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.

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).

Links

Antti Karttunen, Table of n, a(n) for n = 1..100000

Index entries for sequences related to primorial numbers

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); };
A278226(n) = A046523(A276086(n));
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];

Crossrefs

Cf. A002110, A278226, A328768, A374212, A374213.

Cf. A305900, A152822, A328771, A373982, A373991.

Cf. also A374131.

Sequence in context: A099033 A187786 A002330 * A335130 A373595 A358747

Adjacent sequences: A374207 A374208 A374209 * A374212 A374213 A374214

Keyword

nonn

Author

Antti Karttunen, Jun 30 2024

Status

approved