Deceptive Numbers

It's not surprising that I've not come across deceptive numbers before as they are quite light on the ground so to speak. Today, 27613 is the number associated with my diurnal age and it is a deceptive number but the previous such number was 24661 and the next will be 29431. So what constitutes a deceptive number? Here what Numbers Aplenty has to say:

Let us denote with \(R_k\) the repunit \(111\dots 1\) made of \(k\) ones.

It is known that every odd prime \(p\) divides the repunit \(R_{p-1}\).

R. Francis & T. Ray call a composite number \(n\) deceptive if it has the same property, i.e., if it divides the repunit \(R_{n-1}\).

For example, \(91=7 \times 13\) is deceptive because it divides \(R_{90}\).

Francis & Ray have proved that there are infinite deceptive numbers since, if \(n\)  is deceptive, then \(R_n\) is deceptive as well.

Every number greater than 2980 can be written as the sum of deceptive numbers.

Here are the deceptive numbers up to 100,001: 

91, 259, 451, 481, 703, 1729, 2821, 2981, 3367, 4141, 4187, 5461, 6533, 6541, 6601, 7471, 7777, 8149, 8401, 8911, 10001, 11111, 12403, 13981, 14701, 14911, 15211, 15841, 19201, 21931, 22321, 24013, 24661, 27613, 29341, 34133, 34441, 35113, 38503, 41041, 45527, 46657, 48433, 50851, 50881, 52633, 54913, 57181, 63139, 63973, 65311, 66991, 67861, 68101, 75361, 79003, 82513, 83119, 94139, 95161, 97273, 97681, 100001, ...

These numbers comprise OEIS A000864:

 A000864: deceptive nonprimes: composite numbers \(k\) that divide the repunit \(R_{k-1}\).

Yarborough and Anti-Yarborough Primes

I'd not previously heard of a class of primes known as Yarborough primes. My attention was drawn to this class by the fact that 26833, my diurnal age today, is a member of OEIS A296187:

A296187

Yarborough primes that remain Yarborough primes when each of their digits are replaced by their squares.

A Yarborough prime is simply a prime that doesn't contain a zero or a one and clearly 26833 qualifies in that regard. These primes form OEIS A106116:

A106116

Primes with smallest digit > 1.

If we square each of its digits we get 4366499 which is a Yarborough prime. The initial members of the sequence are:

73, 223, 233, 283, 337, 383, 523, 733, 773, 823, 2333, 2683, 2833, 2857, 3323, 3583, 3673, 3733, 3853, 5333, 6673, 6737, 6883, 7333, 7673, 7727, 7877, 8233, 8563, 8623, 22277, 22283, 22727, 23333, 23833, 25237, 25253, 25633, 26227, 26833, 27583, 27827, 27883, 32257

Here is a permalink to a SageMath algorithm that will generate the above sequence. If we can consider squares of digits then why not cubes? This leads to OEIS A296563:

A296563

Yarborough primes that remain Yarborough primes when each of their digits are replaced by their cubes.

The initial members of the sequence are as follows (permalink):

23, 43, 73, 229, 233, 277, 449, 773, 937, 947, 2239, 2243, 2297, 2377, 2777, 3299, 3449, 3727, 3943, 4243, 4423, 4493, 7393, 7723, 7927, 7949, 9227, 9743, 9749, 22277, 22727, 22777, 22943, 23327, 23399, 23497, 23747, 24473, 24733, 27239, 27277, 27427, 27799, 29347, 29443, 29723

There aren't any fourth power Yarborough primes in the range up to one million but there are some fifth power primes in the range up to one million:

683, 2383, 2633, 2663, 6863, 26263, 32833, 36263, 36383, 62233, 63823, 63863, 68633, 68683, 88223, 222883, 232663, 266663, 338383, 386263, 622663, 623683, 632323, 633623, 633883, 663283, 683863, 822223, 828833, 836663, 863833, 866683

The following link mentions the concept of an anti-Yarborough prime and defines it as a prime that contains only zeros and ones e.g. 11 (the first such prime) and 101 (the second such prime). These primes form OEIS A020449:

A020449

Primes whose greatest digit is 1.

The initial members are:

11, 101, 10111, 101111, 1011001, 1100101, 10010101, 10011101, 10100011, 10101101, 10110011, 10111001, 11000111, 11100101, 11110111, 11111101, 100100111, 100111001, 101001001, 101001011, 101100011, 101101111, 101111011, 101111111

I guess the name "Yarborough" derives from bridge where it means "a hand in bridge or whist containing no ace and no card higher than a nine" and is thus useless. The Ace can be assigned the digit 1 and so such a hand would only contain the digits 2 to 9. The name reminds me of a novel that I read in the late sixties called "Yarborough".

Here is an interesting article I found about the book and its author B. H. Friedman.