Undulating number

An undulating number is a natural number, the digits of which in the representation become larger and smaller again in the alternation, for example 979342956 is an undulating number (to the base 10), and whose representation has at least two digits on this base. An undulating number is called smoothly undulating if it only contains two different digits, e.g. 828282828 (to base 10). A double smooth undulating number is a number that is smoothly undulating in two different place value systems , for example 10 is double smooth undulating because it is smoothly undulating in both the binary system ( ) and the ternary system ( ). 10 is the smallest double smooth undulating number. The concept of the undulating number was coined by the science journalist Clifford A. Pickover . ${\displaystyle 10=[1010]_{2}}$ ${\displaystyle 10=[101]_{3}}$

Examples

The following numbers are the first smooth undulating numbers to base 10 that are greater than 100:

101, 121, 131, 141, 151, 161, 171, 181, 191, 202, 212, 232, 242, 252, 262, 272, 282, 292, 303, 313, 323, 343, 353, 363, 373, 383, 393, 404, 414, 424, 434, 454, 464, 474, 484, 494, 505, 515, 525, 535, 545, 565, 575, 585, 595, 606, 616, 626, 636, 646, 656, ... (Follow A046075 in OEIS )

The following numbers are the first undulating prime numbers on base 10 that are less than 500 (the single-digit numbers are listed in the online encyclopedia of the number sequences OEIS and are mentioned here, but are written in brackets because they do not meet the condition, to be at least two digits):

(2, 3, 5, 7), 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 131, 151, 163, 173, 181, 191, 193, 197, 241, 251, 263, 271, 281, 283, 293, 307, 313, 317, 353, 373, 383, 397, 401, 409, 419, 439, 461, 463, 487, 491, ... (sequence A059168 in OEIS )

The following numbers are the first smooth undulating prime numbers with base 10 that are less than 400,000,000:

(2, 3, 5, 7), 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 131, 151, 181, 191, 313, 353, 373, 383, 727, 757, 787, 797, 919, 929, 18181, 32323, 35353, 72727, 74747, 78787, 94949, 95959, 1212121, 1616161, 323232323, 383838383,… (Follow A032758 in OEIS )

The following numbers are the first smooth undulating square numbers to base 10:

(0, 1, 4, 9), 16, 25, 36, 49, 64, 81, 121, 484, 676, 69696, ... (sequence A016073 in OEIS )

There is no nearest smooth undulating square number that has less than a million digits.

The next sequence of numbers represents the first smallest possible smooth undulating numbers based on base 10, in which the nth number is divisible by n (for example, the 17th number of this number sequence is 272, and is actually ): ${\displaystyle a_{i}}$ ${\displaystyle a_{17}=272=16\cdot 17}$

101, 202, 141, 212, 505, 252, 161, 232, 171, 1010, 121, 252, 494, 252, 525, 272, 272, 252, 171, 2020, 252, 242, 161, 696, 525, 494, 2727, 252, 232, 3030, 434, 3232, 363, ... (follow A252664 in OEIS )

The only smooth undulating potency with which less than 100 points has is . ${\displaystyle n^{p}}$ ${\displaystyle 3\leq p\leq 31}$ ${\displaystyle 7^{3}=343}$

A 99-digit smooth undulating prime number is and is written out: ${\displaystyle p=7+720\cdot {\frac {100^{49}-1}{99}}}$

727,272,727,272,727,272,727,272,727,272,727,272,727,272,727,272,727,272,727,272,727,272,727,272,727,272,727,272,727,272,727,272,727

swell

• Clifford A. Pickover: Is there a double smoothly undulating integer? In: Journal of Recreational Mathematics. Volume 22, No. 1, 1990, pp. 77-78.
• Clifford A. Pickover: Dr. Googol's wondrous world of numbers. dtv, Munich 2005, ISBN 3-423-34177-7 .
• WinFfunktion Mathematik plus 18, Handbuch , Kaarst 2010, bhv Publishing, p. 394

Individual evidence

1. ↑ ^{a b c} Eric W. Weisstein: Undulating Number. Wolfram MathWorld, accessed December 6, 2015 .  

Web links

• Eric W. Weisstein : Undulating Number . In: MathWorld (English).