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A181392 Squares in A108571. 3
1, 435343551252544, 543345144355225, 21343774737727744, 525664562544416656, 555621544626466564, 645545246266556416, 656542564646552164, 666524445565146256, 2766717773326766736, 8823883385555888521 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

The last term is 999999999786876856487355368576387875784644 = 999999999893438428238^2. - Giovanni Resta, May 19 2013

There are no squares with 43, 44, or 45 digits. Indeed, numbers of 45 digits have sum of digits 1^1+2^2+...+9^9 = 285, which mod 9 is equal to 6. It is easy to verify that no power can be equal to 6 mod 9, hence there are no squares, cubes, etc. of 45 digits. Similarly, the numbers of 44 and 43 digits can only be obtained by omitting the single 1 or the two 2's, so mod 9 they are equal to 5 and 2, respectively. Again, 2 and 5 are not squares or cubes mod 9, but they can be powers with exponents  k = 5, 7, 11, 13, 17, 19, 23, 25,... (numbers not divisible by 2 or 3). Since 10^(44/k) is at most 6.3*10^8 (for k=5) excluding higher powers by generating them is not a tremendous computational effort, which can be further reduced noticing that certain candidates can be excluded based on their last digits. For example, 9993^5 mod 10000 is 3193, which contains a 1. So no number ending in 9993 can be the base for a 5th power of 44 digits (which should lack the 1). Since 4th powers are squares too, they can have at most 42 digits, and since 10^(42/4) is about 3.16*10^10, it is not difficult to ascertain that no 4th powers belong to A108571. - Giovanni Resta, Jul 26 2015

LINKS

Patrick Wieschollek and Giovanni Resta, Table of n, a(n) for n = 1..1000 (first 160 terms from Patrick Wieschollek)

CROSSREFS

Cf. A108571, A225886.

Sequence in context: A317290 A128769 A261150 * A299799 A175531 A186914

Adjacent sequences:  A181389 A181390 A181391 * A181393 A181394 A181395

KEYWORD

base,nonn,fini

AUTHOR

Patrick Wieschollek, Oct 17 2010

EXTENSIONS

Edited by N. J. A. Sloane, Oct 17 2010

STATUS

approved

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Last modified May 21 02:55 EDT 2019. Contains 323434 sequences. (Running on oeis4.)