A059168 -id:A059168 - 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

A343590 Undulating alternating primes.

Original entry on oeis.org

2, 3, 5, 7, 23, 29, 41, 43, 47, 61, 67, 83, 89, 101, 103, 107, 109, 163, 181, 307, 383, 503, 509, 523, 547, 563, 587, 701, 709, 727, 769, 787, 907, 929, 947, 967, 2141, 2143, 2161, 2309, 2503, 2549, 2707, 2729, 2749, 2767, 2903, 2909, 2927, 2969, 4363, 4507, 4523, 4547, 4549, 4703

Offset: 1

Bernard Schott, Apr 21 2021

Equivalently, primes in which the value of the digits alternately rises or falls (undulating, A059168) and in which the parity of the digits changes in turns (alternating, A030144).

The first 17 terms are the same as A030144; then a(18) = 163 while A030144(18) = 127.

			2143 is a term as it is prime, digits 2, 1, 4 and 3 have even and odd parity alternately, and also alternately fall and rise.

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Intersection of A030144 and A059168.

A343591 is a subsequence.

q[n_] := PrimeQ[n] && AllTrue[Differences[Sign @ Differences[(d = IntegerDigits[n])]], # != 0 &] && AllTrue[Differences @ Mod[d, 2], # != 0 &]; Select[Range[5000], q] (* Amiram Eldar, Apr 21 2021 *)

from sympy import sieve
def sign(n): return (n > 0) - (n < 0)
def ok(p):
  if p < 10: return True
  s = str(p)
  t = set(sign(int(s[i])%2-int(s[i-1])%2)*(-1)**i for i in range(1, len(s)))
  t2 = set(sign(int(s[i])-int(s[i-1]))*(-1)**i for i in range(1, len(s)))
  return (t == {1} or t == {-1}) and (t2 == {1} or t2 == {-1})
def aupto(limit): return [p for p in sieve.primerange(1, limit+1) if ok(p)]
print(aupto(4703)) # Michael S. Branicky, Apr 21 2021

from sympy import isprime
def f(w,dir):
    if dir == 1:
        for s in w:
            for t in range(int(s[-1])+1,10,2):
                yield s+str(t)
    else:
        for s in w:
            for t in range(1-int(s[-1])%2,int(s[-1]),2):
                yield s+str(t)
A343590_list = []
for l in range(5):
    for d in '123456789':
        x = d
        for i in range(1,l+1):
            x = f(x,(-1)**i)
        A343590_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))
            A343590_list.extend([int(p) for p in y if isprime(int(p))]) # Chai Wah Wu, Apr 25 2021

A059170 Strictly undulating primes (digits alternate and differ by 1).

Original entry on oeis.org

2, 3, 5, 7, 23, 43, 67, 89, 101, 787, 32323, 78787, 1212121, 323232323, 989898989, 12121212121, 32323232323, 787878787878787878787, 787878787878787878787878787, 1212121212121212121212121212121212121212121

Offset: 1

N. J. A. Sloane, Feb 14 2001

Of form ababa... with |a-b| = 1.

The next two terms have 95 and 139 digits respectively. - Jayanta Basu, May 09 2013

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

Harvey P. Dale, Table of n, a(n) for n = 1..25 (All terms with less than 1000 digits.)
Patrick De Geest, More undulating primes
C. A. Pickover, "Wonders of Numbers, Adventures in Mathematics, Mind and Meaning," Zentralblatt review

Cf. A032758, A059168, A048398, A033619, A046075, A059758.

