A355000 -id:A355000 - cp's OEIS frontend

A355830 Lexicographically earliest infinite sequence such that a(i) = a(j) => A046523(i) = A046523(j) and A345000(i) = A345000(j) for all i, j >= 1.

1, 2, 2, 3, 2, 4, 2, 5, 6, 4, 2, 7, 2, 4, 4, 8, 2, 9, 2, 7, 10, 4, 2, 11, 12, 10, 13, 7, 2, 14, 2, 15, 4, 4, 16, 17, 2, 4, 4, 18, 2, 14, 2, 19, 20, 10, 2, 21, 6, 22, 10, 19, 2, 23, 4, 18, 4, 4, 2, 24, 2, 4, 20, 25, 16, 14, 2, 7, 4, 14, 2, 26, 2, 4, 27, 28, 16, 14, 2, 29, 30, 4, 2, 24, 4, 10, 4, 31, 2, 32, 4, 33, 4, 34, 16, 35, 2, 36, 20, 37, 2, 38, 2, 18, 14

Offset: 1

Antti Karttunen, Jul 20 2022

nonn

Restricted growth sequence transform of the ordered pair [A046523(n), A345000(n)].

For all i, j: A351235(i) = A351235(j) => a(i) = a(j).

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences related to primorial base

Cf. A003415, A046523, A276086, A345000.

Cf. also A351235, A355000, A355831, A355832, A355836.

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; };
A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
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); };
A345000(n) = gcd(A003415(n),A003415(A276086(n)));
Aux355830(n) = [A046523(n), A345000(n)];
v355830 = rgs_transform(vector(up_to,n,Aux355830(n)));
A355830(n) = v355830[n];

A355836 Lexicographically earliest infinite sequence such that a(i) = a(j) => A046523(i) = A046523(j) and A355442(i) = A355442(j) for all i, j >= 1.

Original entry on oeis.org

1, 2, 3, 4, 3, 5, 3, 6, 7, 8, 3, 9, 3, 8, 5, 10, 3, 11, 3, 12, 5, 8, 3, 13, 14, 8, 15, 16, 3, 17, 3, 18, 19, 8, 20, 21, 3, 8, 5, 22, 3, 23, 3, 12, 24, 8, 3, 25, 14, 26, 5, 16, 3, 27, 20, 28, 5, 8, 3, 29, 3, 8, 30, 31, 20, 23, 3, 12, 5, 32, 3, 33, 3, 8, 34, 16, 19, 23, 3, 35, 36, 8, 3, 37, 20, 8, 5, 38, 3, 29, 19, 12, 19, 8, 20, 39, 3, 12, 9, 40, 3, 23, 3, 28, 41

Offset: 1

Antti Karttunen, Jul 20 2022

nonn

Restricted growth sequence transform of the ordered pair [A046523(n), A355442(n)].

For all i, j: A355835(i) = A355835(j) => a(i) = a(j).

			a(6) = a(15) = a(21) = a(39) = a(51) = a(57) = a(69) = a(87) = a(111) = etc, for an infinite number of other indices k, because for all these k, A355442(k) = 5 and their prime signatures (A101296) are equal, as they are all squarefree semiprimes, A006881.
In contrast, powers of 2 (1, 2, 4, 8, 16, ..., A000079) obtain unique values in this sequence, and in general, for all proper prime powers k (A246547) for which A355442(k) > 1 [that are terms of A355822], the value a(k) is unique in this sequence.

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

Cf. A000079, A003961, A006881, A046523, A101296, A246547, A276086, A355442, A355822, A355835.

Cf. also A355000, A355830.

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; };
A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
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); };
A355442(n) = gcd(A003961(n), A276086(n));
Aux355836(n) = [A046523(n), A355442(n)];
v355836 = rgs_transform(vector(up_to,n,Aux355836(n)));
A355836(n) = v355836[n];

cp's OEIS Frontend

A355830 Lexicographically earliest infinite sequence such that a(i) = a(j) => A046523(i) = A046523(j) and A345000(i) = A345000(j) for all i, j >= 1.

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