A301339 a(n) is the ending digit of a prime number occurring most up to the n-th prime. If a tie exists, then the digits are concatenated in ascending order.

2, 23, 235, 2357, 12357, 3, 37, 37, 3, 3, 3, 37, 137, 3, 37, 3, 3, 3, 37, 137, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 37, 37, 37, 37, 7, 37, 7, 37, 37, 37, 37, 3, 37, 37, 37, 3, 37, 37, 3, 3, 3, 3, 37, 3, 3, 3, 37, 137, 3, 3, 3, 3, 3, 3, 3, 37, 7, 7, 37, 37, 7

Offset

1,1

Comments

1 occurs first at a(2766290)

2 occurs first at a(1)

3 occurs first at a(6)

7 occurs first at a(37)

9 occurs first at a(7153)

13 occurs first at a(45532)

17 occurs first at a(12655)

23 occurs first at a(2)

37 occurs first at a(7)

39 occurs first at a(429687)

79 occurs first at a(7042)

137 occurs first at a(13)

235 occurs first at a(3)

379 occurs first at a(93562)

2357 occurs first at a(4)

12357 occurs first at a(5)

The other possible new terms are 19, 139, 179 and 1379. - Michael S. Branicky, Nov 11 2025

Links

Michael S. Branicky, Table of n, a(n) for n = 1..10000

Caldwell and Honaker, Prime Curios!: 137

Example

a(13) = 137 because primes that end with digits 1, 3, and 7 occur most frequently (exactly three times each) up to the 13th prime.

Mathematica

With[{s = Array[Mod[Prime@ #, 10] &, 73]}, Array[FromDigits@ Last[SplitBy[#, Last]][[All, 1]] &@ SortBy[Tally@ Take[s, #], Last] &, Length@ s]] (* Michael De Vlieger, Apr 21 2018 *)

Prog

(PARI) lista(nn) = {my(p=1, v = vector(9)); for (n=1, nn, p = nextprime(p+1); d = p % 10; v[d] ++; vmax = vecmax(v); s = ""; for (i=1, #v, if (v[i] == vmax, s = concat(s, i)); ); print1(eval(s), ", "); ); } \\ Michel Marcus, Apr 11 2018
(Python)
from gmpy2 import next_prime
from itertools import count, islice
from collections import Counter
def agen(verbose=False): # generator of terms, verbose=True prints first-seen terms
    p, c, ranks = 2, Counter(), set()
    for n in count(1):
        c[str(int(p%10))] += 1
        m = max(c.values())
        r = int("".join(k for k in "123579" if c[k] == m))
        yield r
        if r not in ranks:
            ranks.add(r)
            if verbose: # print terms as in Comments
                print(f"{r} occurs first at a({n})")
        p = next_prime(p)
print(list(islice(agen(), 73))) # Michael S. Branicky, Nov 11 2025

Crossrefs

Cf. A000040.

Sequence in context: A262571 A283560 A159902 * A294268 A118385 A386964

Adjacent sequences: A301336 A301337 A301338 * A301340 A301341 A301342

Keyword

nonn,base

Author

G. L. Honaker, Jr., Mar 27 2018

Extensions

a(14)-a(73) by Chuck Gaydos

Status

approved