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 && Abs[a[p][[1]]-b[p][[1]]] == 1, AppendTo[t,p]]], {n,10^5}]; t (* Jayanta Basu, May 08 2013 *)
t1=Join[{2,3,5,7},Select[Range[10,100],PrimeQ[#]&&Abs[Differences[IntegerDigits[#]]]=={1}&]]; Do[a=n*10+(n-1);b=(n-1)*10+n; t1=Join[t1,Select[Table[(a*10^(2*n+1)-b)/99,{n,25}],PrimeQ]]; If[n<=7,c=n*10+(n+1);d=(n+1)*10+n;t1=Join[t1,Select[Table[(c*10^(2*n+1)-d)/99,{n,25}],PrimeQ]]],{n,1,9,2}]; Sort[t1] (* Jayanta Basu, May 09 2013 *)
 With[{c=Flatten[{#,Reverse[#]}&/@Table[{a,a+1},{a,0,8}],1]},Flatten[ Select[ Table[ FromDigits[PadRight[{},n,#]],{n,50}],PrimeQ]&/@c]]//Union (* Harvey P. Dale, Aug 20 2022 *)

Extended by Patrick De Geest, Feb 25 2001

Offset corrected by Arkadiusz Wesolowski, Sep 13 2011

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

A355301 Normal undulating numbers where "undulating" means that the alternate digits go up and down (or down and up) and "normal" means that the absolute differences between two adjacent digits may differ.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 101, 102, 103, 104, 105, 106, 107, 108, 109, 120, 121, 130, 131, 132, 140, 141, 142, 143, 150

Offset: 1

Bernard Schott, Jun 27 2022

This definition comes from Patrick De Geest's link.
Other definitions for undulating are present in the OEIS (e.g., A033619, A046075).
When the absolute differences between two adjacent digits are always equal (e.g., 85858), these numbers are called smoothly undulating numbers and form a subsequence (A046075).
The definition includes the trivial 1- and 2-digit undulating numbers.
Subsequence of A043096 where the first different term is A043096(103) = 123 while a(103) = 130.
This sequence first differs from A010784 at a(92) = 101, A010784(92) = 102.
The sequence differs from A160542 (which contains 100). - R. J. Mathar, Aug 05 2022

			111 is not a term here, but A033619(102) = 111.
a(93) = 102, but 102 is not a term of A046075.
Some terms: 5276, 918230, 1053837, 263915847, 3636363636363636.
Are not terms: 1331, 594571652, 824327182.

Patrick De Geest, Smoothly Undulating Palindromic Primes, World of Numbers.

Cf. A033619, A043096, A046075.
Cf. A059168 (subsequence of primes).
Differs from A010784, A241157, A241158.
Cf. A355302, A355303, A355304.

isA355301 := proc(n)
    local dgs,i,back,forw ;
    dgs := convert(n,base,10) ;
    if nops(dgs) < 2 then
        return true;
    end if;
    for i from 2 to nops(dgs)-1 do
        back := op(i,dgs) -op(i-1,dgs) ;
        forw := op(i+1,dgs) -op(i,dgs) ;
        if back*forw >= 0 then
            return false;
        end if ;
    end do:
    back := op(-1,dgs) -op(-2,dgs) ;
    if back = 0 then
        return false;
    end if ;
    return true ;
end proc:
A355301 := proc(n)
    option remember ;
    if n = 1 then
        0;
    else
        for a from procname(n-1)+1 do
            if isA355301(a) then
                return a;
            end if;
        end do:
    end if;
end proc:
seq(A355301(n),n=1..110) ; # R. J. Mathar, Aug 05 2022

q[n_] := AllTrue[(s = Sign[Differences[IntegerDigits[n]]]), # != 0 &] && AllTrue[Differences[s], # != 0 &]; Select[Range[0, 100], q] (* Amiram Eldar, Jun 28 2022 *)

isok(m) = if (m<10, return(1)); my(d=digits(m), dd = vector(#d-1, k, sign(d[k+1]-d[k]))); if (#select(x->(x==0), dd), return(0)); my(pdd = vector(#dd-1, k, dd[k+1]*dd[k])); #select(x->(x>0), pdd) == 0; \\ Michel Marcus, Jun 30 2022

A343675 Undulating alternating palindromic primes.

