A071254

[Don Reble]

Numseq fans:

Sequence A071254 says,

    %S 2,8,351,80
    %N Lesser of the smallest pair of consecutive numbers
       divisible by a n'th power.
    %e a(4) = 80 as 80 = 2^4*5 and 81 = 3^4.
    %A Amarnath Murthy (amarnath_murthy at yahoo.com), May 22 2002

I figure the author meant "135" rather than "351". On that basis...


I see three sequences hidden here:
    a) the sequence described by the %N line;
    b) the similar sequence which begins 2,8,135,80; and
    c) one other similar sequence.

Sequence 71254a has a(3)=80, since 80=2^3*10 and 81=3^3*3.
It goes:
    2 8 80 80 1215 16767 76544 636416 3995648 24151040 36315135
    689278976 1487503359 1487503359 155240824832 785129144319
    4857090670592 45922887663615 157197025673216 1375916505694208
    2280241934368767 2280241934368767 2280241934368767
    787449981119234048 3950931357202382847 81339932732586721280
    89241795446991486975 89241795446991486975 89241795446991486975
    73780548369024777453567 368367291072441938345984

Sequence 71254b should be described:
    Lesser of the smallest pair of consecutive numbers divisible
    by an n'th power, but not both divisible by an (n+1)'th power.
It goes as the author intended (I suppose), except for the typo in a(3):
    2 8 135 80 1215 16767 76544 636416 3995648 24151040 36315135
    689278976 11573190656 1487503359 155240824832 785129144319
    4857090670592 45922887663615 157197025673216 1375916505694208
    19656708706009088 129341461907898368 2280241934368767
    787449981119234048 3950931357202382847 81339932732586721280
    934248573630477762560 6051700419017824010240
    89241795446991486975 73780548369024777453567

Sequence 71254c is:
    Lesser of the smallest pair of consecutive numbers divisible
    by an n'th power, but neither divisible by an (n+1)'th power.
It goes:
    2 44 135 80 8991 29888 356480 2316032 14073344 24151040
    326481920 689278976 11573190656 76876660736 314944159743
    2035980763136 28996228218879 55637069004800 766556765683712
    1375916505694208 19656708706009088 129341461907898368
    2280241934368767 787449981119234048 52909644702657019904
    81339932732586721280 934248573630477762560
    6051700419017824010240 36934895082235884470271

All three are somewhat similar. Amarnath, which did you intend?

    -----

In each case, the two consecutive numbers are divisible by 2^n and 3^n.
It is rather unlikely that a higher power would come first, but it
is still conceivable. Anyone wanna try a proof?

--
Don Reble       djr at nk.ca