A033619 -id:A033619 - 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

A046075 Nontrivial undulants; base 10 numbers >100 which are of the form aba, abab, ababa, ..., where a != b.

Original entry on oeis.org

101, 121, 131, 141, 151, 161, 171, 181, 191, 202, 212, 232, 242, 252, 262, 272, 282, 292, 303, 313, 323, 343, 353, 363, 373, 383, 393, 404, 414, 424, 434, 454, 464, 474, 484, 494, 505, 515, 525, 535, 545, 565, 575, 585, 595, 606, 616, 626, 636, 646, 656

Offset: 1

Eric W. Weisstein

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.

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. A033619, A046076.

Cf. A252664.

import Data.Set (fromList, deleteFindMin, insert)
a046075 n = a046075_list !! (n-1)
a046075_list = f $ fromList
               [100 * a + 10 * b + a | a <- [1..9], b <- [0..9], b /= a]
   where f s = m : f (insert (10 * m + div (mod m 100) 10) s')
               where (m, s') = deleteFindMin s
-- Reinhard Zumkeller, Apr 29 2015, May 01 2012

A016073 Undulating squares.

Original entry on oeis.org

0, 1, 4, 9, 16, 25, 36, 49, 64, 81, 121, 484, 676, 69696

Offset: 0

Robert G. Wilson v

Numbers with decimal expansion ababab... which are squares.

"Most mathematicians believe we will never find a larger one" - this has now been proved by David Moews.

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.
Clifford A. Pickover, A Passion for Mathematics, Wiley, 2005; see p. 68.

D. Moews, Home Page [See the paper "No More Undulating Squares", available in LaTeX, DVI and Postscript]
C. A. Pickover, "Wonders of Numbers, Adventures in Mathematics, Mind and Meaning," Zentralblatt review
Eric Weisstein's World of Mathematics, Undulating Number.

Numbers in A033619 that are squares. See A122875 for the square roots.

select(issqr,[$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 &]; A016073 = Reap[Do[Do[wk = wave[n][[k]]; If[IntegerQ[Sqrt[wk]], Sow[wk]], {k, 1, Length[wave[n]]}], {n, 1, 5}]][[2, 1]] (* Jean-François Alcover, Dec 28 2012 *)

A242541 Undulating primes: prime numbers whose digits follow the pattern A, B, A, B, A, B, A, B, ...

Original entry on oeis.org

2, 3, 5, 7, 11, 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

Offset: 1

J. Lowell, May 17 2014

All numbers in this sequence with three or more digits must have an odd number of digits. Any number with an even number of digits that follow this pattern is divisible by a number of the form 1010101...1010101 where the number of digits is one less than the number of digits in the original number.
Union of A004022 and A032758. - Arkadiusz Wesolowski, May 17 2014
Because A may equal B, 11 (and other prime repunits) are terms in this sequence (but not of A032758). - Harvey P. Dale, May 26 2015

			121 = 11*11 is not prime and thus is not a term of this sequence.

Harvey P. Dale, Table of n, a(n) for n = 1..135

Cf. A004022, A032758, A033619.

select(isprime,[$0..99,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..9,2),d=3..9,2)]); # Robert Israel, Jul 08 2016

Select[Union[Flatten[Table[FromDigits[PadRight[{},n,#]],{n,9}]&/@ Tuples[ Range[0,9],2]]],PrimeQ] (* Harvey P. Dale, May 26 2015 *)

from itertools import count, islice
from sympy import isprime, primerange
def agen(): # generator of terms
    yield from primerange(2, 100)
    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 isprime(t))
print(list(islice(agen(), 51))) # Michael S. Branicky, Jun 09 2022

A328142 Elements of cycles for iterations of A329623: n -> |n - concat(sum of adjacent digits of n)|.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 182, 273, 364, 455, 546, 637, 728, 1728, 2637, 3546, 4455, 5364, 6273, 7182, 17182, 26273, 35364, 44455, 53546, 62637, 71728, 171728, 262637, 353546, 444455, 535364, 626273, 717182, 1717182, 2626273, 3535364, 4444455, 5353546, 6262637, 7171728

Offset: 1

M. F. Hasler, Dec 02 2019

Equivalently: range of A328865 \ {-1}.
By a k-cycle (or cycle of length k) of A329623 we mean a vector (x_1, ..., x_k) such that A329623(x_i) = x_{i+1} for i < k, and A329623(x_k) = x_1. We include here the cycles of length k = 1 which are the fixed points of A329623, viz A329623(x_1) = x_1. No cycle with k > 2 is known.
There are 7 infinite subsequences: for initial digit 1 <= d <= 7, alternate digit d and 8-d to form an undulating (A033619) number of arbitrary length L >= 3, then add 11.
The terms with initial digit d > 4 are the larger member of a 2-cycle having a term with d < 4 as smaller member. The terms with d = 4 (and those <= 9) are fixed points. So far no other fixed points or other cycles are known. As far as this remains valid, the terms of this sequence are characterized by A329623(A329623(x)) = x.
Sequence A328279 lists the smallest member of each cycle.

			The single-digit numbers 1, ..., 9 and the numbers f(k) = 4*(10^k-1)/9 + 11, k >= 3, are fixed points of A329623.
Indeed, for f(k) = 4...455 we have A053392(f(k)) = 8...910 = 2*f(k), so A329623(f(k)) = 2*f(k) - f(k) = f(k).
For a(10) = 182, we have A329623(182) = 728 and A329623(728) = 182, so these are members of the 2-cycle (182, 728).
For a(11) = 273, we have A329623(273) = 637 and A329623(637) = 273, so these are members of the 2-cycle (273, 637).
Similarly for all subsequent terms except the f(k) of the form 4...455.

Cf. A329623, A053392 (concatenate sums of adjacent digits of n), A328865, A329624.

See A328279 for the smallest representative of each cycle.

apply( {A328142(n)=if(n>9,fromdigits(vector((n+8)\/7,i,n=if(i>1, 8-n,(n+4)%7+1)))+11,n)}, [1..40]) \\ As far as there are no other terms than those described in COMMENTS. - M. F. Hasler, Dec 06 2019, replacing earlier code.

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

Eric W. Weisstein

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.

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

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

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., 906343609). Smoothly undulating palindromic primes (e.g., 323232323) 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. K. Caldwell, Prime Curios! 906343609 and Prime Curios! 1007.
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, A057333.

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

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

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

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

cp's OEIS Frontend

A032758 Undulating primes (digits alternate).

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A046075 Nontrivial undulants; base 10 numbers >100 which are of the form aba, abab, ababa, ..., where a != b.

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A016073 Undulating squares.

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A242541 Undulating primes: prime numbers whose digits follow the pattern A, B, A, B, A, B, A, B, ...

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A328142 Elements of cycles for iterations of A329623: n -> |n - concat(sum of adjacent digits of n)|.

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

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

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

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