Original entry on oeis.org

2, 3, 5, 7, 101, 181, 383, 727, 787, 929, 10301, 10501, 14341, 16361, 16561, 18181, 30103, 30703, 32323, 36563, 38183, 38783, 70507, 72727, 74747, 78787, 90709, 94949, 96769, 1074701, 1092901, 1212121, 1218121, 1412141, 1616161, 1658561, 1856581, 1878781, 3072703

Offset: 1

Chai Wah Wu, Apr 25 2021

All terms have an odd number of decimal digits.
For n > 3, a(n) is odd and not divisible by 5.
Intersection of A002385, A030144 and A059168.
Subsequence of A343590.

			16361 is a term as it is a palindromic prime, the digits 1, 6, 3, 6 and 1 have odd and even parity alternately, and also alternately rise and fall.

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Cf. A002385, A030144, A059168, A343590.

Union@Flatten[{{2,3,5,7},Array[Select[FromDigits/@Riffle@@@Tuples[{Tuples[{1,3,5,7,9},#],Tuples[{0,2,4,6,8},#-1]}],(s=Union@Partition[Sign@Differences@IntegerDigits@#,2];(s=={{1,-1}}||s=={{-1,1}})&&PrimeQ@#&&PalindromeQ@#)&]&,4]}] (* Giorgos Kalogeropoulos, Apr 26 2021 *)

from sympy import isprime
def f(w):
    for s in w:
        for t in range(int(s[-1])+1,10,2):
            yield s+str(t)
def g(w):
    for s in w:
        for t in range(1-int(s[-1])%2,int(s[-1]),2):
            yield s+str(t)
A343675_list = [2,3,5,7]
for l in range(1,9):
    for d in '1379':
        x = d
        for i in range(1,l+1):
            x = g(x) if i % 2 else f(x)
        A343675_list.extend([int(p+p[-2::-1]) for p in x if isprime(int(p+p[-2::-1]))])
        y = d
        for i in range(1,l+1):
            y = f(y) if i % 2 else g(y)
        A343675_list.extend([int(p+p[-2::-1]) for p in y if isprime(int(p+p[-2::-1]))])

A343676 Number of n-digit undulating alternating primes.

Original entry on oeis.org

4, 9, 23, 49, 175, 321, 1189, 3025, 12111, 28492, 113409, 251513, 1068440, 2629980, 11690210, 28498852, 128871588, 298890814, 1309837146

Offset: 1

Chai Wah Wu, Apr 25 2021

a(n) is the number of n-digit terms in A343590.

a(n) <= A057333(n).

Cf. A002385, A030144, A057333, A059168, A343590, A343675.

def f(w):
    for s in w:
        for t in range(int(s[-1])+1,10,2):
            yield s+str(t)
def g(w):
    for s in w:
        for t in range(1-int(s[-1])%2,int(s[-1]),2):
            yield s+str(t)
def A343676(n):
    c = 0
    for d in '123456789':
        x = d
        for i in range(1,n):
            x = g(x) if i % 2 else f(x)
        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) if i % 2 else g(y)
            c += sum(1 for p in y if isprime(int(p)))
    return c

a(18)-a(19) from Martin Ehrenstein, Apr 27 2021

A343677 Number of (2n+1)-digit undulating alternating palindromic primes.

Original entry on oeis.org

4, 6, 19, 34, 100, 241, 697, 1779, 6590, 16585, 57237, 179291, 591325, 1707010, 6520756, 18271423, 65212230, 210339179, 706823539

Offset: 0

Chai Wah Wu, Apr 25 2021

a(n) is the number of (2n+1)-digit terms in A343675.

a(n) <= A057332(n).

