A003557 -id:A003557 - cp's OEIS frontend

A342001 Arithmetic derivative without its inherited divisor; the arithmetic derivative of n divided by A003557(n), which is a common divisor of both n and A003415(n).

0, 1, 1, 2, 1, 5, 1, 3, 2, 7, 1, 8, 1, 9, 8, 4, 1, 7, 1, 12, 10, 13, 1, 11, 2, 15, 3, 16, 1, 31, 1, 5, 14, 19, 12, 10, 1, 21, 16, 17, 1, 41, 1, 24, 13, 25, 1, 14, 2, 9, 20, 28, 1, 9, 16, 23, 22, 31, 1, 46, 1, 33, 17, 6, 18, 61, 1, 36, 26, 59, 1, 13, 1, 39, 11, 40, 18, 71, 1, 22, 4, 43, 1, 62, 22, 45, 32, 35, 1, 41, 20

Offset: 1

Antti Karttunen, Feb 28 2021

nonn

See also the scatter plot of A342002 that seems to reveal some interesting internal structure in this sequence, not fully explained by the regularity of primorial base expansion used in the latter sequence. - Antti Karttunen, May 09 2022

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A003415, A003557, A341998, A342416, A342459.
Cf. A342002 [= a(A276086(n))], A342463 [= a(A342456(n))], A351945 [= a(A181819(n))], A353571 [= a(A003961(n))].
Cf. A346485 (Möbius transform), A347395 (convolution with Liouville's lambda), A347961 (with itself), and A347234, A347235, A347954, A347959, A347963, A349396, A349612 (for convolutions with other sequences).
Cf. A007947.

Array[#1/#2 & @@ {If[# < 2, 0, # Total[#2/#1 & @@@ FactorInteger[#]]] &@ Abs[#], #/Times @@ FactorInteger[#][[All, 1]]} &, 91] (* Michael De Vlieger, Mar 11 2021 *)

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]));
A342001(n) = (A003415(n) / A003557(n));

from math import prod
from sympy import factorint
def A342001(n):
    q = prod(f:=factorint(n))
    return sum(q*e//p for p, e in f.items()) # Chai Wah Wu, Nov 04 2022

a(n) = A003415(n) / A003557(n).
For all n >= 0, a(A276086(n)) = A342002(n).
a(n) = A342414(n) * A342416(n) = A342459(n) * A342919(n). - Antti Karttunen, Apr 30 2022
Dirichlet g.f.: Dirichlet g.f. of A007947 * Sum_{p prime} p^s/((p^s-1)*(p^s+p-1)) = zeta(s) * Product_{p prime} (1+p^(1-s)-p^(-s)) * Sum_{p prime} p^s/((p^s-1)*(p^s+p-1)). - Sebastian Karlsson, May 05 2022
Sum_{k=1..n} a(k) ~ c * A065464 * Pi^2 * n^2 / 12, where c = Sum_{j>=2} (1/2 + (-1)^j * (Fibonacci(j) - 1/2))*PrimeZetaP(j) = 0.4526952873143153104685540856936425315834753528741817723313791528384... - Vaclav Kotesovec, May 09 2022

A328572 Primorial base expansion of n converted into its prime product form, but with 1 subtracted from all nonzero digits: a(n) = A003557(A276086(n)).

Original entry on oeis.org

1, 1, 1, 1, 3, 3, 1, 1, 1, 1, 3, 3, 5, 5, 5, 5, 15, 15, 25, 25, 25, 25, 75, 75, 125, 125, 125, 125, 375, 375, 1, 1, 1, 1, 3, 3, 1, 1, 1, 1, 3, 3, 5, 5, 5, 5, 15, 15, 25, 25, 25, 25, 75, 75, 125, 125, 125, 125, 375, 375, 7, 7, 7, 7, 21, 21, 7, 7, 7, 7, 21, 21, 35, 35, 35, 35, 105, 105, 175, 175, 175, 175, 525, 525, 875, 875, 875, 875

Offset: 0

Antti Karttunen, Oct 20 2019

Antti Karttunen, Table of n, a(n) for n = 0..2310
Antti Karttunen, Data supplement: n, a(n) computed for n = 0..32768
Index entries for sequences related to primorial base

Cf. A003557, A051903, A085731, A276086, A327860, A328114, A328475, A328571, A328573, A328575, A328577.

