A003557 -id:A003557 - cp's OEIS frontend

A348039 a(n) = gcd(A003557(n), A327564(n)).

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 6, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 12, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 6

Offset: 1

Antti Karttunen, Oct 19 2021

There is interesting regularity in the scatter plot.

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

Cf. A003557, A003959, A003968, A007947, A327564, A348036, A348037, A348038.

Cf. A347960 (positions of terms > 1).

{1}~Join~Array[GCD @@ Map[Times @@ # &, Transpose@ Map[{#1^(#2 - 1), (#1 + 1)^(#2 - 1)} & @@ # &, FactorInteger[#]]] &, 105, 2] (* Michael De Vlieger, Oct 20 2021 *)

A003968(n) = {my(f=factor(n)); for (i=1, #f~, p= f[i, 1]; f[i, 1] = p*(p+1)^(f[i, 2]-1); f[i, 2] = 1); factorback(f); }
A007947(n) = factorback(factorint(n)[, 1]);
A348036(n) = gcd(n, A003968(n));
A348039(n) = (A348036(n)/A007947(n));

A003557(n) = (n/factorback(factorint(n)[, 1]));
A327564(n) = { my(f=factor(n)); for(k=1, #f~, f[k, 1]++; f[k, 2]--); factorback(f); }; \\ From A327564
A348039(n) = gcd(A003557(n), A327564(n));

a(n) = gcd(A003557(n), A327564(n)).
a(n) = A348036(n) / A007947(n).
a(n) = A003557(n) / A348037(n).
a(n) = A327564(n) / A348038(n).

A348496 a(n) = gcd(A018804(n), A347130(n)) / A003557(n).

Original entry on oeis.org

1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 4, 1, 3, 1, 2, 1, 21, 1, 36, 5, 1, 1, 1, 1, 15, 3, 4, 1, 1, 1, 1, 7, 1, 3, 1, 1, 3, 1, 1, 1, 1, 1, 12, 3, 5, 1, 10, 1, 3, 5, 4, 1, 9, 1, 1, 1, 1, 1, 12, 1, 3, 1, 1, 9, 1, 1, 12, 1, 1, 1, 1, 1, 3, 1, 4, 3, 1, 1, 2, 1, 1, 1, 4, 11, 15, 1, 35, 1, 3, 5, 36, 1, 1, 3, 1, 1, 3, 3, 1, 1

Offset: 1

Antti Karttunen, Oct 21 2021

nonn

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A003557, A018804, A347128, A347129, A347130, A348495, A348501 [= a(A276086(n))].

Cf. also A348494, A348498.

Table[GCD[Total@ GCD[n, Range[n]], DivisorSum[n, #*(If[# < 2, 0, # Total[#2/#1 & @@@ FactorInteger[#]]] &[n/#]) &]]/Apply[Times, Map[#1^(#2 - 1) & @@ # &, FactorInteger[n]]], {n, 101}] (* Michael De Vlieger, Oct 21 2021 *)

\\ Needs also code from A348495:
A003557(n) = (n/factorback(factorint(n)[, 1]));
A348496(n) = (A348495(n)/A003557(n));

a(n) = A348495(n) / A003557(n).

a(n) = gcd(A347128(n), A347129(n)).

A349619 Dirichlet convolution of A003415 with the Dirichlet inverse of A003557.

Original entry on oeis.org

0, 1, 1, 3, 1, 3, 1, 7, 5, 5, 1, 7, 1, 7, 6, 15, 1, 9, 1, 13, 8, 11, 1, 15, 9, 13, 19, 19, 1, 14, 1, 31, 12, 17, 10, 17, 1, 19, 14, 29, 1, 20, 1, 31, 24, 23, 1, 31, 13, 25, 18, 37, 1, 27, 14, 43, 20, 29, 1, 30, 1, 31, 34, 63, 16, 32, 1, 49, 24, 34, 1, 33, 1, 37, 34, 55, 16, 38, 1, 61, 65, 41, 1, 44, 20, 43, 30, 71

Offset: 1

Antti Karttunen, Nov 25 2021

nonn

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A003415, A003557, A349340.

Cf. also A349394, A349618, A349620, A349621.

f1[p_, e_] := e/p; d[1] = 0; d[n_] := n*Plus @@ f1 @@@ FactorInteger[n]; f[p_, e_] := -(p - 1)^(e - 1); s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; a[n_] := DivisorSum[n, s[#]*d[n/#] &]; Array[a, 100] (* Amiram Eldar, Nov 25 2021 *)

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A349340(n) = { my(f=factor(n)); prod(i=1, #f~, -((f[i,1]-1)^(f[i,2]-1))); };
A349619(n) = sumdiv(n,d,A003415(n/d)*A349340(d));

a(n) = Sum_{d|n} A003415(n/d) * A349340(d).

