A057332 a(n) is the number of (2n+1)-digit palindromic primes that undulate.

4, 15, 52, 210, 1007, 5156, 25571, 133293, 727082, 3874464, 21072166, 117829671, 654556778

Offset

0,1

Comments

'Undulate' means that the alternate digits are consistently greater than or less than the digits adjacent to them (e.g., 906343609). Smoothly undulating palindromic primes (e.g., 323232323) are a subset and included in the count.

References

C. A. Pickover, "Wonders of Numbers", Oxford New York 2001, Chapter 52, pp. 123-124, 316-317.

Links

Table of n, a(n) for n=0..12.

C. K. Caldwell, Prime Curios! 906343609 and Prime Curios! 1007.

C. A. Pickover, "Wonders of Numbers, Adventures in Mathematics, Mind and Meaning," Zentralblatt review

Eric Weisstein's World of Mathematics, Undulating Number.

Prog

(Python)
from sympy import isprime
from itertools import product
def sign(n): return (n > 0) - (n < 0)
def unds(n):
  s = str(n)
  if len(s) == 1: return True
  signs = set(sign(int(s[i-1]) - int(s[i])) for i in range(1, len(s), 2))
  if len(signs) > 1: return False
  if len(s) % 2 == 0: return signs == {1} or signs == {-1}
  return sign(int(s[-1]) - int(s[-2])) in signs - {0}
def candidate_pals(n): # of length 2n + 1
  if n == 0: yield from [2, 3, 5, 7]; return # one-digit primes
  for rightbutend in product("0123456789", repeat=n-1):
    rightbutend = "".join(rightbutend)
    for end in "1379": # multi-digit primes must end in 1, 3, 7, or 9
      left = end + rightbutend[::-1]
      for mid in "0123456789": yield int(left + mid + rightbutend + end)
def a(n): return sum(1 for p in candidate_pals(n) if unds(p) and isprime(p))
print([a(n) for n in range(6)]) # Michael S. Branicky, Apr 15 2021
(Python)
from sympy import isprime
def f(w, dir):
    if dir == 1:
        for s in w:
            for t in range(int(s[-1])+1, 10):
                yield s+str(t)
    else:
        for s in w:
            for t in range(0, int(s[-1])):
                yield s+str(t)
def A057332(n):
    c = 0
    for d in '123456789':
        x = d
        for i in range(1, n+1):
            x = f(x, (-1)**i)
        c += sum(1 for p in x if isprime(int(p+p[-2::-1])))
        if n > 0:
            y = d
            for i in range(1, n+1):
                y = f(y, (-1)**(i+1))
            c += sum(1 for p in y if isprime(int(p+p[-2::-1])))
    return c # Chai Wah Wu, Apr 25 2021

Crossrefs

Cf. A046075, A033619, A032758, A039944, A016073, A046076, A046077, A057333.

Sequence in context: A027295 A371217 A208722 * A230623 A162978 A171309

Adjacent sequences: A057329 A057330 A057331 * A057333 A057334 A057335

Keyword

nonn,base,more

Author

Patrick De Geest, Sep 15 2000

Extensions

a(5) from Donovan Johnson, Aug 08 2010

a(6)-a(10) from Lars Blomberg, Nov 19 2013

a(11) from Chai Wah Wu, Apr 25 2021

a(12) from Chai Wah Wu, May 02 2021

Status

approved