A030144 -id:A030144 - cp's OEIS frontend

A030141 Numbers in which parity of the decimal digits alternates.

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 16, 18, 21, 23, 25, 27, 29, 30, 32, 34, 36, 38, 41, 43, 45, 47, 49, 50, 52, 54, 56, 58, 61, 63, 65, 67, 69, 70, 72, 74, 76, 78, 81, 83, 85, 87, 89, 90, 92, 94, 96, 98, 101, 103, 105, 107, 109, 121, 123, 125, 127, 129

Offset: 1

Patrick De Geest

An alternating integer is a positive integer for which, in base-10, the parity of its digits alternates.
The number of terms < 10^n (n>=0): 1, 10, 55, 280, 1405, 7030, 35155, ..., . - Robert G. Wilson v, Apr 01 2011
The number of terms between 10^n and 10^(n+1) is 9 * 5^n for n>=0. For n>=0, number of terms < 10^n is 1 + 9 * (5^n-1)/4. - Franklin T. Adams-Watters, Apr 01 2011
A228710(a(n)) = 1. - Reinhard Zumkeller, Aug 31 2013

			121 is alternating and in the sequence because its consecutive digits are odd-even-odd, 1 being odd and 2 even. Of course, 1234567890 is also alternating.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
45th International Mathematical Olympiad (45th IMO), Problem #6 and Solution, Mathematics Magazine, 78 (2005), pp. 247, 250, 251.
Index entries for 10-automatic sequences.

Complement: A228709.
Subsequences: A030142, A030143, A030144, A030147, A030152, A062285.
Cf. A110303, A110304, A110305, A056830, A103181, A228722, A228723.

a030141 n = a030141_list !! (n-1)
a030141_list = filter ((== 1) . a228710) [0..]
-- Reinhard Zumkeller, Aug 31 2013

fQ[n_] := Block[{m = Mod[ IntegerDigits@ n, 2]}, m == Split[m, UnsameQ][[1]]]; Select[ Range[0, 130], fQ] (* Robert G. Wilson v, Apr 01 2011 *)
Select[Range[0,150],FreeQ[Differences[Boole[EvenQ[IntegerDigits[#]]]],0]&] (* Harvey P. Dale, Jul 19 2025 *)

is(n,d=digits(n))=for(i=2,#d, if((d[i]-d[i-1])%2==0, return(0))); 1 \\ Charles R Greathouse IV, Jul 08 2022

from itertools import count
def A030141_gen(startvalue=0): # generator of terms >= startvalue
    return filter(lambda n:all(int(a)+int(b)&1 for a, b in zip(str(n),str(n)[1:])),count(max(startvalue,0)))
A030141_list = list(islice(A030141_gen(),30)) # Chai Wah Wu, Jul 12 2022

from itertools import chain, count, islice
def altgen(seed, digits):
    allowed = "02468" if seed in "13579" else "13579"
    if digits == 1: yield from allowed; return
    for f in allowed: yield from (f + r for r in altgen(f, digits-1))
def agen(): yield from chain(range(10), (int(f+r) for d in count(2) for f in "123456789" for r in altgen(f, d-1)))
print(list(islice(agen(), 65))) # Michael S. Branicky, Jul 12 2022

Offset corrected by Reinhard Zumkeller, Aug 31 2013

A030152 Squares in which parity of digits alternates.

Original entry on oeis.org

0, 1, 4, 9, 16, 25, 36, 49, 81, 121, 169, 256, 361, 529, 676, 729, 961, 1296, 4761, 5476, 6561, 7056, 9216, 12321, 12769, 14161, 16129, 18769, 32761, 34969, 41616, 56169, 69696, 72361, 74529, 76729, 78961, 87616, 96721, 147456, 163216, 181476, 212521

Offset: 1

Patrick De Geest

			1296 is a term as 1, 2, 9 and 6 have odd and even parity alternately.

Giovanni Resta, Table of n, a(n) for n = 1..10000 (first 1000 terms from Reinhard Zumkeller)

Cf. A030142, A030143, A030144, A030152, A030153, A030154, A030159.

