A032758 -id:A032758 - cp's OEIS frontend

A057333 Numbers of n-digit primes that undulate.

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

Offset: 1

Patrick De Geest, Sep 15 2000

'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.

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

C. A. Pickover, "Wonders of Numbers, Adventures in Mathematics, Mind and Meaning," Zentralblatt review
Eric Weisstein's World of Mathematics, Undulating Number.

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

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

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

A077799 Numbers m such that a smoothly undulating palindromic prime of the form (rs*10^m-sr)/99 exists, where r and s are two distinct digits and rs and sr denote concatenations of those digits.

Original entry on oeis.org

3, 5, 7, 9, 11, 15, 17, 21, 23, 25, 27, 31, 33, 37, 39, 43, 45, 51, 55, 57, 63, 65, 71, 77, 81, 83, 89, 95, 99, 109, 133, 139, 143, 145, 149, 161, 163, 195, 209, 219, 225, 229, 237, 243, 245, 277, 315, 357, 479, 513, 515, 537, 551, 561, 567, 583, 627, 849, 857

Offset: 1

Patrick De Geest, Nov 16 2002

Ray Chandler, Table of n, a(n) for n = 1..146
P. De Geest, SUPP's Sorted By Length
Index entries for sequences related to smoothly undulating palindromic primes

Cf. A062209-A062232, A059758, A032758.

Edited by Ray Chandler, Aug 17 2011

Name clarified by Sean A. Irvine, Jun 14 2025

A232066 Periodic primes: primes p whose decimal expansion can be written as sss...st, where s is nonempty string of digits not beginning with 0, there are at least two copies of s, and t (which may be absent) is a prefix of s.

Original entry on oeis.org

11, 18181, 32323, 35353, 72727, 74747, 78787, 94949, 95959, 1041041, 1051051, 1131131, 1191191, 1201201, 1212121, 1221221, 1231231, 1261261, 1281281, 1311311, 1381381, 1401401, 1411411, 1491491, 1501501, 1551551, 1581581, 1616161, 1621621, 1641641, 1671671

Offset: 1

Paul Tek, Nov 17 2013

The terms > 10^4 in A032758 appear in this sequence.

The terms > 10^20 in A231588 appear in this sequence.

			9779779 = 977|977|9,
727272727 = 72|72|72|72|7 = 7272|7272|7.

Paul Tek, Table of n, a(n) for n = 1..5972
Paul Tek, PARI program for this sequence

Cf. A032758, A231588.

A004022 is a subsequence (these are the terms where s=1 and t is absent). - Harvey P. Dale, Jul 09 2019

PARI
```
See Link section.
```

Definition corrected by N. J. A. Sloane, Jul 09 2019

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

A062210 Numbers k such that the smoothly undulating palindromic number (14*10^k - 41)/99 is a prime.

Original entry on oeis.org

11, 277, 479, 583, 1631, 6343, 14689

Offset: 1

Patrick De Geest and Hans Rosenthal (Hans.Rosenthal(AT)t-online.de), Jun 15 2001

Prime versus probable prime status and proofs are given in the author's table.

No further terms < 100000. - Ray Chandler, Aug 17 2011

			11 is in the sequence because (14*10^11 - 41)/99 = 14141414141 is prime.

Cf. A062209-A062232, A059758, A032758.

Edited by Ray Chandler, Aug 17 2011

Name and Example edited by Jon E. Schoenfield, Jun 25 2017

A062216 Numbers k such that the smoothly undulating palindromic number (31*10^k - 13)/99 is a prime.

Original entry on oeis.org

3, 51, 83, 225, 561, 10419, 18255, 43869

Offset: 1

Patrick De Geest and Hans Rosenthal (Hans.Rosenthal(AT)t-online.de), Jun 15 2001

Prime versus probable prime status and proofs are given in the author's table.

No further terms < 100000. - Ray Chandler, Aug 17 2011

			k=51 -> (31*10^51 - 13)/99 = 313131313131313131313131313131313131313131313131313.

Cf. A062209-A062232, A059758, A032758.

