cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A046076 Indices of binary undulants; numbers n such that 2^n contains the alternating sequence of digits 010... or 101...

Original entry on oeis.org

103, 107, 138, 159, 179, 187, 192, 199, 205, 211, 217, 218, 234, 249, 254, 264, 269, 285, 288, 293, 296, 299, 304, 305, 316, 347, 350, 354, 364, 368, 378, 383, 384, 385, 390, 393, 406, 416, 420, 427, 436, 443, 445, 449, 451, 454, 457, 462, 463, 485, 488
Offset: 1

Views

Author

Eric W. Weisstein

Keywords

References

  • C. A. Pickover, ``The Undulation of the Monks.'' Ch. 20 in Keys to Infinity. New York: W.H.Freeman, pp. 159-161 1995.
  • C. A. Pickover, "Wonders of Numbers", Oxford New York 2001, Chapter 52, pp. 123-124, 316-317.

Links

Crossrefs

Cf. A033619, A046075, A046077.

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

Original entry on oeis.org

4, 15, 52, 210, 1007, 5156, 25571, 133293, 727082, 3874464, 21072166, 117829671, 654556778
Offset: 0

Views

Author

Patrick De Geest, Sep 15 2000

Keywords

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

Crossrefs

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

Programs

  • 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

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

A057333 Numbers of n-digit primes that undulate.

Original entry on oeis.org

4, 20, 74, 347, 1743, 8385, 44355, 229952, 1235489, 6629026, 37152645, 202017712, 1142393492, 6333190658
Offset: 1

Views

Author

Patrick De Geest, Sep 15 2000

Keywords

Comments

'Undulate' means that the alternate digits are consistently greater than or less than the digits adjacent to them (e.g., 70769). Smoothly undulating palindromic primes (e.g., 95959) 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

Crossrefs

Cf. A046075, A033619, A032758, A039944, A016073, A046076, A046077, A057332.

Programs

  • 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 A057333(n):
    c = 0
    for d in '123456789':
        x = d
        for i in range(1,n):
            x = f(x,(-1)**i)
        c += sum(1 for p in x if isprime(int(p)))
        if n > 1:
            y = d
            for i in range(1,n):
                y = f(y,(-1)**(i+1))
            c += sum(1 for p in y if isprime(int(p)))
    return c # Chai Wah Wu, Apr 25 2021

Extensions

Offset corrected and a(10)-a(11) from Donovan Johnson, Aug 08 2010
a(12) from Giovanni Resta, Feb 24 2013
a(2) corrected by Chai Wah Wu, Apr 25 2021
a(13)-a(14) from Chai Wah Wu, May 02 2021
Showing 1-3 of 3 results.

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