A241909 -id:A241909 - cp's OEIS frontend

A280489 a(n) = gcd(n,A241909(n)).

1, 2, 1, 1, 1, 3, 1, 1, 3, 1, 1, 1, 1, 1, 3, 1, 1, 3, 1, 5, 3, 1, 1, 1, 1, 1, 1, 1, 1, 15, 1, 1, 3, 1, 1, 1, 1, 1, 3, 1, 1, 3, 1, 1, 5, 1, 1, 1, 1, 5, 3, 1, 1, 3, 1, 7, 3, 1, 1, 5, 1, 1, 1, 1, 1, 3, 1, 1, 3, 5, 1, 1, 1, 1, 15, 1, 1, 3, 1, 1, 1, 1, 1, 7, 1, 1, 3, 1, 1, 3, 1, 1, 3, 1, 1, 1, 1, 1, 1, 25, 1, 3, 1, 1, 15, 1, 1, 1, 1, 5, 3, 1, 1, 3, 1, 1, 1, 1, 1

Offset: 1

Antti Karttunen, Jan 09 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 1..8192

Cf. A241909, A280488, A280491.

(define (A280489 n) (gcd n (A241909 n)))

a(n) = gcd(n,A241909(n)).
a(n) = n / A280488(n).
Other identities. For all n >= 1:
a(A241909(n)) = a(n).

A331596 Number of distinct prime factors of gcd(A122111(n), A241909(n)).

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 2, 2, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 2, 1, 1, 2, 2, 1, 1, 1, 1, 2, 1, 2, 2, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 2, 2, 1, 1, 2, 1, 1, 3

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. A001221, A331596, A331597.

Array[PrimeNu@ 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. *)

A331596(n) = omega(gcd(A122111(n), A241909(n)));

a(n) = A001221(A331596(n)) = A001221(gcd(A122111(n), A241909(n))).

a(n) = A001222(A331597(n)).

A331598 a(n) = A122111(n) / gcd(A122111(n),A241909(n)).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 3, 4, 1, 2, 1, 8, 1, 1, 1, 1, 1, 4, 2, 16, 1, 2, 9, 32, 5, 8, 1, 2, 1, 1, 4, 64, 3, 3, 1, 128, 8, 4, 1, 4, 1, 16, 1, 256, 1, 2, 27, 1, 16, 32, 1, 5, 1, 8, 32, 512, 1, 6, 1, 1024, 2, 1, 2, 8, 1, 64, 64, 2, 1, 3, 1, 2048, 5, 128, 9, 16, 1, 4, 7, 4096, 1, 12, 4, 8192, 128, 16, 1, 10, 3, 256, 256, 16384, 8, 2, 1, 1, 4, 9, 1, 32, 1, 32, 1

Offset: 1

Antti Karttunen, Jan 22 2020

nonn

It appears that these and the terms of A331599 have the same prime signatures, that is, A046523(a(n)) = A046523(A331599(n)) seems to hold for all n. However, the sequences are not equivalence-class-wise same: a(6) = a(12) = 2, whereas A331599(6) = 3 and A331599(12) = 5.

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences computed from indices in prime factorization

Cf. A122111, A241909, A241916, A331594, A331595, A331599.

Array[If[# == 1, 1, #1/GCD[#1, #2] & @@ {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[#]] &, 90] (* Michael De Vlieger, Jan 25 2020, after JungHwan Min at A122111 *)

A331598(n) = { my(u=A122111(n)); u/gcd(u, A241909(n)); };

a(n) = A122111(n)/A331598(n) = A122111(n) / gcd(A122111(n),A241909(n)).

a(n) = A331599(A241916(n)).

A331599 a(n) = A241909(n) / gcd(A122111(n),A241909(n)).

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 1, 1, 2, 9, 1, 5, 1, 27, 1, 1, 1, 1, 1, 25, 3, 81, 1, 7, 4, 243, 2, 125, 1, 5, 1, 1, 9, 729, 2, 5, 1, 2187, 27, 49, 1, 25, 1, 625, 1, 6561, 1, 11, 8, 1, 81, 3125, 1, 3, 1, 343, 243, 19683, 1, 35, 1, 59049, 5, 1, 3, 125, 1, 15625, 729, 5, 1, 7, 1, 177147, 2, 78125, 4, 625, 1, 121, 2, 531441, 1, 245, 9, 1594323, 2187, 2401, 1, 21

Offset: 1

Antti Karttunen, Jan 22 2020

nonn

It appears that these and the terms of A331598 have the same prime signatures, that is, A046523(a(n)) = A046523(A331598(n)) seems to hold for all n.