43869 from Ray Chandler, Sep 30 2010

Edited by Ray Chandler, Aug 17 2011

A062220 Numbers k such that the smoothly undulating palindromic number (38*10^k - 83)/99 is a prime.

Original entry on oeis.org

3, 9, 15, 17, 21, 57, 4233, 4335, 13221, 26447, 29897, 91997

Offset: 1

Patrick De Geest and Hans Rosenthal (Hans.Rosenthal(AT)t-online.de), Jun 15 2001

Prime versus probable prime status and proofs are given in the author's table.

No further terms < 100000. - Ray Chandler, Aug 17 2011

			k=21 -> (38*10^21 - 83)/99 = 383838383838383838383.

Cf. A062209-A062232, A059758, A032758.

a(12)=91997 from Ray Chandler, Jul 29 2011

Edited by Ray Chandler, Aug 17 2011

A062231 Numbers k such that the smoothly undulating palindromic number (97*10^k - 79)/99 is a prime.

Original entry on oeis.org

9, 27, 45, 237

Offset: 1

Patrick De Geest and Hans Rosenthal (Hans.Rosenthal(AT)t-online.de), Jun 15 2001

Prime versus probable prime status and proofs are given in the author's table.

No further terms < 100000. - Ray Chandler, Aug 17 2011

			k=27 -> (97*10^27 - 79)/99 = 979797979797979797979797979.

Patrick De Geest, SUPP Reference Table - 979
Index entries for sequences related to smoothly undulating palindromic primes

Cf. A062209-A062232, A059758, A032758.

Select[Range[1,239,2],PrimeQ[FromDigits[PadRight[{},#,{9,7}]]]&] (* Harvey P. Dale, Mar 24 2021 *)

Edited by Ray Chandler, Aug 17 2011

A062211 Numbers k such that the smoothly undulating palindromic number (15*10^k - 51)/99 is a prime.

Original entry on oeis.org

3, 15, 63, 89, 245, 583, 1791, 2123, 7233, 24787, 44653

Offset: 1

Patrick De Geest and Hans Rosenthal (Hans.Rosenthal(AT)t-online.de), Jun 15 2001

Prime versus probable prime status and proofs are given in the author's table.

No further terms < 100000. - Ray Chandler, Aug 17 2011

			k=15 -> (15*10^15 - 51)/99 = 151515151515151.

Cf. A062209-A062232, A059758, A032758.

a(11)=44653 from Ray Chandler, Nov 11 2010

Edited by Ray Chandler, Aug 17 2011

A062212 Numbers k such that the smoothly undulating palindromic number (16*10^k - 61)/99 is a prime.

Original entry on oeis.org

7, 55, 109, 145, 229, 961

Offset: 1

Patrick De Geest and Hans Rosenthal (Hans.Rosenthal(AT)t-online.de), Jun 15 2001

Prime versus probable prime status and proofs are given in the author's table.

No further terms < 100000. - Ray Chandler, Aug 17 2011

			k=7 -> (16*10^7 - 61)/99 = 1616161.

Cf. A062209-A062232, A059758, A032758.

Edited by Ray Chandler, Aug 17 2011

cp's OEIS Frontend

A057333 Numbers of n-digit primes that undulate.

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A077799 Numbers m such that a smoothly undulating palindromic prime of the form (rs*10^m-sr)/99 exists, where r and s are two distinct digits and rs and sr denote concatenations of those digits.

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A232066 Periodic primes: primes p whose decimal expansion can be written as sss...st, where s is nonempty string of digits not beginning with 0, there are at least two copies of s, and t (which may be absent) is a prefix of s.

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PARI

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A343591 Smoothly undulating alternating primes.

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Maple

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A062210 Numbers k such that the smoothly undulating palindromic number (14*10^k - 41)/99 is a prime.

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A062216 Numbers k such that the smoothly undulating palindromic number (31*10^k - 13)/99 is a prime.

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A062220 Numbers k such that the smoothly undulating palindromic number (38*10^k - 83)/99 is a prime.

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A062231 Numbers k such that the smoothly undulating palindromic number (97*10^k - 79)/99 is a prime.

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Mathematica