Sequence Numbers that Show Powers of a Number

9/16/2015

Sequence Number that show Powers of a Number:

“1,000,000 – n” will produce a sequence number that shows “the powers of n”, beginning with n⁰, n¹, n², etc., written in 6 digit strings.

“1,000,000,000,000 – n” will produce a Sequence Number that produces a list of the powers of n, beginning with n⁰, n¹, n², etc., written in 12 digit strings.

Examples:

Powers of 3: 1,000,000 – 3 = 999,997.  So the Sequence Number is 999,997 1/999997 = 0. 000001  000003  000009  000027  000081  000243  000729  002187  006561  019683  059049  177147  … This Sequence number produce a list of the powers of three, written in 6 digit strings, from 30 through 311 (12 non-zero terms).

Powers of 23: 1,000,000 – 23 = 999,977.  So the Sequence Number is 999,977. 1/999977 = 0. 000001  000023  000529  012167  … This Sequence Number produces a list of the powers of 23, written in six digit strings, from 230 to 233.  Since this only produces 4 terms, we would want to rewrite the Sequence number so that more terms could be shown. 1,000,000,000 – 23 = 999,999,977.  So our new Sequence number is 999,999,977 1/999999977 = 0. 000000001  000000023  000000529  000012167  000279841  006436343  Written in nine digit strings. This sequence now shows 6 valid and accurate terms, and 999,999,977 is considered a valid Sequence Number. Compare with OEIS sequence A009967.

“1,000,000,000,000 – n” will produce a sequence of “the powers of n”, beginning with n⁰, n¹, n², etc., written in 12 digit strings.

Example:

1,000,000,000,000 – 95 = 999,999,999,905.  

Powers of 95 999,999,999,905 1/999999999905 = 0. 000000000001  000000000095  000000009025  000000857375  000081450625  007737809375  … Written in 12 digit strings. Accurate to the sixth non-zero term.

“1,000,000,000,000,000,000 – n” will produce a sequence of “the powers of n”, beginning with n⁰, n¹, n², etc., written in 18 digit strings.

Example:

1,000,000,000,000,000,000 – 123 = 999,999,999,999,999,877.

Powers of 123 The sequence number that produces powers of 123 in 18 digit strings is 999,999,999,999,999,877 1/999999999999999877 = 0. 000000000000000001  000000000000000123  000000000000015129  000000000001860867  000000000228886641  000000028153056843  000003462825991689  000425927596977747  … Written in 18 digit strings. Accurate to the eighth non-zero term.

We are able to continue using this pattern to expand the sequence to show whatever size digit strings are needed or to lengthen the number of terms which are accurate.

David