A392975 - A contribution to the On-Line Encyclopedia of Integer Sequences (OEIS)

 

(Image: Michael De Vlieger)

I recently proposed a new sequence for the On-Line Encyclopedia of Integer Sequences (OEIS). 

A Sisyphus sequence: a(0) = 0, a(1) = 1, a(2) = 2; for n > 2, a(n) is the smallest unused positive integer having the same greatest prime factor as the sum of the previous terms.

Sisyphus is a figure in Greek mythology, known as the cunning King of Corinth who was punished by Zeus to eternally roll a massive boulder up a hill in Tartarus, only for it to roll back down each time. Numerical sequences are sometimes called Sisyphus sequences when their terms climb and climb, but then drop repeatedly. 

The image above shows this process for this sequence for the first 10^6 terms. Below is a graph for how the sequence begins (first 300 terms):

One interesting feature of the sequence is the low points all appear to be prime. They begin:  7, 13, 17, 19, 23, 29, 59, 107, 137, 173, 257, 293, 467, 503...

This trend continues for at least the first 200,000 terms of the sequence.

Another feature of the sequence is that beginning with the third term, the common greatest prime factor repeats in runs of 3 or more. These factors begin:  3, 3, 3, 7, 7, 7, 7, 13, 13, 13, 13... and this trend continues for at least the first 200,000 terms of the sequence.

(Image: Sisyphus (1548–49) by Titian, Prado Museum, Madrid, Spain)

Update 4/3/2026: the sequence was published A392975 - OEIS .

Update 4/14/2026: a companion sequence, A396326, was published: Low points in A392975 (having the property of being all prime numbers).

The Sisyphus Sequence (A350877)

(Image: Sisyphus (1548–49) by Titian, Prado Museum, Madrid, Spain)

Sisyphus, in Greek mythology, was cursed upon death to roll a large boulder up a hill only for it to roll down once it neared the top, repeating for eternity. 

The integer sequence, OEIS:  (https://oeis.org/A350877) starts 1, 3, 6, 3, 8, 4, 2, 1, 8 ,4, 2, 1, 12, 6, 3, 16, 8, 4, 2, 1... . The formal definition for the sequence is: start the sequence S with a(1) = 1 and extend S with a(n)/2 when a(n) is even, otherwise with a(n) + the smallest prime not yet added. The sequence was named Sisyphus because the sequence climbs only to drop back to 1.

Additional notes on the sequence:  Michael De Vlieger's OEIS sequence analysis: A350877.

A proposed variant of this sequence is to begin the sequence with n, integers 1 and greater, and define a(n) as the number of steps required to reach 1. This new sequence is: 

0, 1, 5, 2, 979, 6, 8, 3, 13, 980, 6, 7, 209, 9, 980, 4, 8, 14, 9, 981, 10, 7, 14, 8, 980, 210, 7, 10, 20, 981, 210, 5, 14, 9, 981, 15, 13, 10, 9, 982, 210, 11, 10, 8, 806, 15, 11, 9, 9, 981, 15, 211, 980, 8, 981, 11, 11, 21, 8, 982, 16, 211, 21, 6, 981, 15, 211...

a(1) = 0 because no further steps are required to reach 1.

a(2) = 1 because in one step, 2/2 =1, 1 is reached.

a(3) = 5 the steps proceed: 5, 8, 4, 2, 1 (5 steps to reach 1).

An interesting feature of this new sequence is how terms tend to cluster around certain values - from just the first 67 terms above, see terms clustered around 11, 210, 980.

This clustering feature is shown below in the scatterplot of the first 4000 terms:

Michael De Vlieger produced a log-log scatterplot of the first 65536 terms, which shows how this clustering continues:

Sequence published: https://oeis.org/A388038

Narcissistic Numbers

 (Image: Narciso (Jan Cossiers), CC BY-SA 4.0 <https://creativecommons.org/licenses/by-sa/4.0>, via Wikimedia Commons)

In Greek mythology, Narcissus was known for his beauty. He rejected advances from all suitors, both men and women, and instead fell in love with his own reflection. The legend gives us the source of the words narcissist and narcissistic. 

There are also numbers known as Narcissistic Numbers. In the On-Line Encyclopedia of Integer Sequences (OEIS), sequence A005188 defines such numbers to be:  m-digit positive numbers equal to the sum of the m-th powers of their digits.

For example, the number 153 has 3 digits. The sum of 1^3 + 5^3 + 3^3 = 153. The sequence of Narcissistic Numbers begins: 1, 2, 3, 4, 5, 6, 7, 8, 9, 153, 370, 371, 407...

Narcissistic Numbers.

Münchhausen Numbers

 

(Image: Statue of Baron Münchhausen riding half of a horse - https://commons.wikimedia.org/wiki/User:Franzfoto)

Baron Münchhausen is a fictional character known for making exaggerated stories (other spellings include Munchhausen and Munchausen). In addition to the fables of the Baron, the name is used to describe Münchhausen syndrome and Münchhausen syndrome by proxy, in which someone fabricates a story of their illness or the illness of one they care for to gain attention or sympathy.

