A380943 Contribution to the OEIS (Saint Patrick Primes)

 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.

+40
0

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

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OFFSET

1,1

COMMENTS

Member primes could be named Saint Patrick primes because the date of Saint Patrick's Day, March 17 (3/17), produces the terms 173 and 317.

LINKS

James C. McMahon, Table of n, a(n) for n = 1..1000

Michael De Vlieger, Plot a(n) = p<>q at (x,y) = (pi(p), pi(q)) showing all terms with pi(p) <= 40 and pi(q) <= 40, labeling p in red and q in blue.

Michael De Vlieger, Plot a(n) = p<>q at (x,y) = (pi(p), pi(q)) showing all terms with pi(p) <= 2000 and pi(q) <= 2000.

EXAMPLE

Primes 173 and 317 are members because they are formed by the concatenation of 17 & 3 and 3 & 17, respectively.

While the concatenation of 13 and 7 forms the prime 137, it is not a member because the concatenation of 7 and 13 is 713, which is not prime.

MATHEMATICA

lim=3300; plim=Max[FromDigits[Rest[IntegerDigits[lim]]], FromDigits[Drop[IntegerDigits[lim], -1]]]; p=Prime[Range[PrimePi[plim]]]; p2=Subsets[p, {2}]; fp[{a_, b_}]:=FromDigits[Join[IntegerDigits[a], IntegerDigits[b]]]; rfp[{a_, b_}]:=FromDigits[Join[IntegerDigits[b], IntegerDigits[a]]]; pabba=Select[p2, PrimeQ[fp[#]]&&PrimeQ[rfp[#]]&]; pp1=fp/@pabba; pp2=rfp/@pabba; Select[Union[Join[pp1, pp2]], #<=lim&]

PROG

(Python)

from sympy import isprime

def ok(n):

if not isprime(n): return False

s = str(n)

return any(s[i]!='0' and isprime(int(s[:i])) and isprime(int(s[i:])) and isprime(int(s[i:]+s[:i])) for i in range(1, len(s)))

print([k for k in range(3300) if ok(k)]) # Michael S. Branicky, Apr 05 2025

CROSSREFS

Subsequence of A019549 and A105184.

Cf. A133187.

KEYWORD

nonn,base,new

AUTHOR

James C. McMahon, Apr 03 2025

STATUS

approved

See post: https://jamesmacmath.blogspot.com/2025/03/saint-patrick-another-prime-day.html

Also, Michael De Vlieger made interesting graphics for this sequence: https://oeis.org/A380943/a380943.png and https://oeis.org/A380943/a380943_1.png

Smart Crows and Geometry

 

(Image: https://www.iconfinder.com/SatawatIcon)

Crows are recognized as very intelligent. Listen to this from NPR: https://www.npr.org/2025/04/11/nx-s1-5355849/a-new-study-finds-crows-can-recognize-geometric-shapes

Or read the summary here: https://www.npr.org/2025/04/12/nx-s1-5359438/a-crows-math-skills-include-geometry

Also summarized in ScienceAdvances: https://www.science.org/doi/10.1126/sciadv.adt3718

Another post on math skills of animals: https://jamesmacmath.blogspot.com/2025/03/counting-critters-whos-got-numbers-game.html

Some researchers believe birds developed intelligence in a way independent of how humans and primates did. 

Also see Live Science article: Which animals can count and understand simple math?

Counting Critters: Who’s Got the Numbers Game on Lock?

Ever wonder if animals can count? I mean, not like they’re sitting there with a pencil stub and a ledger, tallying up their groceries, but do they get numbers? Turns out, some of them do—and they’re better at it than I am when I’m half-asleep trying to split a restaurant bill. Nature’s got a few math whizzes, and I’ve been digging into which ones stand out. Spoiler: it’s not just the usual suspects.

Take chimpanzees. These furry cousins of ours are basically showing off at this point. I read about studies—fancy ones, like from Kyoto University—where chimps zip through number tasks faster than humans. They can eyeball sets of dots up to 9 and pick the bigger one without breaking a sweat. Imagine them in the wild, sizing up rival groups or eyeballing a fruit haul. Makes me wonder if they’d judge my banana stash—three’s enough, right? Probably not.

Then there’s crows. I’ve always liked their vibe—those sleek, black tricksters perched on telephone wires like they’re plotting something. Turns out, they’re counting too. Scientists say they can handle numbers up to 4 or 5, and some even grok the idea of zero. Zero! I barely got that in grade school. Next time I see one hopping around, I’ll picture it mentally tallying acorns. “Three’s a party, four’s a feast,” it might caw to itself.

Dolphins, though—those sea geniuses—they’ve got a knack for it too. They’ll pick the bigger pile of fishy treats in lab tests, no problem, up to about 5. I can see why: swimming in pods, keeping track of friends or foes, numbers matter. Makes me think of the time I tried counting waves at the beach and lost track at, what, 7? Dolphins would’ve smirked.

And bees! Tiny, buzzing bees! I couldn’t believe this one—research says they count up to 4 and even do basic math. Add one, subtract one, all to find the best flowers. Their waggle dance is like a GPS with a side of algebra. Next time I’m lost in the weeds (literally or otherwise), I’ll wish I had a bee’s brain.

Elephants round out the list. Big, lumbering, and apparently good with headcounts. They can tell 5 rumbles from 10 when listening to calls—handy for spotting trouble or rallying the herd. I once saw an elephant at the zoo, trunk swinging, and now I’m retroactively impressed. It wasn’t just staring at me; it was probably counting my bad hair day.

So who’s the champ? Hard to say. Chimps and crows flex the abstract stuff, bees punch above their weight, and elephants keep it practical. Me, I’m just glad I don’t have to compete. I’d be the guy fumbling with “how many coffee cups is too many?” (Answer: there’s no limit.) Nature’s got its own calculators, and I’m here for it. Which one’s your favorite?

(Post and graphic produced by https://grok.com/chat)

Read more about Crows: https://jamesmacmath.blogspot.com/2025/04/smart-crows-and-geometry.html

References:

https://www.elephanttrust.org/research (Amboseli project overview; search for cognitive or acoustic studies.)

https://www.science.org/doi/10.1126/sciadv.aav0961 (Direct link to the paper "Numerical ordering of zero in honey bees.")

https://dolphins.org/research

https://www.ox.ac.uk/news/science-blog/2018/02/crows-can-count-zero (Oxford University news article summarizing the study by Nieder et al.)

https://www.pri.kyoto-u.ac.jp/research/e-behavior/e-cognition.html

Can Giraffes count? Giraffes Can Do Math