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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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,4

COMMENTS

In binary representation of a(n), the distances between successive 1's (one more than the lengths of intermediate 0-runs) 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 1-bit is at rightmost position (bit-0), 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 1-bit comes 3-1 = 2 positions left of the rightmost 1, thus a(4) = 2^0 + 2^(3-1) = 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^(1-1) + 2^(1-1+1) + 2^(1-1+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

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Last modified December 6 09:37 EST 2019. Contains 329800 sequences. (Running on oeis4.)