A320009 -id:A320009 - cp's OEIS frontend

A323073 Number of iterations of A049820(x) = x - A000005(x) needed to reach either zero or such x that x and A049820(x) are coprime, when starting from x = n.

0, 0, 1, 0, 0, 0, 2, 0, 1, 3, 3, 0, 3, 0, 4, 0, 0, 0, 4, 0, 5, 0, 5, 0, 1, 0, 6, 0, 6, 0, 6, 0, 7, 0, 7, 0, 1, 0, 8, 0, 8, 0, 8, 0, 9, 1, 9, 0, 9, 0, 10, 0, 10, 0, 10, 0, 10, 0, 11, 0, 10, 0, 12, 1, 0, 0, 12, 0, 13, 0, 13, 0, 11, 0, 14, 1, 14, 0, 14, 0, 14, 0, 15, 0, 12, 0, 16, 0, 15, 0, 15, 0, 17, 0, 16, 0, 13, 0

Offset: 0

Antti Karttunen, Jan 05 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 0..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..110880

Cf. A000005, A009191, A049820, A155043, A259934, A286540, A320009, A322974, A322987, A322996.

Cf. A046642 (positions of zeros after the initial a(0)=0).

A323073(n) = if(!n,0,my(nn=(n-numdiv(n))); if(1==gcd(n,nn),0,1+A323073(nn)));

A323073(n) = if(!n,0,for(j=0,oo,my(nn=(n-numdiv(n))); if((0==nn)||(1==gcd(n,nn)),return(j+(2==n)),n = nn)));

a(0) = 0; for n > 0, if A009191(n) == 1, a(n) = 0, otherwise a(n) = 1 + a(n-A000005(n)).

a(n) <= A155043(n).

A320002 a(0) = 1; for n > 0, a(n) = A002828(n) * a(n-A002828(n)), where A002828(n) is the least number of squares that add up to n.

Original entry on oeis.org

1, 1, 2, 3, 3, 6, 9, 12, 18, 18, 36, 54, 54, 108, 162, 216, 216, 432, 432, 648, 864, 1296, 1944, 2592, 3888, 3888, 7776, 11664, 15552, 23328, 34992, 46656, 69984, 104976, 139968, 209952, 209952, 419904, 629856, 839808, 1259712, 1679616, 2519424, 3779136, 5038848, 7558272, 11337408, 15116544, 22674816, 22674816, 45349632

Offset: 0

Antti Karttunen, Nov 24 2018

nonn

Product of A002828(x) computed over all x encountered when map x -> x - A002828(x) is iterated, starting from x = n, until 0 is reached.
Sequence is monotonic because A255131 is monotonic.
All terms are 3-smooth (A003586).

Antti Karttunen, Table of n, a(n) for n = 0..121

Cf. A002828, A003586, A255131, A276573.

Cf. also A320008, A320009.

Nest[Append[#1, #2 #1[[-#2]] ] & @@ {#, If[First@ # > 0, 1, Length@ First@ Split@ # + 1] &@ SquaresR[Range@ 4, Length@ #]} &, {1}, 50] (* Michael De Vlieger, Nov 25 2018, after Harvey P. Dale at A002828 *)

istwo(n:int) = { my(f); if(n<3, return(n>=0); ); f=factor(n>>valuation(n, 2)); for(i=1, #f[, 1], if(bitand(f[i, 2], 1)==1&&bitand(f[i, 1], 3)==3, return(0))); 1 };
isthree(n:int) = { my(tmp=valuation(n, 2)); bitand(tmp, 1)||bitand(n>>tmp, 7)!=7 };
A002828(n) = if(issquare(n), !!n, if(istwo(n), 2, 4-isthree(n))); \\ From A002828
A255131(n) = (n-A002828(n));
A320002(n) = { my(m=1, v); while(n>0, v = A002828(n); m *= v; n -= v); (m); };

A320002(n) = if(0==n,1,A002828(n)*A320002(n-A002828(n)));

a(0) = 1; for n > 0, a(n) = A002828(n) * a(n-A002828(n)).