Block[{b = MixedRadix[Reverse@ Prime@ Range@ 12]}, Array[#1/(Times @@ #2[[All, 1]]) & @@ {#1, FactorInteger[#]} &[Times @@ Power @@@ #] &@ Transpose@ {Prime@ Range@ Length@ #, Reverse@ #} &@ IntegerDigits[#, b] &, 87, 0]] (* Michael De Vlieger, Mar 12 2021 *)

A328572(n) = { my(m=1, p=2); while(n, if(n%p, m *= p^((n%p)-1)); n = n\p; p = nextprime(1+p)); (m); };

a(n) = A003557(A276086(n)).
a(n) = A276086(n) / A328571(n).
a(n) = A328475(n) / A328573(n).
For all n >= 1, 1+A051903(a(n)) = A328114(n).
a(n) = A085731(A276086(n)) = gcd(A276086(n), A327860(n)). - Antti Karttunen, Feb 28 2021

A291751 Lexicographically earliest such sequence a that a(i) = a(j) => A003557(i) = A003557(j) and A048250(i) = A048250(j), for all i, j.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 6, 11, 12, 13, 13, 14, 10, 15, 16, 17, 18, 19, 13, 20, 21, 22, 23, 24, 25, 26, 18, 27, 28, 29, 28, 30, 31, 32, 33, 34, 22, 35, 36, 37, 38, 26, 28, 39, 40, 41, 26, 42, 29, 43, 26, 44, 45, 46, 32, 47, 48, 35, 49, 50, 51, 52, 53, 54, 35, 52, 26, 55, 56, 57, 58, 59, 35, 60, 45, 61, 62, 63, 51, 64, 65, 66, 67, 68, 46, 69, 70, 47, 71

Offset: 1

Antti Karttunen, Sep 06 2017

nonn

Restricted growth sequence transform of A291750, which means that this is the lexicographically least sequence a, such that for all i, j: a(i) = a(j) <=> A291750(i) = A291750(j) <=> A003557(i) = A003557(j) and A048250(i) = A048250(j). That this is equal to the definition given in the title follows because any such lexicographically least sequence satisfying relation <=> is also the least sequence satisfying relation => with the same parameters.

Sigma (A000203) and psi (A001615) are functions of this sequence. See comments in A291750 for the reason. For example, to find the value of A001615(n) when we know just a(n), but without knowing n, let m be the least i for which a(i) = a(n); then A001615(n) = A003991(A291750(m)) = A003557(m) * A048250(m).

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

Cf. A001615, A003557, A048250, A291750, A291752, A294877, A295300, A295886, A295887, A295888, A319698, A322021.

Differs from A286603 for the first time at n = 25, where a(25) = 21, while A286603(25) = 14.

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) = n/factorback(factor(n)[, 1]); \\ From A003557
A048250(n) = if(n<1, 0, sumdiv(n, d, if(core(d)==d, d)));
A291750(n) = (1/2)*(2 + ((A003557(n)+A048250(n))^2) - A003557(n) - 3*A048250(n));
v291751 = rgs_transform(vector(65537,n,A291750(n)));
A291751(n) = v291751[n];

Name changed and comments added by Antti Karttunen, Nov 24 2018

A291750 Compound filter: a(n) = P(A003557(n), A048250(n)), where P(n,k) is sequence A000027 used as a pairing function.

Original entry on oeis.org

1, 4, 7, 8, 16, 67, 29, 19, 18, 154, 67, 80, 92, 277, 277, 53, 154, 94, 191, 173, 497, 631, 277, 109, 50, 862, 75, 302, 436, 2557, 497, 169, 1129, 1432, 1129, 142, 704, 1771, 1541, 214, 862, 4561, 947, 668, 328, 2557, 1129, 179, 98, 236, 2557, 905, 1432, 199, 2557, 355, 3161, 4006, 1771, 2630, 1892, 4561, 564, 593, 3487, 10297, 2279, 1487, 4561, 10297, 2557

Offset: 1

Antti Karttunen, Sep 04 2017

nonn

A000203 (sigma(n)) is a function of this sequence, because formula
  A000203(n) = A092261(n) * A295294(n)
can be rewritten as a relation:
  A000203(n) = A000203(A057521(n)) * A048250(n) / A048250(A057521(n)),
where A057521(n) = A064549(A003557(n)), thus A000203(n) is a function of A003557(n) and A048250(n), the values that are packed here into a(n).
A001615 (Dedekind's psi) is a function of this sequence, because it can be written as A001615(n) = A003557(n)*A048250(n).

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A000027, A000203, A001615, A003557, A048250, A291751 (rgs-version of this filter).

A003557(n) = n/factorback(factor(n)[, 1]); \\ This function from Charles R Greathouse IV, Nov 17 2014
A048250(n) = if(n<1, 0, sumdiv(n, d, if(core(d)==d, d)));
A291750(n) = (1/2)*(2 + ((A003557(n)+A048250(n))^2) - A003557(n) - 3*A048250(n));

a(n) = (1/2)*(2 + ((A003557(n) + A048250(n))^2) - A003557(n) - 3*A048250(n)).

