login
A374211
Lexicographically earliest infinite sequence such that for all i, j >= 1, a(i) = a(j) => f(i) = f(j), with f(1) = 1, and for n > 1, f(n) = [A278226(A328768(n)), A374212(n), A374213(n)], where A328768 is the first primorial based variant of the arithmetic derivative, and A374212 and A374213 are its 2- and 3-adic valuations.
2
1, 2, 3, 4, 5, 6, 5, 7, 7, 8, 5, 9, 5, 10, 11, 12, 5, 13, 5, 14, 15, 16, 5, 17, 7, 8, 18, 19, 5, 16, 5, 20, 21, 22, 23, 24, 5, 25, 26, 27, 5, 28, 5, 29, 30, 31, 5, 32, 7, 33, 17, 34, 5, 35, 36, 37, 38, 39, 5, 40, 5, 10, 41, 23, 42, 43, 5, 29, 44, 45, 5, 46, 5, 47, 48, 49, 50, 51, 5, 52, 53, 54, 5, 44, 55, 16, 34, 56, 5, 57, 58, 26, 15, 59, 60, 20, 5, 61, 62, 29
OFFSET
1,2
COMMENTS
Restricted growth sequence transform of the function f given in the definition.
For all i, j >= 1:
A305900(i) = A305900(j) => a(i) = a(j),
a(i) = a(j) => A152822(i) = A152822(j),
a(i) = a(j) => A373982(i) = A373982(j) => A328771(i) = A328771(j),
a(i) = a(j) => A373991(i) = A373991(j).
PROG
(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; };
A002110(n) = prod(i=1, n, prime(i));
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); }; \\ From A046523
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A328768(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]*A002110(primepi(f[i, 1])-1)/f[i, 1]));
Aux374211(n) = if(1==n, n, my(u=A328768(n)); [A278226(u), valuation(u, 2), valuation(u, 3)]);
v374211 = rgs_transform(vector(up_to, n, Aux374211(n)));
A374211(n) = v374211[n];
KEYWORD
nonn
AUTHOR
Antti Karttunen, Jun 30 2024
STATUS
approved