Cf. A002385, A030144, A057332, A059168, A343590.

from sympy import isprime
def f(w):
    for s in w:
        for t in range(int(s[-1])+1,10,2):
            yield s+str(t)
def g(w):
    for s in w:
        for t in range(1-int(s[-1])%2,int(s[-1]),2):
            yield s+str(t)
def A343677(n):
    if n == 0:
        return 4
    c = 0
    for d in '1379':
        x = d
        for i in range(1,n+1):
            x = g(x) if i % 2 else f(x)
        c += sum(1 for p in x if isprime(int(p+p[-2::-1])))
        y = d
        for i in range(1,n+1):
            y = f(y) if i % 2 else g(y)
        c += sum(1 for p in y if isprime(int(p+p[-2::-1])))
    return c

a(17)-a(18) from Chai Wah Wu, May 02 2021

A309488 Primes whose decimal expansion is of the form d_1+0+d_2+0+d_3+0+...+0+d_k where d_i are digits with 1 <= d_i <= 9, k > 1 and + stands for concatenation.

Original entry on oeis.org

101, 103, 107, 109, 307, 401, 409, 503, 509, 601, 607, 701, 709, 809, 907, 10103, 10301, 10303, 10501, 10601, 10607, 10709, 10903, 10909, 20101, 20107, 20201, 20407, 20507, 20509, 20707, 20807, 20809, 20903, 30103, 30109, 30203, 30307, 30403, 30509, 30703, 30707, 30803, 30809

Offset: 1

Bernard Schott, Aug 04 2019

The terms of this sequence have necessarily an odd number >= 3 of digits.
There is only one term whose digits > 0 are all equal: 101.
The only cyclops primes (A134809) of this sequence are the first 15 terms from 101 to 907.
The first palindromes of this sequence are 101, 10301, 10501, 10601, 30103, 30203, 30403, 30703, 30803, ...
Intersection with A309101 = {503, 10103, 10303, 10903, ...}.

			103 is the smallest term with two distinct digits > 0.
10607 is the smallest term with three distinct digits > 0.

Cf. A000040, A002385, A134809, A309101.

Subsequence of A059168 (undulating primes).

sol:=[]; m:=1; for p in PrimesInInterval(101,50000) do  v:=Reverse(Intseq(p)); test:=0; for u in [1..#v-1] do if u mod 2 eq 0 and v[u] eq 0 and v[u+1] ne 0 then test:=test+1; end if; end for; if test eq (#v-1)/2 then sol[m]:=p; m:=m+1; end if; end for; sol; // Marius A. Burtea, Aug 04 2019

aQ[n_] := PrimeQ[n] && OddQ[(m = Length[(d = IntegerDigits[n])])] && Flatten@Position[d, ?(# == 0 &)] == Range[2, m, 2]; Select[Range[100, 31000], aQ] (* _Amiram Eldar, Aug 04 2019 *)

eva(n) = subst(Pol(n), x, 10)
f(n) = my(d=digits(n)); eva(vector(2*#d-1, k, if (k%2, d[1+k\2]))) \\ from Michel Marcus
terms(n) = my(i=0); for(k=10, oo, if(i>=n, break); if(vecmin(digits(k)) > 0, my(iz=f(k)); if(ispseudoprime(iz), print1(iz, ", "); i++)))
/* Print initial 40 terms as follows: */
terms(40) \\ Felix Fröhlich, Aug 08 2019

cp's OEIS Frontend

A032758 Undulating primes (digits alternate).

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Mathematica

Python

Extensions

A343590 Undulating alternating primes.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Python

Python

A059170 Strictly undulating primes (digits alternate and differ by 1).

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Mathematica

Extensions

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

Views

Author

Keywords

Comments

Links

Crossrefs

Formula

Extensions

A355301 Normal undulating numbers where "undulating" means that the alternate digits go up and down (or down and up) and "normal" means that the absolute differences between two adjacent digits may differ.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

A343675 Undulating alternating palindromic primes.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Python

A343676 Number of n-digit undulating alternating primes.

Views

Author

Keywords

Comments

Crossrefs

Programs

Python

Extensions

A343677 Number of (2n+1)-digit undulating alternating palindromic primes.

Views

Author