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A300250
Restricted growth sequence transform of A297174: a filter sequence recording the prime signatures of divisors of n, with divisors ordered by their magnitude.
8
1, 2, 2, 3, 2, 4, 2, 5, 3, 4, 2, 6, 2, 4, 4, 7, 2, 8, 2, 9, 4, 4, 2, 10, 3, 4, 5, 9, 2, 11, 2, 12, 4, 4, 4, 13, 2, 4, 4, 14, 2, 15, 2, 9, 6, 4, 2, 16, 3, 8, 4, 9, 2, 17, 4, 14, 4, 4, 2, 18, 2, 4, 6, 19, 4, 15, 2, 9, 4, 11, 2, 20, 2, 4, 8, 9, 4, 15, 2, 21, 7, 4, 2, 22, 4, 4, 4, 23, 2, 24, 4, 9, 4, 4, 4, 25, 2, 8, 9, 26, 2, 15, 2, 23, 11
OFFSET
1,2
COMMENTS
This sequence gives a coarser partitioning of natural numbers than A290110, and finer than A101296:
For all i, j:
A290110(i) = A290110(j) => a(i) = a(j) => A101296(i) = A101296(j).
LINKS
EXAMPLE
Divisors of 462 are 1, 2, 3, 6, 7, 11, 14, 21, 22, 33, 42, 66, 77, 154, 231, 462.
Divisors of 858 are 1, 2, 3, 6, 11, 13, 22, 26, 33, 39, 66, 78, 143, 286, 429, 858.
If one takes the smallest prime-signature representative (A046523) of each these, one gets in both cases [1, 2, 2, 6, 2, 2, 6, 6, 6, 6, 30, 30, 6, 30, 30, 210]. E.g. 462 = 2*3*7*11 and 858 = 2*3*11*13, which both have the same prime signature as 210 = 2*3*5*7. And similarly for all the other divisors, from which follows that a(462) = a(858).
On the other hand, for 12 = 2*2*3 the divisors are 1, 2, 3, 2*2, 2*3, 2*2*3, and for 18 = 2*3*3 the divisors are 1, 2, 3, 2*3, 3*3, 2*3*3, and because the prime signatures differ both in the fourth and in the fifth places, a(18) != a(12).
PROG
(PARI)
up_to = 65537;
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; };
write_to_bfile(start_offset, vec, bfilename) = { for(n=1, length(vec), write(bfilename, (n+start_offset)-1, " ", vec[n])); }
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); };
write_to_bfile(1, rgs_transform(vector(up_to, n, A297174(n))), "b300250.txt");
CROSSREFS
Differs from similar A290110 for the first time at n=858.
Sequence in context: A077462 A324203 A290110 * A319357 A237433 A236515
KEYWORD
nonn
AUTHOR
Antti Karttunen, Mar 07 2018
STATUS
approved