that will also probably
change:

%I A074980
%S A074980
6,10,14,34,42,46,50,58,62,66,70,78,82,86,90,102,110,114,122,130,134,
%T A074980 146,150,158,162,166,178,182,194
%N A074980 Numbers which are not of the form m^p - n^q where p = 2 or 3, q =
2 or 3.
%C A074980 Take d=m^p - n^q, (p,q = 2,3) that is difference between
cube/square of any m and cube/square of any n. Presumably the absolute value
of d can not take the values 6, 10, 14, 34, 42, ... in the sequence.
%e A074980 First squares and cubes are: 1, 4, 9, 16, 25, 36, 49, 64, 81,
100, 8, 27, 64. Take differences between any two terms. The absolute values
of these differences can not be (presumably) equal to numbers in the
sequence, e.g. 6, or 10.
%Y A074980 Cf. A074981.
%K A074980 nonn,new
%O A074980 1,1
%A A074980 Zakir F. Seidov (seidovzf at yahoo.com), Oct 07 2002
%E A074980 Probably wrong. Will be modified. - njas

Although I don't think this has received much attention yet.
Maybe someone could check it.

Neil