A030152 -id:A030152 - cp's OEIS frontend

A068879 Duplicate of A030152.

1, 4, 9, 16, 25, 36, 49, 81, 121, 169, 256, 361, 529, 676, 729, 961, 1296, 4761, 5476

Offset: 1

dead

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

A030144 Primes in which parity of digits alternates.

Original entry on oeis.org

2, 3, 5, 7, 23, 29, 41, 43, 47, 61, 67, 83, 89, 101, 103, 107, 109, 127, 149, 163, 167, 181, 307, 347, 349, 367, 383, 389, 503, 509, 521, 523, 541, 547, 563, 569, 587, 701, 709, 727, 743, 761, 769, 787, 907, 929, 941, 947, 967, 983, 2129, 2141, 2143, 2161, 2309

Offset: 1

Patrick De Geest

			2129 is a term as 2, 1, 2 and 9 have even and odd parity alternately.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Intersection of A000040 and A030141.

Cf. A030143, A030152.

a030144 n = a030144_list !! (n-1)
a030144_list = filter ((== 1) . a228710) a000040_list
-- Reinhard Zumkeller, Aug 31 2013

Join[{2,3,5,7},Select[Prime[Range[400]],Union[Abs[Differences[Boole/@ EvenQ[ IntegerDigits[#]]]]] == {1}&]] (* Harvey P. Dale, Jul 26 2011 *)

A010051(a(n)) * A228710(a(n)) = 1. - Reinhard Zumkeller, Aug 31 2013

Offset corrected by Reinhard Zumkeller, Aug 31 2013

A068882 Triangular numbers with property that digits alternate in parity.

Original entry on oeis.org

1, 3, 6, 10, 21, 36, 45, 78, 105, 210, 276, 325, 496, 561, 630, 703, 741, 903, 1830, 2145, 2701, 5050, 6105, 6903, 8385, 9870, 10585, 12561, 14365, 18145, 18721, 23436, 25878, 29890, 30381, 32385, 36585, 38503, 38781, 41616, 47278, 50721

Offset: 1

Amarnath Murthy, Mar 19 2002

			1830 is a term as 1, 8, 3 and 0 have odd and even parity alternately.

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

Cf. A068876, A030152.

altQ[n_] := n < 10 || Union[ Total /@ Partition[ Mod[ IntegerDigits@n, 2], 2, 1]] == {1}; t[n_] := n (n + 1)/2; Select[ t@ Range@ 400, altQ] (* Giovanni Resta, Aug 17 2018 *)

More terms from Sascha Kurz, Mar 23 2002

A030151 Numbers k such that in k^2 the parity of digits alternates.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 9, 11, 13, 16, 19, 23, 26, 27, 31, 36, 69, 74, 81, 84, 96, 111, 113, 119, 127, 137, 181, 187, 204, 237, 264, 269, 273, 277, 281, 296, 311, 384, 404, 426, 461, 463, 539, 574, 584, 606, 661, 673, 677, 689, 726, 736, 764, 819

Offset: 1

Patrick De Geest

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

Cf. A030152, A030153, A030154, A030159.

upto=2000;tfs=Flatten[Table[{True,False},{Ceiling[IntegerLength[ upto]]}]]; altparQ[n_]:= Module[{n2=n^2},MemberQ[Partition[ tfs,IntegerLength[n2],1], EvenQ/@IntegerDigits[n2]]]; Join[{0},Select[Range[upto],altparQ]] (* Harvey P. Dale, May 12 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

A030160 Cubes in which parity of digits alternates.

Original entry on oeis.org

0, 1, 8, 27, 125, 216, 343, 729, 5832, 12167, 614125, 658503, 1030301, 1092727, 4741632, 8741816, 27270901, 27818127, 47832147, 381078125, 4767078987, 14905098181, 21670967872, 496981290961, 874545616547, 903670125632

Offset: 1

Patrick De Geest

The number ((1+10^2)*(1+10^8)*(1+10^32)*(1+10^128)*(1+10^512))^3 is a term of the sequence. - Giovanni Resta, Aug 16 2018

Giovanni Resta, Table of n, a(n) for n = 1..350

Cf. A030159, A030161, A030162, A030152.

altQ[n_] := n < 10 || Union[Total /@ Partition[ Mod[ IntegerDigits@n, 2], 2, 1]] == {1}; Select[ Range[0, 10^4]^3, altQ[#] &] (* Giovanni Resta, Aug 16 2018 *)