There are also numbers known as Münchhausen Numbers. These are numbers that equal the sum of each of their digits raised to their own power. For example, in base ten, the numbers include 1 and 3435. 1^1=1, 3^3 +4^4+3^3+5^5=3435. This sequence of numbers is A046253 in the On-Line Encyclopedia of Integer Sequences (OEIS). Another sequence of other Münchhausen Numbers is A166623, which lists such numbers in multiple bases.

Achilles Numbers

 

(Photo: Closeup of Achilles thniskon in Corfu Achilleion  - Dr.K., CC BY-SA 3.0 <https://creativecommons.org/licenses/by-sa/3.0>, via Wikimedia Commons)

In Greek mythology, Achilles, a central figure in Homer's Iliad, was known as the greatest Greek warrior. By legend, Achilles was invulnerable in all of his body except for one heel. According to myth, when his mother Thetis dipped him in the river Styx as an infant, she held him by one of his heels, leaving it untouched by the waters and thus his only vulnerable body part. To this day, Achilles' heel refers to a point of weakness that can lead to downfall.

There is also a series of numbers named for Achilles. Sequence A052486 of the On-Line Encyclopedia of Integer Sequences (OEIS) describes Achilles numbers as "powerful but imperfect: if n = Product(p_i^e_i) then all e_i > 1 (i.e., powerful), but the highest common factor of the e_i is 1, i.e., not a perfect power."

This definition uses the names of other types of numbers. 

Powerful numbers:  if a prime p divides n then p^2 must also divide n (also called squareful, square full, square-full or 2-powerful numbers). This is sequence A001694 in the OEIS.

Perfect numbers: Numbers k such that k is equal to the sum of the proper divisors of k. See OEIS: A000396.

Perfect power: Numbers of the form m^k where m > 0 and k >= 2.  See OEIS: A001597.

The first few Achilles numbers are: 72, 108, 200, 288, 392.

This post was inspired by the comment left by a reader to a recent post about the number 108. He recognized that 108 is also an Achilles number.

Also see: A388293, Cubefull Achilles numbers.

Saint Patrick - Another Prime Day

(Image: By Internet Archive Book Images -  https://archive.org/stream/irelandscrownoft00ryan/irelandscrownoft00ryan#page/n61/mode/1up, No restrictions, https://commons.wikimedia.org/w/index.php?curid=41964622)

A recent post about Julius Caesar introduced the concept of Caesar Primes. Caesar's birthday is the 13th of July (the 7th month), and 137 is a prime number. Caesar Primes are primes formed by the concatenation of q and p, where q > p are also primes.

As we celebrate Saint Patrick's Day on March 17, we note that 317 and 173 are prime numbers. This suggests a sequence of primes similar to the Caesar Primes. Other pairs of primes formed by two primes include: {37, 73}, {113, 311}, {313,331},{337,733}, {359,593}...

Update 4/8/2025. I submitted a proposed sequence to the On-Line Encyclopedia of Integer Sequences (OEIS). It is currently under review as https://oeis.org/draft/A380943, Primes written in decimal representation by the concatenation of primes p and q such that the concatenation of q and p also forms a prime.

The sequence begins: 37, 73, 113, 173, 197, 311, 313, 317, 331, 337, 359, 367, 373, 593, 617, 673, 719, 733, 761, 797, 977, 1093, 1097, 1123, 1277, 1319, 1361, 1373, 1783, 1913, 1931, 1979, 1997, 2293, 2297, 2311, 2347, 2389, 2713, 2837, 2971, 3109, 3119, 3137, 3191, 3229, 3271...

Update 4/13/2025 - the proposed sequence was published in the OEIS: https://oeis.org/A380943.

Julius Caesar and Caesar Primes

 

(Image:https://commons.wikimedia.org/wiki/File:Retrato_de_Julio_C%C3%A9sar_(26724093101).jpg)

March 15 is also known as the Ides of March, when Julius Caesar was assassinated in 44 BCE. Caesar was born on July 12 or 13, 100 BCE (records vary on the date of July 12 or July 13). For the purpose of this post, July 13 is consistent with the numbers known as Caesar Primes. One such number is 137 - the concatenation of two primes, 13 and 7 (with Caesar's birthday being the 13th day of the 7th month).

The list of Caesar Primes is found in the On-Line Encyclopedia of Integer Sequences (OEIS) as sequence: https://oeis.org/A133187. The sequence consists of the prime numbers formed by the concatenation of q and p, where q > p are also primes.

I wrote a Mathematica  to produce this sequence:

lim=2700;plim=Max[FromDigits[Rest[IntegerDigits[lim]]],FromDigits[Drop[IntegerDigits[lim],-1]]];f2p[{p_,q_}]:=FromDigits[Join[IntegerDigits[q],IntegerDigits[p]]];p=Prime[Range[PrimePi[plim]]];p2=Subsets[p,{2}];Union[Select[f2p/@p2,PrimeQ[#]&&#<=lim&]]

The first 46 Caesar Primes are: 53,  73, 113, 137, 173, 193, 197, 233, 293, 313, 317, 373, 433, 593, 613, 617, 673, 677, 733, 797, 977, 1013, 1033, 1093, 1097, 1277, 1373, 1493, 1637, 1733, 1913, 1933, 1973, 1993, 1997, 2113, 2237, 2273, 2293, 2297, 2311, 2333, 2393, 2417, 2633, 2693.