

A297174


An auxiliary sequence for computing A300250. See comments and examples.


4



0, 1, 1, 5, 1, 19, 1, 69, 5, 19, 1, 2123, 1, 19, 19, 4165, 1, 2131, 1, 2125, 19, 19, 1, 4228171, 5, 19, 69, 2125, 1, 526631, 1, 2101317, 19, 19, 19, 268706123, 1, 19, 19, 4228237, 1, 526643, 1, 2125, 2123, 19, 1, 550026380363, 5, 2131, 19, 2125, 1, 4229203, 19, 4228237, 19, 19, 1, 8798249190555, 1, 19, 2123, 17181970501, 19, 526643, 1, 2125
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OFFSET

1,4


COMMENTS

In binary representation of a(n), the distances between successive 1's (one more than the lengths of intermediate 0runs) from the right record the prime signature ranks (A101296) of successive divisors of n, as ordered from the smallest divisor (> 1) to the largest divisor (= n).


LINKS

Antti Karttunen, Table of n, a(n) for n = 1..4096
Index entries for sequences related to binary expansion of n


EXAMPLE

a(1) = 0 by convention (as 1 has no prime divisors).
a(p) = 1 for any prime p.
For any n > 1, the least significant 1bit is at rightmost position (bit0), signifying the smallest prime factor of n, which is always the least divisor > 1.
For n = 4 = 2*2, the next divisor of 4 after 2 is 4, for which A101296(4) = 3, thus the second least significant 1bit comes 31 = 2 positions left of the rightmost 1, thus a(4) = 2^0 + 2^(31) = 1+4 = 5.
For n = 6 with divisors d = 2, 3 and 6 larger than one, for which A101296(d)1 gives 1, 1 and 3, thus a(6) = 2^(11) + 2^(11+1) + 2^(11+1+3) = 2^0 + 2^1 + 2^4 = 19.
For n = 12 with divisors d = 2, 3, 2*2, 2*3, 2*2*3 larger than one, A101296(d)1 gives 1, 1, 2, 3 and 5 thus a(12) = 2^0 + 2^(0+1) + 2^(0+1+2) + 2^(0+1+2+3) + 2^(0+1+2+3+5) = 2123.
For n = 18 with divisors d = 2, 3, 2*3, 3*3, 2*3*3 larger than one, A101296(d)1 gives 1, 1, 3, 2, and 5 thus a(18) = 2^0 + 2^(0+1) + 2^(0+1+3) + 2^(0+1+3+2) + 2^(0+1+3+2+5) = 2131.


PROG

(PARI)
up_to = 4096;
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; };
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); }; \\ From A046523.
v101296 = rgs_transform(vector(up_to, n, A046523(n)));
A101296(n) = v101296[n];
A297174(n) = { my(s=0, i=1); fordiv(n, d, if(d>1, i += (A101296(d)1); s += 2^i)); (s); };


CROSSREFS

Cf. A101296, A300250 (restricted growth sequence transform of this sequence).
Cf. also A292258, A294897.
Sequence in context: A326121 A008971 A151335 * A226605 A055584 A193861
Adjacent sequences: A297171 A297172 A297173 * A297175 A297176 A297177


KEYWORD

nonn


AUTHOR

Antti Karttunen, Mar 07 2018


STATUS

approved



