A242541 -id:A242541 - cp's OEIS frontend

A032758 Undulating primes (digits alternate).

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, 131, 151, 181, 191, 313, 353, 373, 383, 727, 757, 787, 797, 919, 929, 18181, 32323, 35353, 72727, 74747, 78787, 94949, 95959, 1212121, 1616161, 323232323, 383838383

Offset: 1

Patrick De Geest, May 15 1998

Sometimes called "smoothly undulating primes", to distinguish them from A059168.

C. A. Pickover, "Keys to Infinity", Wiley 1995, p. 159,160.
C. A. Pickover, "Wonders of Numbers", Oxford New York 2001, Chapter 52, pp. 123-124, 316-317.

Michael S. Branicky, Table of n, a(n) for n = 1..131 (terms 1..100 from Sean A. Irvine)
C. A. Pickover, "Wonders of Numbers, Adventures in Mathematics, Mind and Meaning," Zentralblatt review
Charles W. Trigg, Nine-digit patterned palindromic primes, Crux Mathematicorum, Vol. 7, No. 6, June - July 1981, pp. 168-170.

Cf. A033619, A030291, A059168, A242541, A004022.

a[n_] := DeleteDuplicates[Take[IntegerDigits[n],{1,-1,2}]]; b[n_] := DeleteDuplicates[Take[IntegerDigits[n],{2,-1,2}]]; t={}; Do[p=Prime[n]; If[p<10, AppendTo[t,p], If[Length[a[p]] == Length[b[p]] == 1 && a[p][[1]] != b[p][[1]], AppendTo[t,p]]], {n,3*10^7}]; t (* Jayanta Basu, May 04 2013 *)

from itertools import count, islice
from sympy import isprime, primerange
def agen(): # generator of terms
    yield from (p for p in primerange(2, 100) if p != 11)
    yield from (t for t in (int((A+B)*d2+A) for d2 in count(1) for A in "1379" for B in "0123456789" if A != B) if isprime(t))
print(list(islice(agen(), 51))) # Michael S. Branicky, Jun 09 2022

Sequence corrected by Juri-Stepan Gerasimov, Jan 28 2010

Offset corrected by Arkadiusz Wesolowski, Sep 13 2011

A033619 Undulating numbers (of form abababab... in base 10).

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 101, 111, 121, 131, 141, 151

Offset: 1

N. J. A. Sloane

C. A. Pickover, "Keys to Infinity", Wiley 1995, p. 159,160.
C. A. Pickover, "Wonders of Numbers", Oxford New York 2001, Chapter 52, pp. 123-124, 316-317.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
C. A. Pickover, "Wonders of Numbers, Adventures in Mathematics, Mind and Meaning," Zentralblatt review
Eric Weisstein's World of Mathematics, Undulating Number

Cf. A016073, A046075, A122875, A242541.

import Data.Set (fromList, deleteFindMin, insert)
a033619 n = a033619_list !! (n-1)
a033619_list = [0..9] ++ (f $ fromList [10..99]) where
   f s = m : f (insert (m * 10 + h) s') where
     h = div (mod m 100) 10
     (m,s') = deleteFindMin s
-- Reinhard Zumkeller, May 01 2012

$0..9,seq(seq(seq(a*(10^(d+1)-10^(d+1 mod 2))/99 + b*(10^d - 10^(d mod 2))/99, b=0..9),a=1..9),d=2..6); # Robert Israel, Jul 08 2016

wave[1] = Range[0, 9]; wave[2] = Range[10, 99]; wave[n_] := wave[n] = Select[ Union[ Flatten[ {id = IntegerDigits[#]; FromDigits[ Prepend[id, id[[2]]]], FromDigits[ Append[id, id[[-2]]]]} & /@ wave[n-1]]], 10^(n-1) < # < 10^n & ]; Flatten[ Table[ wave[n], {n, 1, 3}]] (* Jean-François Alcover, Jun 19 2012 *)

from itertools import count, islice
def agen(): # generator of terms
    yield from range(10)
    for d in count(2):
        q, r = divmod(d, 2)
        for a in "123456789":
            for b in "0123456789":
                yield int((a+b)*q + a*r)
print(list(islice(agen(), 106))) # Michael S. Branicky, Mar 28 2022

A092696 Smoothly undulating palindromic primes of the form (12*10^n-21)/99.

Original entry on oeis.org

1212121, 12121212121, 1212121212121212121212121212121212121212121

Offset: 1

Rick L. Shepherd, Mar 04 2004

The De Geest link calls these smoothly undulating palindromic primes. Corresponding n are given in A062209. Equivalently, primes of the form 1212...121: Decimal digits "12" repeated k times with 1 appended (or "21" repeated k times with 1 prefixed). Corresponding k are given in A056803. The next term, a(4), has "12" repeating A056803(4) = 69 times and length A062209(4) = 2*A056803(4) + 1 = 139 decimal digits.

