A059168 Primes in which digits alternately rise and fall (or vice versa); sometimes called undulating primes.

2, 3, 5, 7, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 131, 151, 163, 173, 181, 191, 193, 197, 241, 251, 263, 271, 281, 283, 293, 307, 313, 317, 353, 373, 383, 397, 401, 409, 419, 439, 461, 463, 487, 491

Offset: 1

Author

N. J. A. Sloane, Feb 14 2001

Keywords

• nonn
• base
• easy

References

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

Links

• Robert Israel, Table of n, a(n) for n = 1..10000
• C. A. Pickover, "Wonders of Numbers, Adventures in Mathematics, Mind and Meaning," Zentralblatt review

Crossrefs

Cf. A032758, A059170.

Programs

Maple

extend:= proc(n) local L,j;
  L:= convert(n,base,10);
  if (L[-1] < L[-2]) xor (nops(L)::odd) then
    seq(10*n+j,j=0..L[1]-1)
  else
    seq(10*n+j,j=L[1]+1..9)
  fi
end proc:
und[2]:= [seq(seq(10*i+j,j=subs(i=NULL,[$0..9])),i=1..9)]:
for i from 3 to 4 do und[i]:= map(extend,und[i-1]) od:
select(isprime, [2,3,5,7,seq(op(und[i],i=2..4)]); # Robert Israel, Nov 15 2018

Mathematica

d[n_]:=Differences[IntegerDigits[n]]; mQ[n_]:=MemberQ[d[n],0]==False; a[n_]:=DeleteDuplicates[Sign[Take[d[n],{1,-1,2}]]]; b[n_]:=DeleteDuplicates[Sign[Take[d[n],{2,-1,2}]]]; t={}; Do[p=Prime[n]; If[mQ[p],If[Length[IntegerDigits[p]]<=2,AppendTo[t,p],If[Length[a[p]]==Length[b[p]]==1 && a[p][[1]]!=b[p][[1]],AppendTo[t,p]]]],{n,95}]; t (* Jayanta Basu, May 08 2013 *)
Table[Which[p<10,p,p<100&&Differences[IntegerDigits[p]]!={0},p,p>100&&Union[Total/@ Partition[Sign[Differences[IntegerDigits[p]]],2,1]]=={0},p,True,Nothing],{p,Prime[ Range[ 150]]}] (* Harvey P. Dale, Aug 07 2023 *)

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)
A059168_list = []
for l in range(5):
    for d in '123456789':
        x = d
        for i in range(1,l+1):
            x = f(x,(-1)**i)
        A059168_list.extend([int(p) for p in x if isprime(int(p))])
        if l > 0:
            y = d
            for i in range(1,l+1):
                y = f(y,(-1)**(i+1))
            A059168_list.extend([int(p) for p in y if isprime(int(p))]) # Chai Wah Wu, Apr 25 2021

Extensions

More terms from Larry Reeves (larryr(AT)acm.org), Feb 15 2001

Offset changed by Robert Israel, Nov 15 2018