Intersection of A000290 and A030141.

a030152 n = a030152_list !! (n-1)
a030152_list = filter ((== 1) . a228710) a000290_list
-- Reinhard Zumkeller, Aug 31 2013

i := 0:for a from 1 to 1000 do b := a^2:g := ceil(log(b+1)/log(10)):iss := true:for j from 1 to g-1 do if((b mod 2)=1) then if((floor(b/10^j) mod 2)=((-1)^(j+1)+1)/2) then iss := false:end if:else if((floor(b/10^j) mod 2)=((-1)^j+1)/2) then iss := false:end if:end if:end do: if(iss=true) then i := i+1:c[i] := b:end if:end do:q := seq(c[k],k=1..i-1); # Sascha Kurz, Mar 23 2002

altQ[n_] := n < 10 || Union[Total /@ Partition[ Mod[ IntegerDigits@n, 2], 2, 1]] == {1}; Select[ Range[0, 500]^2, altQ[#] &] (* Giovanni Resta, Aug 16 2018 *)

A010052(a(n)) * A228710(a(n)) = 1. - Reinhard Zumkeller, Aug 31 2013

Edited by N. J. A. Sloane, Aug 31 2009 at the suggestion of R. J. Mathar

Offset corrected by Reinhard Zumkeller, Aug 31 2013

A068876 Smallest n-digit prime with property that digits alternate in parity.

Original entry on oeis.org

2, 23, 101, 2129, 10103, 210101, 1010129, 21010127, 101010167, 2101010147, 10101010163, 210101010187, 1010101010341, 21010101010147, 101010101010323, 2101010101010141, 10101010101010141, 210101010101010323, 1010101010101010143

Offset: 1

Amarnath Murthy, Mar 19 2002

			a(4) = 2129 as 2, 1, 2 and 9 have even and odd parity alternately.

Robert Israel, Table of n, a(n) for n = 1..700

Cf. A030144, A068877, A056830.

alp:= proc(n) local L,d;
L:= convert(n,base,10);
d:= nops(L);
if d::even then L:= L + map(op, [[0,1]$(d/2)]) else L:= L + map(op, [[0,1]$((d-1)/2),[0]]) fi;
nops(convert(L mod 2, set))=1
end proc:
f:= proc(d) local s;
  if d::even then s:= 2*10^(d-1)+(10^d-1)/99-1
  else s:= (10^(d+1)-1)/99-1
  fi;
  do s:= nextprime(s);
     if alp(s) then return s fi
  od
end proc:
seq(f(d),d=1..20); # Robert Israel, Aug 14 2018

fQ[n_] := Block[{m = Mod[ IntegerDigits@ n, 2]}, m == Split[m, UnsameQ][[1]]]; f[n_] := Block[{c = 1 + 100 (100^Ceiling[n/2 - 1] - 1)/99, k}, k = If[ OddQ@ n, c, 2*10^(n - 1) + c]; k = NextPrime[k - 1]; While[ !fQ@ k, k = NextPrime@ k]; k]; Array[f, 21] (* Robert G. Wilson v, Apr 01 2011 *)

concat = lambda x: Integer(''.join(map(str,x)),base=10)
def A068876(n):
    dd = {0:range(0,10,2), 1: range(1,10,2)}
    for d0 in [1..9]:
        if n % 2 == 0 and d0 % 2 == 1: continue # optimization
        ds = [dd[(d0+1+i) % 2] for i in range(n-1)]
        for dr in cartesian_product(ds):
            c = concat([d0]+dr)
            if is_prime(c): return c  # D. S. McNeil, Apr 02 2011

a(9)-a(13) corrected and a(14)-a(19) from Donovan Johnson, Apr 01 2011

A030147 Palindromes in which parity of digits alternates.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 101, 121, 141, 161, 181, 212, 232, 252, 272, 292, 303, 323, 343, 363, 383, 414, 434, 454, 474, 494, 505, 525, 545, 565, 585, 616, 636, 656, 676, 696, 707, 727, 747, 767, 787, 818, 838, 858, 878, 898, 909, 929, 949