A295300 Lexicographically earliest sequence such that a(i) = a(j) => f(i) = f(j), where f(n) = [A003557(n), A046523(n), A048250(n)].

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 32, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 44, 49, 50, 51, 44, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 58, 62, 65, 66, 67, 68, 69, 70, 58, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 80

Offset: 1

Antti Karttunen, Nov 19 2017

nonn

Restricted growth sequence transform of A291752.
For all i, j:
  a(i) = a(j) => A291751(i) = A291751(j),
  a(i) = a(j) => A326199(i) = A326199(j) => A294877(i) = A294877(j),
  a(i) = a(j) => A322021(i) = A322021(j),
  a(i) = a(j) => A295888(i) = A295888(j),
  a(i) = a(j) => A296090(i) = A296090(j).

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

Cf. A003557, A046523, A048250, A101296, A286360, A291750, A291751, A291752, A291757, A291758, A295888, A296090, A322021, A326199.

Cf. also A305801, A305900, A323368, A324400.

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; };
A003557(n) = n/factorback(factor(n)[, 1]); \\ From A003557
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ From A046523
A048250(n) = if(n<1, 0, sumdiv(n, d, if(core(d)==d, d)));
A291750(n) = (1/2)*(2 + ((A003557(n)+A048250(n))^2) - A003557(n) - 3*A048250(n));
Aux295300(n) = (1/2)*(2 + ((A046523(n) + A291750(n))^2) - A046523(n) - 3*A291750(n));
v295300 = rgs_transform(vector(up_to,n,Aux295300(n)));
A295300(n) = v295300[n];

Name changed and the comments section added by Antti Karttunen, Jul 13 2019

A344025 Lexicographically earliest infinite sequence such that a(i) = a(j) => A003415(i) = A003415(j) and A003557(i) = A003557(j), for all i, j >= 1.

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 5, 6, 7, 2, 8, 2, 9, 10, 11, 2, 12, 2, 13, 14, 15, 2, 16, 17, 18, 19, 20, 2, 21, 2, 22, 23, 24, 25, 26, 2, 27, 28, 29, 2, 30, 2, 31, 32, 33, 2, 34, 35, 36, 37, 38, 2, 39, 28, 40, 41, 21, 2, 42, 2, 43, 44, 45, 46, 47, 2, 48, 49, 50, 2, 51, 2, 52, 53, 54, 46, 55, 2, 56, 57, 58, 2, 59, 41, 60, 61, 62, 2, 63, 37, 64

Offset: 1

Antti Karttunen, May 07 2021

nonn

Restricted growth sequence transform of the ordered pair [A003415(n), A003557(n)], where A003415(n) is the arithmetic derivative of n, and A003557(n) is n divided by its largest squarefree divisor.
For all i, j:
  parent(i) = parent(j) => a(i) = a(j),
  a(i) = a(j) => A342001(i) = A342001(j),
  a(i) = a(j) => A369051(i) = A369051(j) => A085731(i) = A085731(j).
Where "parent" can be any of the sequences A351236, A351260, A353520, A353521, A369050, for example.

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

Cf. A003415, A003557, A085731, A342001, A369051.

Cf. also A305800, A351236, A351260, A353520, A353521, A369050, and also A353523.

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]));
A003557(n) = (n/factorback(factorint(n)[, 1]));
Aux344025(n) = [A003415(n), A003557(n)];
v344025 = rgs_transform(vector(up_to, n, Aux344025(n)));
A344025(n) = v344025[n];

A295887 Filter sequence combining A003557(n) and A173557(n); the restricted growth sequence transform of A291756.

Original entry on oeis.org

1, 1, 2, 3, 4, 2, 5, 6, 7, 4, 8, 9, 10, 5, 11, 12, 13, 7, 14, 15, 10, 8, 16, 17, 18, 10, 19, 20, 21, 11, 22, 23, 24, 13, 25, 26, 27, 14, 25, 28, 29, 10, 30, 31, 32, 16, 33, 34, 35, 18, 36, 37, 38, 19, 29, 39, 27, 21, 40, 41, 42, 22, 43, 44, 45, 24, 46, 47, 48, 25, 49, 50, 51, 27, 52, 53, 42, 25, 54, 55, 56, 29, 57, 37, 58, 30, 59, 60, 61, 32, 51, 62, 42, 33, 51