Offset changed by Giovanni Resta, Aug 16 2018

A030153 Numbers k such that in k and k^2 the parity of digits alternates.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 9, 16, 23, 27, 36, 69, 74, 81, 96, 127, 181, 187, 296, 874, 2327, 2369, 2723, 2727, 2763, 3816, 4589, 5874, 6563, 6589, 6727, 8323, 10181, 12723, 18163, 18587, 21236, 21274, 29236, 29274, 30127, 43296, 52361, 78163, 87616

Offset: 1

Patrick De Geest

Giovanni Resta, Table of n, a(n) for n = 1..1226 (derived from A030154 b-file)

Cf. A030151, A030152, A030154, A030156, A030157, A030158.

altQ[n_] := n < 10 || Union[Total /@ Partition[ Mod[ IntegerDigits@n, 2], 2, 1]] == {1}; Select[ Range[0, 10^5], altQ[#] && altQ[#^2] &] (* Giovanni Resta, Aug 16 2018 *)

A030154 Squares such that in n and sqrt(n) the parity of digits alternates.

Original entry on oeis.org

0, 1, 4, 9, 16, 25, 36, 49, 81, 256, 529, 729, 1296, 4761, 5476, 6561, 9216, 16129, 32761, 34969, 87616, 763876, 5414929, 5612161, 7414729, 7436529, 7634169, 14561856, 21058921, 34503876, 43072969, 43414921, 45252529, 69272329

Offset: 1

Patrick De Geest

The more digits there are in n, the lower the likelihood that the parity of n's digits will strictly alternate. Thus, the terms of the sequence become increasingly rare as n gets larger. - Harvey P. Dale, Aug 05 2018

For n > 3 the last digit of a(n) isn't 0 or 4. - David A. Corneth, Aug 05 2018

Andrew Howroyd, Table of n, a(n) for n = 1..1226 (first 101 terms from Harvey P. Dale, terms 102..223 from David A. Corneth)

Cf. A030141, A030151, A030152, A030153.

pdaQ[n_]:=Module[{a=Mod[IntegerDigits[n],2],b=Mod[IntegerDigits[ Sqrt[ n]],2]},Length[ Split[a]] ==IntegerLength[n]&&Length[Split[b]]== IntegerLength[ Sqrt[n]]]; Join[{0},Select[Range[8500]^2,pdaQ]] (* Harvey P. Dale, Aug 05 2018 *)

alternating(n)={my(v=digits(n)%2);0==#select(i->v[i]==v[i-1],[2..#v])}
{ for(n=0, 10^5, if(alternating(n^2) && alternating(n), print1(n^2, ", "))) } \\ Andrew Howroyd, Aug 05 2018

\\ for larger n: requires alternating function above
upto(n)={local(R=List([0])); my(recurse(s,b)=if(b0&&alternating(k^2\b), listput(R, k)); self()(k, 10*b)))))); recurse(0,1); listsort(R); Vec(R)}
apply(n->n^2, upto(sqrtint(10^12))) \\ Andrew Howroyd, Aug 05 2018

Offset changed by David A. Corneth, Aug 05 2018

A030156 Odd squares in which parity of digits alternates.

Original entry on oeis.org

1, 9, 25, 49, 81, 121, 169, 361, 529, 729, 961, 4761, 6561, 12321, 12769, 14161, 16129, 18769, 32761, 34969, 56169, 72361, 74529, 76729, 78961, 96721, 212521, 214369, 290521, 436921, 452929, 458329, 474721, 670761, 690561, 1038361

Offset: 1

Patrick De Geest

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

Cf. A030151, A030152, A030155, A030157, A030158.

altQ[n_] := n < 10 || Union[Total /@ Partition[ Mod[ IntegerDigits@n, 2], 2, 1]] == {1}; Select[ Range[1, 2000, 2]^2, altQ[#] &] (* Giovanni Resta, Aug 16 2018 *)

Offset changed by Giovanni Resta, Aug 16 2018

cp's OEIS Frontend

A068879 Duplicate of A030152.

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A030141 Numbers in which parity of the decimal digits alternates.

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A030144 Primes in which parity of digits alternates.

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A068882 Triangular numbers with property that digits alternate in parity.

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A030151 Numbers k such that in k^2 the parity of digits alternates.

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A030147 Palindromes in which parity of digits alternates.

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A030160 Cubes in which parity of digits alternates.

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A030153 Numbers k such that in k and k^2 the parity of digits alternates.

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