A077462 Prime factor configuration patterns.

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

Offset: 0

Author

Michael Somos, Nov 07 2002

Keywords

• nonn
• easy

Comments

Call two numbers equivalent if they have the same prime factorization exponents (in the same order). This sequence enumerates the equivalence classes.

A055932(a(n)) = A071364(n). - David Wasserman, Dec 21 2004

From Antti Karttunen, Jun 13 2018: (Start)

After a(0) = 0, this is the restricted growth sequence transform of A071364. The latter sequence is an "ordered variant" of A046523, and because A101296 is the rgs-transform of A046523, it follows that for all i, j: a(i) = a(j) => A101296(i) = A101296(j).

(End)

Examples

			12 = 2^2*3^1 has exponents {2,1}, and is the first number with that pattern, so its value is one more than the largest previous value; a(12) = 6. Contrast that with 18 = 2^1*3^2 having exponents {1,2}, which is different from {2,1}, so a(18) is not equal to a(12). - _Franklin T. Adams-Watters_, Aug 01 2012
		

Links

• Antti Karttunen, Table of n, a(n) for n = 0..100000 (terms 0..10000 from T. D. Noe)
• S. Whealton, Prime Factor Configuration Pattern Numbers.

Crossrefs

Cf. A037916, A071364, A101296, A290110, A300250.

One more than A079616.

Programs

Mathematica

fList = {{0}}; Join[{0, 1}, Table[e = Transpose[FactorInteger[n]][[2]]; pos = Position[fList, e]; If[pos == {}, AppendTo[fList, e]; Length[fList], pos[[1, 1]]], {n, 2, 100}]] (* T. D. Noe, Aug 01 2012 *)

PARI

a(n)=local(vn); if(n<1,return(0)); vn=factor(n)[,2]; for(i=1,n,if(vn==factor(i)[,2],return(#Set(vector(i,j,factor(j)[,2])))))

PARI

up_to = 100000;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
A071364(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = prime(i)); factorback(f); }; \\ From A071364
v077462 = rgs_transform(vector(up_to,n,A071364(n)));
A077462(n) = if(!n,n,v077462[n]); \\ Antti Karttunen, Jun 13 2018