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences computed from indices in prime factorization

Cf. A122111, A241909, A241916, A331595, A331598.

Array[If[# == 1, 1, #2/GCD[#1, #2] & @@ {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[#]] &, 90] (* Michael De Vlieger, Jan 24 2020, after JungHwan Min at A122111 *)

A331599(n) = { my(u=A241909(n)); u/gcd(A122111(n), u); };

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

a(n) = A331598(A241916(n)).

A331731 Odd part of A331595(n), where A331595(n) = gcd(A122111(n), A241909(n)).

Original entry on oeis.org

1, 1, 1, 3, 1, 3, 1, 5, 3, 3, 1, 5, 1, 3, 9, 7, 1, 15, 1, 5, 9, 3, 1, 7, 3, 3, 5, 5, 1, 15, 1, 11, 9, 3, 9, 7, 1, 3, 9, 7, 1, 15, 1, 5, 25, 3, 1, 11, 3, 45, 9, 5, 1, 7, 27, 7, 9, 3, 1, 7, 1, 3, 25, 13, 27, 15, 1, 5, 9, 45, 1, 11, 1, 3, 15, 5, 9, 15, 1, 11, 7, 3, 1, 7, 27, 3, 9, 7, 1, 7, 27, 5, 9, 3, 27, 13, 1, 135, 25, 7, 1, 15, 1, 7, 75

Offset: 1

Antti Karttunen, Jan 25 2020

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537
Index entries for sequences computed from indices in prime factorization

Cf. A122111, A241909, A331595, A331600.

Cf. also A322865, A331732.

PARI
```
A331731(n) = A000265(A331595(n));
```

a(n) = A000265(A331595(n)).

A369032 LCM-transform of permutation A241909.

Original entry on oeis.org

1, 2, 2, 3, 2, 3, 2, 5, 1, 3, 2, 5, 2, 3, 1, 7, 2, 1, 2, 5, 1, 3, 2, 7, 1, 3, 1, 5, 2, 1, 2, 11, 1, 3, 1, 1, 2, 3, 1, 7, 2, 1, 2, 5, 1, 3, 2, 11, 1, 1, 1, 5, 2, 1, 1, 7, 1, 3, 2, 1, 2, 3, 1, 13, 1, 1, 2, 5, 1, 1, 2, 1, 2, 3, 1, 5, 1, 1, 2, 11, 1, 3, 2, 1, 1, 3, 1, 7, 2, 1, 1, 5, 1, 3, 1, 13, 2, 1, 1, 1, 2, 1, 2, 7, 1

Offset: 1

Antti Karttunen, Jan 12 2024

nonn

See comments in A368900.

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

Cf. A014963, A122111, A368900, A369031, A369033.

up_to = 2^18;
LCMtransform(v) = { my(len = length(v), b = vector(len), g = vector(len)); b[1] = g[1] = 1; for(n=2,len, g[n] = lcm(g[n-1],v[n]); b[n] = g[n]/g[n-1]); (b); };
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));
v369032 = LCMtransform(vector(up_to,i,A241909(i)));
A369032(n) = v369032[n];

A014963(n) = { ispower(n, , &n); if(isprime(n), n, 1); };
A369032(n) = A014963(A241909(n));

a(1) = 1, for n > 1, a(n) = lcm {1..A241909(n)} / lcm {1..A241909(n-1)}.

a(n) = A014963(A241909(n)). [A241909 satisfies the property S defined in A368900]

A279356 Permutation of natural numbers: a(n) = A241909(A249818(n)).

Original entry on oeis.org

1, 2, 4, 3, 8, 9, 16, 5, 6, 27, 32, 25, 64, 81, 18, 7, 128, 15, 256, 125, 10, 243, 512, 49, 12, 729, 54, 625, 1024, 75, 2048, 11, 50, 2187, 36, 35, 4096, 6561, 162, 343, 8192, 375, 16384, 3125, 14, 19683, 32768, 121, 24, 45, 30, 15625, 65536, 21, 20, 2401, 250, 59049, 131072, 245, 262144, 177147, 486, 13, 108, 1875, 524288, 78125, 98, 225

Offset: 1

Antti Karttunen, Dec 12 2016

nonn

Antti Karttunen, Table of n, a(n) for n = 1..818
Index entries for sequences that are permutations of the natural numbers

Inverse: A279355.

Cf. A241909, A249818, A279354.

a(n) = A241909(A249818(n)).