Cf. A056803 (number of 12's (or 21's)), A062209 (corresponding decimal digit lengths).

Cf. A032758, A033619, A030291, A059168, A242541, A004022.

a(n) = (4*10^A062209(n)-7)/33. - M. F. Hasler, Jul 30 2015

Edited by M. F. Hasler, Jul 30 2015

A343591 Smoothly undulating alternating primes.

Original entry on oeis.org

2, 3, 5, 7, 23, 29, 41, 43, 47, 61, 67, 83, 89, 101, 181, 383, 727, 787, 929, 18181, 32323, 72727, 74747, 78787, 94949, 1212121, 1616161, 323232323, 383838383, 727272727, 929292929, 989898989, 12121212121, 14141414141, 32323232323, 383838383838383, 38383838383838383, 72727272727272727

Offset: 1

Bernard Schott, Apr 21 2021

Equivalently, numbers that are primes, smoothly undulating = in which the digits alternate: ababab... with a <> b (A032758) and alternating = in which parity of the digits alternates (A030144).
Charles W. Trigg was the first to use the word 'smoothly' for these integers.
If we note a(ba) the terms where the substring (ba) is repeated in their decimal expansion, there exist only 16 possibilities with a odd <> 5 and b even <> 0 to get such primes. Indeed, there exist primes of the form 1(21), 1(41), 1(61), 1(81), 3(23), 3(83), 7(27), 7(47), 7(87), 9(29), 9(49), 9(89). There do not exist terms of the form 3(63), 7(67), 9(69), as they are always composite.
Now, what about possible terms of the form 3(43)? If (43) is repeated 3k times, 3(43) is divisible by 3; if (43) is repeated 3k+1 times, 3(43) is divisible by 7; so if such a prime exists, then the substring (43) must be repeated 3k+2 times, but it is not known if such smoothly undulating prime 3(43) exists and if it exists, (43) must be repeated at least 9302 times, so k >= 3100 (link).
Some properties:
-> Every term has two digits or an odd number of digits.
-> All terms with an odd number of digits are palindromic (A059758).
-> Only 2 and the nine 2-digit terms begin with an even digit.

			1616161 is a term as it is prime and the digits 1 and 6 have odd and even parity and alternate.

Charles W. Trigg, Special Palindromic Primes, Journal of Recreational Mathematics, 4 (July 1971) 169-170.

Robert Israel, Table of n, a(n) for n = 1..69
Mathematics Stack Exchange, (Unsolved) In this infinite sequence, no term is a prime: prove/disprove.

Intersection of A030144 and A032758.
Subsequence of A343590.
Cf. A059758, A242541.

f:= proc(n) local i,a,b,c,d;
  c:= add(10^i,i=1..n-1,2);
  d:= add(10^i,i=0..n-1,2);
  if n = 2 then op(select(isprime,[seq(seq(a*c+b*d, b=[1,3,7,9]),a=[2,4,6,8])]))
    else op(select(isprime, [seq(seq(a*c+b*d, a=[0,2,4,6,8]),b=[1,3,7,9])]))
  fi
end proc:
f(1):= (2,3,5,7):
map(f, [1,2,seq(i,i=3..17,2)]); # Robert Israel, Nov 09 2023

f[o_,e_,n_,m_] := FromDigits @ Riffle[ConstantArray[o,n], ConstantArray[e,n-m]]; seq[n_,m_] := Module[{o = Range[1,9,2], e = Range[0,8,2]}, Select[Union[f @@@ Join[Tuples[{o, e, {n}, {m}}], Tuples[{Rest @ e, o, {n}, {m}}]]], PrimeQ]]; s = seq[1, 1]; Do[s = Join[s, seq[m, Boole[m > 1]]], {m, 1, 10}]; s (* Amiram Eldar, Apr 21 2021 *)

from sympy import isprime
def agenthru(maxdigits):
  if maxdigits >= 1: yield from [2, 3, 5, 7]
  for digits in [2]*(maxdigits >= 2) + list(range(3, maxdigits+1, 2)):
    hlf, odd = (digits+1)//2, digits%2
    d1range = "1379" if digits%2 == 1 else "123456789"
    d2range = "1379" if digits%2 == 0 else "0123456789"
    for d1 in d1range:
      for d2 in d2range:
        if int(d1)%2 == int(d2)%2: continue
        t = int("".join([*sum(zip(d1*hlf, d2*(digits-hlf)), ())]+[d1*odd]))
        if isprime(t): yield t
print([p for p in agenthru(17)]) # Michael S. Branicky, Apr 21 2021

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A032758 Undulating primes (digits alternate).

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A033619 Undulating numbers (of form abababab... in base 10).

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A092696 Smoothly undulating palindromic primes of the form (12*10^n-21)/99.

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A343591 Smoothly undulating alternating primes.

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