A030151 -id:A030151 - cp's OEIS frontend

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

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

A030159 Numbers n such that in n^3 the parity of digits alternates.

Original entry on oeis.org

0, 1, 2, 3, 5, 6, 7, 9, 18, 23, 85, 87, 101, 103, 168, 206, 301, 303, 363, 725, 1683, 2461, 2788, 7921, 9563, 9668, 20606, 28443, 29501, 45168, 46701, 49501, 63556, 78206, 80901, 90009, 167861, 168069, 208288, 278636, 331841, 375121, 440468

Offset: 1

Patrick De Geest

A simple heuristic argument suggests that this sequence (albeit rather sparse) is infinite. The numbers of terms of k digits, for k=1..14, are 8, 4, 8, 6, 10, 14, 20, 18, 33, 23, 42, 37, 46, 77, respectively. The 5 numbers obtained multiplying the first h=1..5 terms of (1+10^2, 1+10^8, 1+10^32, 1+10^128, 1+10^512), are all member of the sequence. The largest one is a number of 683 digits whose alternating cube has 2047 digits. - Giovanni Resta, Aug 16 2018

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

Cf. A030160, A030161, A030162, A030151.

n3pdaQ[n_]:=Module[{pty=Boole[EvenQ/@IntegerDigits[n^3]],len= IntegerLength[ n^3]}, pty== PadRight[{},len,{1,0}]||pty==PadRight[ {}, len, {0,1}]]; Join[{0},Select[Range[450000],n3pdaQ]] (* Harvey P. Dale, Mar 26 2018 *)

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

A030155 Odd n such that in n^2 the parity of digits alternates.

Original entry on oeis.org

1, 3, 5, 7, 9, 11, 13, 19, 23, 27, 31, 69, 81, 111, 113, 119, 127, 137, 181, 187, 237, 269, 273, 277, 281, 311, 461, 463, 539, 661, 673, 677, 689, 819, 831, 1019, 1027, 1111, 1113, 1119, 1127, 1137, 1269, 1277, 1287, 1361, 1369, 1377, 1863, 1969

Offset: 1

Patrick De Geest

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

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

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

Offset changed by Giovanni Resta, Aug 16 2018

cp's OEIS Frontend

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

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Extensions

A030156 Odd squares in which parity of digits alternates.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Extensions

A030159 Numbers n such that in n^3 the parity of digits alternates.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

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

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A030155 Odd n such that in n^2 the parity of digits alternates.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Extensions