Offset: 1

Patrick De Geest

All terms have an odd number of digits. - Alonso del Arte, Jan 31 2020

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index entries for sequences related to palindromes

Intersection of A002113 and A030141.
Subsequence of A001633.
Cf. A030143, A030144, A030152.

a030147 n = a030147_list !! (n-1)
a030147_list = filter ((== 1) . a228710) a002113_list
-- Reinhard Zumkeller, Aug 31 2013

palQ[n_, b_:10] := (IntegerDigits[n, b] == Reverse[IntegerDigits[n, b]]); alternParQ[n_, b_:10] := (Union[BlockMap[Xor @@ # &, OddQ[IntegerDigits[n, b]], 2, 1]] == {True}); Join[Range[0, 9], Select[Range[1000], palQ[#] && alternParQ[#] &]] (* Alonso del Arte, Feb 02 2020 *)
Join[Range[0,9],Select[Range[100000],PalindromeQ[#]&&Union[Total/@Partition[Boole[ EvenQ[ IntegerDigits[ #]]],2,1]] =={1}&]] (* Harvey P. Dale, Jul 04 2023 *)

def isPal(n: Int) = (n.toString == n.toString.reverse)
def alternsPar(n: Int): Boolean = {
  val dPars = Integer.toString(n).toList.map(_ % 2 == 0)
  val scanPars = (dPars zip dPars.tail).map{ case (x, y) => x ^ y }
  scanPars.toSet == Set(true)
}
(0 to 9) ++: (10 to 999).filter(isPal).filter(alternsPar) // Alonso del Arte, Feb 02 2020

A136522(a(n)) * A228710(a(n)) = 1. - Reinhard Zumkeller, Aug 31 2013

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

A323578 Primes with distinct digits for which parity of digits alternates.

Original entry on oeis.org

2, 3, 5, 7, 23, 29, 41, 43, 47, 61, 67, 83, 89, 103, 107, 109, 127, 149, 163, 167, 307, 347, 349, 367, 389, 503, 509, 521, 523, 541, 547, 563, 569, 587, 701, 709, 743, 761, 769, 907, 941, 947, 967, 983, 2143, 2309

Offset: 1

Bernard Schott, Jan 18 2019

There are 4426 terms (found by David A. Corneth) in this sequence, which is a subsequence of A030144.

The largest prime of this sequence is 987654103 which is also the largest prime with distinct digits in A029743.

			2143 is a term as 2, 1, 4 and 3 have even and odd parity alternately and these four digits are all distinct.

David A. Corneth, Table of n, a(n) for n = 1..4426 (Complete sequence)
Chris K. Caldwell and G. L. Honaker, Jr., 987654103, Prime Curios!

Intersection of A030144 and A029743.

Cf. A000040, A030141.

{2}~Join~Select[Prime@ Range@ 350, And[Max@ Tally[#][[All, -1]] == 1, AllTrue[#[[Range[2, Length[#], 2] ]], EvenQ], AllTrue[#[[Range[1, Length[#], 2] ]], OddQ]] &@ Reverse@ IntegerDigits@ # &] (* Michael De Vlieger, Jan 19 2019 *)

allTerms() = {my(res = List([2])); c = vector(10); odd = [1, 3, 5, 7, 9]; even = [0, 2, 4, 6, 8]; for(i = 0, 119, pi = numtoperm(5, i); vi = vector(5, k, odd[pi[k]]); for(j = 0, 119, pj = numtoperm(5, j); vj = vector(5, k, even[pj[k]]); for(m = 1, 5, c[2*m] = vi[m]; c[2*m - 1] = vj[m]; ); cv = fromdigits(c); for(m = 1, 10, if(isprime(cv % 10^m), listput(res, cv % 10^m); ) ) ) ); listsort(res, 1); res } \\ David A. Corneth, Jan 18 2019

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]))])

