Saint Patrick - Another Prime Day

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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

 

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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.

Sublime Numbers

 

(Image: Grok)

The numbers 12 and 6086555670238378989670371734243169622657830773351885970528324860512791691264 are the two entries in the OEIS sequence: A081357, Sublime Numbers. 

In number theory, a sublime number is a positive integer which has a perfect number of positive factors (including itself), and whose positive factors add up to another perfect number.

The first entry, 12, has a perfect number of positive factors (6): 1, 2, 3, 4, 6, and 12, and the sum of these is again a perfect number: 1 + 2 + 3 + 4 + 6 + 12 = 28.

The second entry, above, is the product: (2¹²⁶)(2⁶¹ − 1)(2³¹ − 1)(2¹⁹ − 1)(2⁷ − 1)(2⁵ − 1)(2³ − 1).