A351944 a(n) = A003557(A181819(n)).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 2, 1, 1, 1, 4, 1, 1, 2, 2, 2, 3, 1, 2, 2, 1, 1, 4, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 2, 1, 2, 1, 2, 1, 1, 2, 4, 1, 1, 2, 4, 1, 1, 1, 2, 1, 1, 2, 4, 1, 1, 1, 2, 1, 2, 2, 2, 2, 1, 1, 2, 2, 1, 2, 2, 2, 1, 1, 1, 1, 3, 1, 4, 1, 1, 4

Offset: 1

Antti Karttunen, Apr 02 2022

nonn

Antti Karttunen, Table of n, a(n) for n = 1..20000
Index entries for sequences computed from exponents in factorization of n

Cf. A003557, A181819, A329601, A351945.

Cf. also A295879.

A003557(n) = (n/factorback(factorint(n)[, 1]));
A181819(n) = factorback(apply(e->prime(e),(factor(n)[,2])));
A351944(n) = A003557(A181819(n));

a(n) = A181819(n) / A329601(n) = A003557(A181819(n)).

A369008 a(n) = A085731(n) / A003557(n).

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Jan 15 2024

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

Cf. A003415, A003557, A083345, A085731, A342001, A369009.

Cf. A342090 (positions of terms > 1).

f[p_, e_] := If[Divisible[e, p], p, 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Jan 20 2024 *)

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A003557(n) = (n/factorback(factorint(n)[, 1]));
A369008(n) = { my(u=A003415(n)); (gcd(n,u)/A003557(n)); };

A369008(n) = if(1==n, n, my(f=factor(n)); for(i=1, #f~, if((f[i, 2]%f[i, 1]), f[i, 1] = 1, f[i, 2] = 1)); factorback(f));

Multiplicative with a(p^e) = p if p|e, otherwise a(p^e) = 1.
For n > 1, a(n) = A342001(n) / A083345(n).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Product_{p prime} ((p^(p+1) + p^2 - 3*p +1)/(p*(p^p-1))) = 1.22775972725472961826... . - Amiram Eldar, Jan 20 2024

A255425 a(n) = A003557(A255334(n)) = A255334(n) / A255424(n).

Original entry on oeis.org

36, 36, 36, 36, 36, 36, 36, 180, 36, 576, 36, 36, 36, 36, 36, 36, 36, 36, 36, 36, 36, 36, 36, 36, 36, 36, 36, 36, 36, 36, 36, 36, 36, 36, 36, 396, 900, 36, 36, 36, 36, 36, 36, 576, 36, 36, 36, 36, 36, 36, 468, 36, 36, 36, 36, 36, 36, 36, 36, 36, 36, 576, 36, 36

Offset: 1

Antti Karttunen, Apr 06 2015

nonn

Antti Karttunen, Table of n, a(n) for n = 1..6589

Cf. A003557, A254791, A255334, A255424, A255426.

a(n) = A003557(A255334(n)) = A255334(n) / A255424(n).

For all n, a(n) > 1 and a(n) < A255426(n).

A255426 a(n) = A003557(A255423(n)) = A255423(n) / A255424(n).

Original entry on oeis.org

49, 49, 49, 49, 49, 49, 49, 245, 49, 676, 49, 49, 49, 49, 49, 49, 49, 49, 49, 49, 49, 49, 49, 49, 49, 49, 49, 49, 49, 49, 49, 49, 49, 49, 49, 539, 1225, 49, 49, 49, 49, 49, 49, 676, 49, 49, 49, 49, 49, 49, 637, 49, 49, 49, 49, 49, 49, 49, 49, 49, 49, 676, 49, 49

Offset: 1

Antti Karttunen, Apr 06 2015

nonn

Antti Karttunen, Table of n, a(n) for n = 1..6589

Cf. A003557, A254791, A255423, A255424, A255425.

a(n) = A003557(A255423(n)) = A255423(n) / A255424(n).

For all n, a(n) > A255425(n) > 1.

A319698 Filter sequence combining A003557(n) [n divided by largest squarefree divisor of n] with A319697(n) [sum of even squarefree divisors of n].