A030145 Primes such that in p^2 the parity of digits alternates.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 19, 23, 31, 113, 127, 137, 181, 269, 277, 281, 311, 461, 463, 661, 673, 677, 1019, 1277, 1361, 1973, 2287, 2339, 2377, 2411, 2423, 2689, 2731, 4673, 5023, 5081, 5261, 6563, 6577, 8311, 9013, 9437, 9439, 10181, 10463

Offset: 1

Patrick De Geest

Giovanni Resta, Table of n, a(n) for n = 1..10000

Cf. A030144, A030146, A030151.

id[n_]:=IntegerDigits[n]; t={}; Do[p=Prime[n]; If[Length[id[p]]==1,AppendTo[t,p], If[Union[Abs[Differences[Boole/@EvenQ[id[p^2]]]]]=={1}, AppendTo[t,p]]], {n,1300}]; t (* Jayanta Basu, May 07 2013 *)
pdaQ[n_]:=FreeQ[Differences[Boole[EvenQ[IntegerDigits[n^2]]]],0]; Select[ Prime[ Range[1300]],pdaQ] (* Harvey P. Dale, Jul 30 2019 *)

a(n)^2 = A030146(n). - Giovanni Resta, Aug 16 2018

Offset corrected by Giovanni Resta, Aug 16 2018

A030150 Palindromic primes in which parity of digits alternates.

Original entry on oeis.org

2, 3, 5, 7, 101, 181, 383, 727, 787, 929, 10301, 10501, 12721, 14341, 14741, 16361, 16561, 18181, 30103, 30703, 32323, 34543, 36563, 38183, 38783, 70507, 72727, 74747, 76367, 78787, 90709, 94349, 94949, 96769, 98389, 1074701

Offset: 1

Patrick De Geest

Giovanni Resta, Table of n, a(n) for n = 1..10000

Cf. A030144, A030147, A030148, A030149.

altQ[n_] := n < 10 || Union[Total /@ Partition[ Mod[ IntegerDigits@ n, 2], 2, 1]] == {1}; palQ[n_] := Block[{d = IntegerDigits[n]}, d == Reverse@ d]; Select[ Prime@ Range@ 10000, palQ[#] && altQ[#] &] (* Giovanni Resta, Aug 16 2018 *)
Join[{2,3,5,7},Select[Prime[Range[85000]],PalindromeQ[#]&&Union[Total/@Partition[ Boole[ EvenQ/@IntegerDigits[#]],2,1]]=={1}&]] (* Harvey P. Dale, Jul 10 2023 *)

A068877 Largest n-digit prime with property that digits alternate in parity.

Original entry on oeis.org

7, 89, 983, 8969, 98981, 898987, 9898921, 89898983, 989898989, 8989898969, 98989898981, 898989898987, 9898989898901, 89898989898967, 989898989898943, 8989898989898969, 98989898989898981, 898989898989898943, 9898989898989898789

Offset: 1

Amarnath Murthy, Mar 19 2002

			a(4) = 8969 as 8, 9, 6 and 9 have even and odd parity alternately.

Giovanni Resta, Table of n, a(n) for n = 1..500

Cf. A030144, A068876.

concat = lambda x: Integer(''.join(map(str,x)),base=10)
def A068877(n):
    dd = {0:range(0,10,2)[::-1], 1: range(1,10,2)[::-1]}
    for d0 in [1..9][::-1]:
        if n % 2 == 0 and d0 % 2 == 1: continue # optimization
        ds = [dd[(d0+1+i) % 2] for i in range(n-1)]
        for dr in cartesian_product(ds):
            c = concat([d0]+dr)
            if is_prime(c): return c  # [D. S. McNeil, Apr 02 2011]

a(15)-a(19) from Donovan Johnson, Apr 01 2011

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A030141 Numbers in which parity of the decimal digits alternates.

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A030152 Squares in which parity of digits alternates.

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A068876 Smallest n-digit prime with property that digits alternate in parity.

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A030147 Palindromes in which parity of digits alternates.

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A343590 Undulating alternating primes.

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A323578 Primes with distinct digits for which parity of digits alternates.

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A343675 Undulating alternating palindromic primes.

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