A373989 -id:A373989 - cp's OEIS frontend

A374033 Lexicographically earliest infinite sequence such that a(i) = a(j) => A278226(A373985(i)) = A278226(A373985(j)), for all i, j >= 1.

1, 1, 1, 2, 1, 2, 1, 1, 2, 2, 1, 1, 1, 2, 3, 3, 1, 1, 1, 1, 3, 2, 1, 2, 2, 2, 4, 1, 1, 1, 1, 1, 2, 2, 2, 5, 1, 2, 2, 2, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 3, 1, 1, 2, 6, 7, 3, 2, 1, 8, 1, 2, 1, 2, 6, 1, 1, 1, 2, 1, 1, 4, 1, 2, 1, 1, 6, 1, 1, 1, 7, 2, 1, 9, 2, 2, 3, 2, 1, 2, 6, 1, 3, 2, 2, 5, 1, 1, 10, 5, 1, 1, 1, 2, 1

Offset: 1

Antti Karttunen, Jun 27 2024

nonn

Restricted growth sequence transform of A278226(A373985(n)).
For all i, j >= 1:
  A305800(i) = A305800(j) => a(i) = a(j) => A373989(i) = A373989(j).

Antti Karttunen, Table of n, a(n) for n = 1..100000
Index entries for sequences related to primorial numbers

Cf. A002110, A046523, A108951, A276086, A278226, A373158, A373985.
Cf. A305800, A373989.
Cf. also A373982, A373983, A374032.

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; };
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };
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));
A373985(n) = { my(f=factor(n),m=1,s=0); for(i=1, #f~, my(x=prod(i=1,primepi(f[i, 1]),prime(i))); s += f[i, 2]*x; m *= x^f[i, 2]); gcd(m,s); };
v374033 = rgs_transform(vector(up_to, n, A278226(A373985(n))));
A374033(n) = v374033[n];

A374034 a(n) = A276150(gcd(A276085(n), A328768(n))), where A276150 is the digit sum in primorial base, A276085 is the primorial base log-function, and A328768 is the first primorial based variant of arithmetic derivative.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 2, 1, 1, 2, 2, 1, 1, 1, 2, 2, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 3, 2, 1, 2, 1, 1, 1, 2, 3, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 2, 2, 1, 1, 2, 2, 4, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 2, 4, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 2, 1, 1, 6, 2, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3

Offset: 1

Antti Karttunen, Jun 27 2024

nonn

Antti Karttunen, Table of n, a(n) for n = 1..100000
Index entries for sequences related to primorial numbers

Cf. A002110, A276085, A276150, A328768, A374031.

Cf. also A328771, A373989.

A002110(n) = prod(i=1,n,prime(i));
A276085(n) = { my(f = factor(n)); sum(k=1, #f~, f[k, 2]*A002110(primepi(f[k, 1])-1)); };
A276150(n) = { my(s=0, p=2, d); while(n, d = (n%p); s += d; n = (n-d)/p; p = nextprime(1+p)); (s); };
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]));
A374034(n) = A276150(gcd(A276085(n), A328768(n)));

a(n) = A276150(A374031(n)).

Apparently, a(n) <= A328771(n) for all n >= 1.

cp's OEIS Frontend

A374033 Lexicographically earliest infinite sequence such that a(i) = a(j) => A278226(A373985(i)) = A278226(A373985(j)), for all i, j >= 1.

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