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 A057332 a(n) is the number of (2n+1)-digit palindromic primes that undulate. 4
 4, 15, 52, 210, 1007, 5156, 25571, 133293, 727082, 3874464, 21072166, 117829671, 654556778 (list; graph; refs; listen; history; text; internal format)
 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

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Last modified August 10 05:56 EDT 2024. Contains 375044 sequences. (Running on oeis4.)