differences of two prime powers

[T. D. Noe]

Here are the solutions with minimum r^i.  The definition of sequences needs
to be improved.  For instance 2^4 = 4^2.  I think we should use the
smallest power -- which means all the powers are prime, a natural condition.

r={3,3,2,2,3,0,2,3,5,13,3,4,7,0,4,5,5,3,3,6,5,7,3,5,5,3,6,2,15,83,2,6,7,0,6,
10,4,37,4,7,7,0,22,5,7,17,2,7,9,0,10,14,9,9,4,4,11,0,30,4,5,0,4,10,9,0,34,10
,13,0,14,9,9,3,10,5,9,0,2,9,15,0,42,10,11,0,16,13,11,0,10,10,5,11,12,10,15,5
,10,5}

i={2,3,2,3,2,0,3,2,2,3,3,2,2,0,2,2,2,3,3,2,2,2,3,2,3,3,2,5,2,2,5,2,2,0,2,2,3
,2,3,2,2,0,2,3,2,2,7,2,2,0,2,2,3,2,3,3,2,0,2,3,3,0,3,2,2,0,2,2,2,0,2,2,2,5,2
,3,2,0,7,2,2,0,2,2,2,0,2,2,2,0,2,2,3,2,2,2,2,3,2,3}

s={2,5,1,2,2,0,1,1,4,3,4,2,6,0,1,3,2,3,2,4,2,3,2,1,10,1,3,2,14,19,1,2,4,0,1,
4,3,11,5,3,2,0,21,9,2,3,9,1,2,0,7,12,26,3,3,2,4,0,29,2,4,0,1,6,4,0,33,2,10,0
,5,3,2,13,5,7,2,0,7,1,12,0,41,4,6,0,13,9,2,0,3,2,2,3,7,2,2,3,1,5}

j={3,2,2,2,2,0,2,2,2,7,2,2,2,0,2,2,3,2,3,2,2,3,2,2,2,2,2,2,2,3,2,2,2,0,2,3,3
,3,2,2,3,0,2,2,2,5,2,2,5,0,2,2,2,3,2,3,3,0,2,2,3,0,2,2,2,0,2,5,2,0,3,2,3,2,2
,2,2,0,2,2,2,0,2,2,2,0,2,2,5,0,2,3,5,3,2,2,7,3,2,2}

Tony


>I am adding these 4 sequences.  They have been mentioned several times
>on this mailing list by several correspondents.  Could people check
>and extend them please?     Neil
>
>%I A075788
>%S A075788 3,3,2,2,3,0,4,4,6,13,3,4,7,0,4,5,5,3,3,6,5,7,3,7,5,3,6,2,15
>%N A075788 If n can be written in the form r^i-s^j (see A074981), where
>r,s,i,j are integers with r>0, s>0, i>1, j>1 choose the representation
>with smallest r^i (in case of ties, minimize i and j); or if n is not of
>this form set r=s=i=j=0; sequence gives values of r.
>%C A075788 The zeros are only conjectures (cf. A074981).
>%C A075788 Use 4^2 rather than 2^4, etc.
>%e A075788 1 = 3^2 - 2^3, 2 = 3^3 - 5^2, 3 = 2^2 - 1^2, 4 = 2^3 - 2^2, etc.
>%O A075788 0,1
>%K A075788 nonn,hard,more
>%A A075788 Zakir F. Seidov (seidovzf at yahoo.com), Oct 13 2002
>
>%I A075789
>%S A075789 2,3,2,3,2,0,2,2,2,3,3,2,2,0,2,2,2,3,3,2,2,2,3,2,3,3,2,5,2
>%N A075789 If n can be written in the form r^i-s^j (see A074981), where
>r,s,i,j are integers with r>0, s>0, i>1, j>1 choose the representation
>with smallest r^i (in case of ties, minimize i and j); or if n is not of
>this form set r=s=i=j=0; sequence gives values of i.
>%C A075789 The zeros are only conjectures (cf. A074981).
>%C A075789 Use 4^2 rather than 2^4, etc.
>%e A075789 1 = 3^2 - 2^3, 2 = 3^3 - 5^2, 3 = 2^2 - 1^2, 4 = 2^3 - 2^2, etc.
>%O A075789 0,1
>%K A075789 nonn,hard,more
>%A A075789 Zakir F. Seidov (seidovzf at yahoo.com), Oct 13 2002
>
>%I A075790
>%S A075790 2,5,1,2,2,0,3,2,3,3,4,2,6,0,1,3,2,3,2,4,2,3,2,5,10,1,3,2,14
>%N A075790 If n can be written in the form r^i-s^j (see A074981), where
>r,s,i,j are integers with r>0, s>0, i>1, j>1 choose the representation
>with smallest r^i (in case of ties, minimize i and j); or if n is not of
>this form set r=s=i=j=0; sequence gives values of s.
>%C A075790 The zeros are only conjectures (cf. A074981).
>%C A075790 Use 4^2 rather than 2^4, etc.
>%e A075790 1 = 3^2 - 2^3, 2 = 3^3 - 5^2, 3 = 2^2 - 1^2, 4 = 2^3 - 2^2, etc.
>%O A075790 0,1
>%K A075790 nonn,hard,more
>%A A075790 Zakir F. Seidov (seidovzf at yahoo.com), Oct 13 2002
>
>%I A075791
>%S A075791 3,2,2,2,2,0,2,3,3,7,2,2,2,0,2,2,3,2,3,2,2,3,2,2,2,2,2,2,2
>%N A075791 If n can be written in the form r^i-s^j (see A074981), where
>r,s,i,j are integers with r>0, s>0, i>1, j>1 choose the representation
>with smallest r^i (in case of ties, minimize i and j); or if n is not of
>this form set r=s=i=j=0; sequence gives values of j.
>%C A075791 The zeros are only conjectures (cf. A074981).
>%C A075791 Use 4^2 rather than 2^4, etc.
>%e A075791 1 = 3^2 - 2^3, 2 = 3^3 - 5^2, 3 = 2^2 - 1^2, 4 = 2^3 - 2^2, etc.
>%O A075791 0,1
>%K A075791 nonn,hard,more
>%A A075791 Zakir F. Seidov (seidovzf at yahoo.com), Oct 13 2002