A367735 Prime numbers wherein digit values increase, decrease, and finally increase.

1201, 1213, 1217, 1301, 1303, 1307, 1319, 1327, 1409, 1423, 1427, 1429, 1439, 1523, 1549, 1601, 1607, 1609, 1613, 1619, 1627, 1637, 1657, 1709, 1723, 1747, 1759, 1801, 1823, 1847, 1867, 1879, 1901, 1907, 1913, 1949, 1979, 2309, 2417, 2423, 2437, 2503, 2539

Offset: 1

Author

James S. DeArmon, Jan 24 2024

Keywords

• nonn
• base
• fini

Comments

Terms must have at least 4 digits.

There are 3287310 terms, with the last being 1245678987653210123456789. - Michael S. Branicky, Jan 26 2024

Examples

			The first term is 1201: increases 1-2, decreases 2-0, then increases 0-1.  An example 7-digit term is 1215679.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..10000
• Michael S. Branicky, Python program producing full sequence
• James S. DeArmon, Python program for A367735

Crossrefs

Cf. A062351, A062352, A156116.

Programs

Maple

q:= proc(n) local i, l, s;
      l, s:= convert(n, base, 10), 1;
      for i to nops(l)-1 while s<5 do s:=
       `if`(l[i]=l[i+1], 5,
       `if`(l[i]>l[i+1], [2$2, 4$2][s], [5, 3$2, 5][s]))
      od; is(s=4)
    end:
select(isprime and q, [$1..15000])[];  # Alois P. Heinz, Jan 26 2024

Python

from sympy import isprime
from itertools import combinations, islice
def agen(): # generator of terms
    for d in range(4, 28):
        print(d)
        passed = set()
        for d1 in range(2, min(d-2, 9)+1):
            for c1 in combinations("123456789", d1):
                for d2 in range(1, min(d-d1-1, 10)+1):
                    digits2 = list(map(str, range(int(c1[-1])-1, -1, -1)))
                    for c2 in combinations(digits2, d2):
                        digits3 = list(map(str, range(int(c2[-1])+1, 11)))
                        for c3 in combinations(digits3, d - d1 - d2):
                            t = int("".join(c1 + c2 + c3))
                            if isprime(t):
                                passed.add(t)
        yield from sorted(passed)
print(list(islice(agen(), 63))) # Michael S. Branicky, Jan 26 2024