Original entry on oeis.org

1, 2, 1, 3, 1, 4, 1, 5, 6, 7, 1, 8, 1, 9, 1, 10, 1, 11, 1, 12, 1, 13, 1, 14, 15, 16, 17, 18, 1, 19, 1, 20, 1, 21, 1, 22, 1, 23, 1, 24, 1, 25, 1, 26, 6, 19, 1, 27, 28, 29, 1, 30, 1, 31, 1, 32, 1, 33, 1, 34, 1, 25, 6, 35, 1, 36, 1, 37, 1, 36, 1, 38, 1, 39, 15, 40, 1, 41, 1, 42, 43, 44, 1, 45, 1, 46, 1, 47, 1, 48, 1, 34, 1, 36, 1, 49, 1, 50, 6, 51, 1, 52, 1, 53, 1

Offset: 1

Antti Karttunen, Oct 31 2018

nonn

Restricted growth sequence transform of ordered pair [A003557(n), A319697(n)].

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

Cf. A003557, A319697.

Cf also A291750, A291751.

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; };
A003557(n) = { my(f=factor(n)); for (i=1, #f~, f[i, 2] = f[i, 2]-1); factorback(f); }; \\ From A003557
A319697(n) = sumdiv(n, d, (!(d%2))*issquarefree(d)*d);
v319698 = rgs_transform(vector(up_to,n,[A003557(n),A319697(n)]));
A319698(n) = v319698[n];

A322318 a(n) = gcd(A003557(n), A048250(n)).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 3, 1, 2, 1, 1, 1, 4, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 6, 1, 1, 1, 2, 1, 1, 1, 2, 3, 1, 1, 4, 1, 1, 1, 2, 1, 3, 1, 4, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 12, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 4, 1, 3, 1, 2, 1, 1, 1, 4, 1, 1, 3, 2, 1, 1, 1, 2, 1

Offset: 1

Antti Karttunen, Dec 05 2018

nonn

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A001615, A003557, A048250, A322319.

Cf. also A066086, A322320.

a[n_] := If[n == 1, 1, Module[{f=FactorInteger[n]}, GCD[ Times@@ (First[#] ^(Last[#]-1)& /@  f), Times@@((#+1)& @@@ f)]]]; Array[a, 120] (* Amiram Eldar, Dec 05 2018 *)

A003557(n) = { my(f=factor(n)); for(i=1, #f~, f[i, 2] = f[i, 2]-1); factorback(f); }; \\ From A003557
A048250(n) = factorback(apply(p -> p+1, factor(n)[, 1]));
A322318(n) = gcd(A048250(n), A003557(n));

a(n) = gcd(A003557(n), A048250(n)).

a(n) = A001615(n) / A322319(n).

A322351 a(n) = min(A003557(n), A173557(n)).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 4, 1, 2, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 4, 1, 1, 1, 2, 3, 1, 1, 2, 6, 4, 1, 2, 1, 2, 1, 4, 1, 1, 1, 2, 1, 1, 3, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 5, 2, 1, 1, 1, 4, 2, 1, 1, 2, 1, 1, 1, 4, 1, 3, 1, 2, 1, 1, 1, 2, 1, 6, 3, 4, 1, 1, 1, 4, 1

Offset: 1

Antti Karttunen, Dec 05 2018

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A000010, A003557, A173557, A322320, A322321, A322352, A322355.

a[n_] := If[n == 1, 1, Module[{f=FactorInteger[n]}, Min[ Times@@ (First[#]^(Last[#]-1)& /@  f), Times@@((#-1)& @@@ f)]]]; Array[a, 120] (* Amiram Eldar, Dec 05 2018 *)

A003557(n) = { my(f=factor(n)); for (i=1, #f~, f[i, 2] = f[i, 2]-1); factorback(f); }; \\ From A003557
A173557(n) = factorback(apply(p -> p-1, factor(n)[, 1]));
A322351(n) = min(A003557(n), A173557(n));

a(n) = min(A003557(n), A173557(n)).

a(n) = A000010(n) / A322352(n).

cp's OEIS Frontend

A348039 a(n) = gcd(A003557(n), A327564(n)).

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A348496 a(n) = gcd(A018804(n), A347130(n)) / A003557(n).

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A349619 Dirichlet convolution of A003415 with the Dirichlet inverse of A003557.

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A351944 a(n) = A003557(A181819(n)).

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A369008 a(n) = A085731(n) / A003557(n).

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A255425 a(n) = A003557(A255334(n)) = A255334(n) / A255424(n).

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A255426 a(n) = A003557(A255423(n)) = A255423(n) / A255424(n).

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A319698 Filter sequence combining A003557(n) [n divided by largest squarefree divisor of n] with A319697(n) [sum of even squarefree divisors of n].

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A322318 a(n) = gcd(A003557(n), A048250(n)).

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