A280488 a(n) = n / A280489(n) = n / gcd(n,A241909(n)).

Original entry on oeis.org

1, 1, 3, 4, 5, 2, 7, 8, 3, 10, 11, 12, 13, 14, 5, 16, 17, 6, 19, 4, 7, 22, 23, 24, 25, 26, 27, 28, 29, 2, 31, 32, 11, 34, 35, 36, 37, 38, 13, 40, 41, 14, 43, 44, 9, 46, 47, 48, 49, 10, 17, 52, 53, 18, 55, 8, 19, 58, 59, 12, 61, 62, 63, 64, 65, 22, 67, 68, 23, 14, 71, 72, 73, 74, 5, 76, 77, 26, 79, 80, 81, 82, 83, 12, 85, 86, 29, 88, 89, 30, 91, 92, 31

Offset: 1

Antti Karttunen, Jan 09 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 1..8192

Cf. A241909, A280489, A280490, A278520.

Scheme
```
(define (A280488 n) (/ n (A280489 n)))
```

a(n) = n / A280489(n).

A331594 Number of prime factors (with multiplicity) of A331598(n), where A331598(n) = A122111(n) / gcd(A122111(n),A241909(n)).

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 1, 2, 0, 1, 0, 3, 0, 0, 0, 0, 0, 2, 1, 4, 0, 1, 2, 5, 1, 3, 0, 1, 0, 0, 2, 6, 1, 1, 0, 7, 3, 2, 0, 2, 0, 4, 0, 8, 0, 1, 3, 0, 4, 5, 0, 1, 0, 3, 5, 9, 0, 2, 0, 10, 1, 0, 1, 3, 0, 6, 6, 1, 0, 1, 0, 11, 1, 7, 2, 4, 0, 2, 1, 12, 0, 3, 2, 13, 7, 4, 0, 2, 1, 8, 8, 14, 3, 1, 0, 0, 2, 2, 0, 5, 0, 5, 0

Offset: 1

Antti Karttunen, Jan 25 2020

nonn

Apparently also the number of prime factors (with multiplicity) of A331599(n).

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

Cf. A001222, A331598, A331599.

Array[If[# == 1, 0, PrimeOmega[#1/GCD[#1, #2]] & @@ {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 25 2020, after JungHwan Min at A122111 *)

PARI
```
A331594(n) = bigomega(A331598(n));
```

a(n) = A001222(A331598(n)).

A331601 a(n) = A002487(A241909(n)).

Original entry on oeis.org

1, 1, 1, 2, 1, 4, 1, 3, 2, 8, 1, 7, 1, 14, 4, 3, 1, 4, 1, 11, 8, 22, 1, 9, 2, 64, 3, 43, 1, 18, 1, 5, 14, 110, 4, 9, 1, 162, 22, 47, 1, 34, 1, 127, 7, 440, 1, 13, 2, 12, 64, 191, 1, 8, 8, 97, 110, 1002, 1, 23, 1, 752, 11, 5, 14, 112, 1, 1249, 162, 16, 1, 17, 1, 610, 4, 897, 4, 220, 1, 111, 3, 4882, 1, 121, 22, 5494, 440, 281, 1, 26, 8, 7623, 1002

Offset: 1

Antti Karttunen, Jan 22 2020

nonn

Cf. A002487, A241909, A331732.

Cf. also A323901, A323902, A323903, A331600.

A002487(n) = { my(a=1, b=0); while(n>0, if(bitand(n, 1), b+=a, a+=b); n>>=1); (b); }; \\ From A002487
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));
A331601(n) = A002487(A241909(n));

a(n) = A002487(A241909(n)).

a(n) = A002487(A331732(n)).

cp's OEIS Frontend

A280489 a(n) = gcd(n,A241909(n)).

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A331596 Number of distinct prime factors of gcd(A122111(n), A241909(n)).

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Formula

A331598 a(n) = A122111(n) / gcd(A122111(n),A241909(n)).

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Formula

A331599 a(n) = A241909(n) / gcd(A122111(n),A241909(n)).

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A331731 Odd part of A331595(n), where A331595(n) = gcd(A122111(n), A241909(n)).

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A369032 LCM-transform of permutation A241909.

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PARI

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A279356 Permutation of natural numbers: a(n) = A241909(A249818(n)).

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A280488 a(n) = n / A280489(n) = n / gcd(n,A241909(n)).

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Formula

A331594 Number of prime factors (with multiplicity) of A331598(n), where A331598(n) = A122111(n) / gcd(A122111(n),A241909(n)).

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