A320008 a(0) = 1; for n > 0, a(n) = A000120(n) * a(n-A000120(n)), where A000120(n) is the binary weight of n.

Original entry on oeis.org

1, 1, 1, 2, 2, 4, 4, 6, 6, 12, 12, 18, 24, 36, 54, 72, 72, 144, 144, 216, 288, 432, 648, 864, 1296, 1944, 2592, 3456, 5832, 7776, 10368, 12960, 12960, 25920, 25920, 38880, 51840, 77760, 116640, 155520, 233280, 349920, 466560, 622080, 1049760, 1399680, 1866240, 2332800, 3732480, 5598720, 6998400, 9331200, 16796160, 22394880, 27993600, 34992000

Offset: 0

Antti Karttunen, Nov 24 2018

nonn

Sequence is monotonic because A011371 is.

Antti Karttunen, Table of n, a(n) for n = 0..133
Index entries for sequences related to binary expansion of n

Cf. A000120, A011371, A179016.

Cf. also A320002, A320009.

Nest[Append[#1, #2 #1[[-#2]] ] & @@ {#, Total@ IntegerDigits[Length@ #, 2]} &, {1}, 55] (* Michael De Vlieger, Nov 25 2018 *)

A320008(n) = if(0==n,1,hammingweight(n)*A320008(n-hammingweight(n)));

a(0) = 1; for n > 0, a(n) = A000120(n) * a(n-A000120(n)).

A320016 a(1) = a(2) = 1; for n > 2, a(n) = A000005(n) * a(A000005(n)), where A000005(n) gives the number of divisors of n.

Original entry on oeis.org

1, 1, 2, 6, 2, 24, 2, 24, 6, 24, 2, 144, 2, 24, 24, 10, 2, 144, 2, 144, 24, 24, 2, 192, 6, 24, 24, 144, 2, 192, 2, 144, 24, 24, 24, 54, 2, 24, 24, 192, 2, 192, 2, 144, 144, 24, 2, 240, 6, 144, 24, 144, 2, 192, 24, 192, 24, 24, 2, 1728, 2, 24, 144, 14, 24, 192, 2, 144, 24, 192, 2, 1728, 2, 24, 144, 144, 24, 192, 2, 240, 10, 24, 2, 1728

Offset: 1

Antti Karttunen, Nov 24 2018

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10080

Cf. A000005, A036459, A060937.

Cf. also A320009, A320118.

a:=[1,1];; for n in [3..100] do a[n]:=Tau(n)*a[Tau(n)]; od; a; # Muniru A Asiru, Nov 24 2018

Nest[Append[#1, #2 #1[[#2]] ] & @@ {#, DivisorSigma[0, Length@ # + 1]} &, {1, 1}, 82] (* Michael De Vlieger, Nov 25 2018 *)

A320016(n) = if(n<=2,1,numdiv(n)*A320016(numdiv(n)));

a(1) = a(2) = 1; for n > 2, a(n) = A000005(n) * a(A000005(n)), where A000005(n) gives the number of divisors of n.

cp's OEIS Frontend

A323073 Number of iterations of A049820(x) = x - A000005(x) needed to reach either zero or such x that x and A049820(x) are coprime, when starting from x = n.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

PARI

Formula

A320002 a(0) = 1; for n > 0, a(n) = A002828(n) * a(n-A002828(n)), where A002828(n) is the least number of squares that add up to n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

A320008 a(0) = 1; for n > 0, a(n) = A000120(n) * a(n-A000120(n)), where A000120(n) is the binary weight of n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A320016 a(1) = a(2) = 1; for n > 2, a(n) = A000005(n) * a(A000005(n)), where A000005(n) gives the number of divisors of n.

Views

Author

Keywords

Links

Crossrefs

Programs

GAP

Mathematica

PARI

Formula