A062352 -id:A062352 - cp's OEIS frontend

A062351 Palindromic primes with strictly increasing digits up to the middle and then strictly decreasing.

2, 3, 5, 7, 11, 131, 151, 181, 191, 353, 373, 383, 787, 797, 12421, 12721, 12821, 13831, 13931, 14741, 17971, 34543, 34843, 35753, 1235321, 1245421, 1257521, 1268621, 1278721, 1456541, 1469641, 1489841, 1579751, 1589851, 3479743

Offset: 1

Amarnath Murthy, Jun 23 2001

The last term of the finite series is a(63) = 123467898764321.

			13831 belongs to the sequence as it is a palindromic prime in which the digits are increasing up to the middle digit 8 and then decreasing.

Nathaniel Johnston, Table of n, a(n) for n = 1..63 (complete sequence)
Patrick De Geest, World of Palindromic Primes

Cf. A343524 (strictly increasing palindromes), A062352, A084836.

from sympy import isprime
from itertools import combinations
def agen():
  for digits in range(1, 19):
    for left in combinations("123456789", (digits+1)//2):
      left = "".join(left)
      yield int(left + (left[:digits//2])[::-1])
print(list(filter(isprime, agen()))) # Michael S. Branicky, Apr 25 2021

Corrected and edited by Patrick De Geest, Jun 07 2003

A084836 Palindromic primes with nondecreasing digits up to the middle and then nonincreasing.

Original entry on oeis.org

2, 3, 5, 7, 11, 131, 151, 181, 191, 353, 373, 383, 787, 797, 11311, 11411, 12421, 12721, 12821, 13331, 13831, 13931, 14741, 15551, 16661, 17971, 19991, 33533, 34543, 34843, 35753, 77977, 78887, 79997, 1114111, 1117111, 1123211, 1126211, 1129211, 1134311

Offset: 1

Patrick De Geest, Jun 07 2003

Michael S. Branicky, Table of n, a(n) for n = 1..10000
Patrick De Geest, World of Palindromic Primes

Cf. A084837, A062351, A062352.

from sympy import isprime
from itertools import count, islice, combinations_with_replacement as mc
def agen():
    yield from (2, 3, 5, 7, 11)
    for d in count(2):
        ndni = (int("".join(m+m[:-1][::-1])) for m in mc("123456789", d))
        yield from filter(isprime, ndni)
print(list(islice(agen(), 40))) # Michael S. Branicky, Jun 26 2022

a(39) and beyond from Michael S. Branicky, Jun 26 2022

A084837 Palindromic primes with nonincreasing digits up to the middle and then nondecreasing.

Original entry on oeis.org

2, 3, 5, 7, 11, 101, 313, 727, 757, 919, 929, 31013, 72227, 73037, 73237, 74047, 75557, 76367, 76667, 77377, 77477, 91019, 93139, 93239, 94049, 94349, 96269, 96469, 97379, 97579, 98389, 98689, 3211123, 3222223, 3310133, 3321233, 3331333, 7100017, 7300037, 7310137

Offset: 1

Patrick De Geest, Jun 07 2003

Michael S. Branicky, Table of n, a(n) for n = 1..10000
Patrick De Geest, World of Palindromic Primes

Cf. A084836, A062351, A062352.

from sympy import isprime
from itertools import count, islice, combinations_with_replacement as mc
def agen(): # generator of terms
    yield from (2, 3, 5, 7, 11)
    for d in count(2):
        nind = (int("".join(m+m[:-1][::-1])) for m in mc("9876543210", d))
        yield from sorted(filter(isprime, nind))
print(list(islice(agen(), 40))) # Michael S. Branicky, Jun 26 2022

a(38) and beyond from Michael S. Branicky, Jun 26 2022

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

Original entry on oeis.org

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

James S. DeArmon, Jan 24 2024

Terms must have at least 4 digits.

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

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

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

Cf. A062351, A062352, A156116.

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

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

A371378 Prime numbers wherein digit values decrease, increase, and finally decrease.

Original entry on oeis.org

1021, 1031, 1051, 1061, 1063, 1087, 1091, 1093, 1097, 2053, 2063, 2081, 2083, 2087, 2131, 2141, 2143, 2153, 2161, 3041, 3061, 3083, 3121, 3163, 3181, 3187, 3191, 3251, 3253, 3271, 4021, 4051, 4073, 4091, 4093, 4153, 4231, 4241, 4243, 4253, 4261, 4271, 4273, 4283

Offset: 1

James S. DeArmon, Mar 20 2024

Terms must have at least 4 digits. The sequence is finite.

There are 3136837 terms, with the last being 98765432101234567987654321. - Michael S. Branicky, Mar 20 2024

Alois P. Heinz, Table of n, a(n) for n = 1..10000
Michael S. Branicky, Alternative Python program for full sequence A371378
James S. DeArmon, LISP Code for A371378

Cf. A062351, A062352, A156116, A367735.

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]Alois P. Heinz, Mar 21 2024

Select[Prime[Range[600]], SplitBy[Sign[Differences[IntegerDigits[#]]], Sign][[;; , 1]] == {-1, 1, -1} &] (* Amiram Eldar, Mar 21 2024 *)

from sympy import isprime
from itertools import combinations, islice
def agen(): # generator of terms
    for d in range(4, 29):
        print(d)
        passed = set()
        for d1 in range(2, min(d-2, 11)+1):
            for c1 in combinations("9876543210", d1):
                for d2 in range(1, min(d-d1-1, 10)+1):
                    digits2 = list(map(str, range(int(c1[-1])+1, 10)))
                    for c2 in combinations(digits2, d2):
                        digits3 = list(map(str, range(int(c2[-1])-1, -1, -1)))
                        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, Mar 20 2024

More terms from Michael S. Branicky, Mar 20 2024

cp's OEIS Frontend

A062351 Palindromic primes with strictly increasing digits up to the middle and then strictly decreasing.

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A084836 Palindromic primes with nondecreasing digits up to the middle and then nonincreasing.

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A084837 Palindromic primes with nonincreasing digits up to the middle and then nondecreasing.

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

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A371378 Prime numbers wherein digit values decrease, increase, and finally decrease.

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Maple

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