A241909 -id:A241909 - cp's OEIS frontend

A331597 a(n) = A007947(A331595(n)).

1, 2, 2, 3, 2, 3, 2, 5, 3, 3, 2, 5, 2, 3, 6, 7, 2, 15, 2, 5, 6, 3, 2, 7, 3, 3, 5, 5, 2, 15, 2, 11, 6, 3, 6, 7, 2, 3, 6, 7, 2, 15, 2, 5, 10, 3, 2, 11, 3, 15, 6, 5, 2, 7, 6, 7, 6, 3, 2, 7, 2, 3, 10, 13, 6, 15, 2, 5, 6, 15, 2, 11, 2, 3, 15, 5, 6, 15, 2, 11, 7, 3, 2, 7, 6, 3, 6, 7, 2, 7, 6, 5, 6, 3, 6, 13, 2, 15, 10, 7, 2, 15, 2, 7, 30

Offset: 1

Antti Karttunen, Jan 22 2020

nonn

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences computed from indices in prime factorization

Cf. A007947, A064989, A122111, A241909, A331595, A331596.

Array[Times @@ FactorInteger[#][[All, 1]] &@ If[# == 1, 1, GCD @@ {Block[{k = #, m = 0}, Times @@ Power @@@ Table[k -= m; k = DeleteCases[k, 0]; {Prime@ Length@ k, m = Min@ k}, Length@ Union@ k]] &@ Catenate[ConstantArray[PrimePi[#1], #2] & @@@ #], Function[t, Times @@ Prime@ Accumulate[If[Length@ t < 2, {0}, Join[{1}, ConstantArray[0, Length@ t - 2], {-1}]] + ReplacePart[t, Map[#1 -> #2 & @@ # &, #]]]]@ ConstantArray[0, Transpose[#][[1, -1]]] &[# /. {p_, e_} /; p > 0 :> {PrimePi@ p, e}]} &@ FactorInteger[#]] &, 105] (* Michael De Vlieger, Jan 24 2020, after JungHwan Min at A122111 *)

A331597(n) = factorback(factorint(gcd(A122111(n), A241909(n)))[, 1]);

a(n) = A007947(A331595(n)) = A007947(gcd(A122111(n), A241909(n))).

A331280 Lexicographically earliest infinite sequence such that a(i) = a(j) => A278220(i) = A278220(j) for all i, j.

Original entry on oeis.org

1, 2, 3, 2, 4, 3, 5, 2, 6, 4, 7, 3, 8, 5, 9, 2, 10, 6, 11, 4, 12, 7, 13, 3, 9, 8, 6, 5, 14, 9, 15, 2, 16, 10, 17, 6, 18, 11, 19, 4, 20, 12, 21, 7, 9, 13, 22, 3, 12, 9, 23, 8, 24, 6, 25, 5, 26, 14, 27, 9, 28, 15, 12, 2, 29, 16, 30, 10, 31, 17, 32, 6, 33, 18, 34, 11, 25, 19, 35, 4, 6, 20, 36, 12, 37, 21, 38, 7, 39, 9, 40, 13, 41, 22, 42, 3, 43, 12, 16, 9, 44, 23, 45, 8, 46

Offset: 1

Antti Karttunen, Jan 17 2020

nonn

Restricted growth sequence transform of A278220(n) (= A046523(A241909(n))).

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences computed from indices in prime factorization

Cf. A046523, A241909, A278220.

Cf. also A286621, A331299.

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; };
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ This function from A046523
A241909(n) = if(1==n||isprime(n),2^primepi(n),my(f=factor(n),h=1,i,m=1,p=1,k=1); while(k<=#f~, p = nextprime(1+p); i = primepi(f[k,1]); m *= p^(i-h); h = i; if(f[k,2]>1, f[k,2]--, k++)); (p*m));
A278220(n) = A046523(A241909(n));
v331280 = rgs_transform(vector(up_to, n, A278220(n)));
A331280(n) = v331280[n];

A331730 Lexicographically earliest infinite sequence such that a(i) = a(j) => f(i) = f(j), where f(n) = A331595(n) for all other n, except for odd primes p, f(p) = 0.

Original entry on oeis.org

1, 2, 3, 4, 3, 4, 3, 5, 4, 4, 3, 5, 3, 4, 6, 7, 3, 8, 3, 5, 6, 4, 3, 7, 4, 4, 5, 5, 3, 8, 3, 9, 6, 4, 6, 7, 3, 4, 6, 7, 3, 8, 3, 5, 10, 4, 3, 9, 4, 11, 6, 5, 3, 7, 12, 7, 6, 4, 3, 7, 3, 4, 10, 13, 12, 8, 3, 5, 6, 11, 3, 9, 3, 4, 8, 5, 6, 8, 3, 9, 7, 4, 3, 7, 12, 4, 6, 7, 3, 7, 12, 5, 6, 4, 12, 13, 3, 14, 10, 7, 3, 8, 3, 7, 15

Offset: 1

Antti Karttunen, Jan 25 2020

nonn

For all i, j:
  A305801(i) = A305801(j) => a(i) = a(j),
  a(i) = a(j) => A331597(i) = A331597(j) => A331596(i) = A331596(j),
  a(i) = a(j) => A331731(i) = A331731(j) => A331600(i) = A331600(j).

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A122111, A241909, A305801, A331594, A331595, A331596, A331597, A331600, A331731.

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; };
A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
A122111(n) = if(1==n,n,prime(bigomega(n))*A122111(A064989(n)));
A241909(n) = if(1==n||isprime(n),2^primepi(n),my(f=factor(n),h=1,i,m=1,p=1,k=1); while(k<=#f~, p = nextprime(1+p); i = primepi(f[k,1]); m *= p^(i-h); h = i; if(f[k,2]>1, f[k,2]--, k++)); (p*m));
A331595(n) = gcd(A122111(n), A241909(n));
Aux331730(n) = if((n%2)&&isprime(n),0,A331595(n));
v331730 = rgs_transform(vector(up_to, n, Aux331730(n)));
A331730(n) = v331730[n];

cp's OEIS Frontend

A331597 a(n) = A007947(A331595(n)).

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A331280 Lexicographically earliest infinite sequence such that a(i) = a(j) => A278220(i) = A278220(j) for all i, j.

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A331730 Lexicographically earliest infinite sequence such that a(i) = a(j) => f(i) = f(j), where f(n) = A331595(n) for all other n, except for odd primes p, f(p) = 0.

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