Offset: 1

Antti Karttunen, Dec 03 2017

nonn

First define function f(n) = (1/2)*(2 + ((A003557(n) + A173557(n))^2) - A003557(n) - 3*A173557(n)), or in short, f(n) = P(A003557(n),  A173557(n)), where P(n,k) is triangular table sequence A000027 used as an injective N x N -> N pairing function. Then apply the restricted growth sequence transform to the sequence f(1), f(2), f(3), ... See the example-section.
This is also the restricted growth sequence transform of sequence A291756, as A291756(n) = P(A003557(n), A000010(n)), where again P(n,k) is sequence A000027 used as a pairing function. Given either an ordered pair (A003557(n),A000010(n)) or (A003557(n),A173557(n)), the other one can be computed because A000010(n) = A003557(n)*A173557(n).
Note that the exact pairing function P used is not important, as long as it provides an injective mapping N x N -> N. So instead of Cantor's mapping we could as well used bit-interleaving A054238 (Morton code) to pack together A003557(n) and A173557(n), or equally, A000010(n) and A003557(n).

			The first ten terms of the sequence f(n) = (1/2)*(2 + ((A003557(n) + A173557(n))^2) - A003557(n) - 3*A173557(n)) are 1, 1, 2, 3, 7, 2, 16, 10, 9, 7. When we assign to each newly occurring term the least unused number k so far (starting by giving k=1 for the initial term, this k increases by one for each new distinct term produced by f(n) when n grows), and for each repeated term the same number it was given the previous time (equal to the number it was given for the first time), we obtain 1, 1, 2, 3, 4, 2, 5, 6, 7, 4, the first 10 terms of this sequence. Note how f(10) = 7 gets 4 because when seven occurred for the first time at f(5), it was the 4th distinct new number in that sequence.
This is also true for the sequence A291756 although there the terms are different: 1, 1, 2, 5, 7, 2, 16, 25, 31, 7.

Antti Karttunen, Table of n, a(n) for n = 1..65537
Eric Weisstein's World of Mathematics, Pairing Function
Wikipedia, Pairing Function

Cf. A000010, A003557, A054238, A173557, A291750, A291756, A295300, A295886.

allocatemem(2^30);
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; };
write_to_bfile(start_offset,vec,bfilename) = { for(n=1, length(vec), write(bfilename, (n+start_offset)-1, " ", vec[n])); }
A003557(n) = { my(f=factor(n)); for (i=1, #f~, f[i, 2] = max(0,f[i, 2]-1)); factorback(f); };
A173557(n) = my(f=factor(n)[, 1]); prod(k=1, #f, f[k]-1); \\ This function from Michel Marcus, Oct 31 2017
Anotsubmitted7(n) = (1/2)*(2 + ((A003557(n)+A173557(n))^2) - A003557(n) - 3*A173557(n));
write_to_bfile(1,rgs_transform(vector(up_to,n,Anotsubmitted7(n))),"b295887.txt");

A344592 a(n) = A003557(A276086(A108951(n))).

Original entry on oeis.org

1, 1, 1, 3, 1, 5, 1, 1, 1, 7, 1, 125, 1, 11, 16807, 15, 1, 35, 1, 343, 161051, 13, 1, 25, 9317, 17, 1, 1331, 1, 2401, 1, 1, 371293, 19, 253333223, 42875, 1, 23, 1419857, 1, 1, 1, 1, 2197, 14641, 29, 1, 49, 371293, 6684099653, 2476099, 4913, 1, 55, 37349, 19487171, 6436343, 31, 1, 5929, 1, 37, 20449, 21, 582622237229761, 1792160394037

Offset: 1

Antti Karttunen, May 26 2021

nonn

Cf. A003557, A085731, A108951, A276086, A324886, A328572, A329047, A342002, A342920, A346091, A346092 [= A276085(a(n))].

Cf. A344591 (positions of ones), A344593 (rgs-transform).

Block[{b = MixedRadix[Reverse@ Prime@ Range@ 20]}, Array[#/(Times @@ FactorInteger[#][[All, 1]]) &@ Apply[Times, Power @@@ #] &@ Transpose@ {Prime@ Range@ Length@ #, Reverse@ #} &@ IntegerDigits[#, b] &@ Apply[Times, Map[(Times @@ Prime@ Range@ PrimePi@ #1)^#2 & @@ # &, FactorInteger[#]]] &, 66]] (* Michael De Vlieger, Jul 14 2021 *)

A034386(n) = prod(i=1, primepi(n), prime(i));
A108951(n) = { my(f=factor(n)); prod(i=1, #f~, A034386(f[i, 1])^f[i, 2]) }; \\ From A108951
A328572(n) = { my(m=1, p=2); while(n, if(n%p, m *= p^((n%p)-1)); n = n\p; p = nextprime(1+p)); (m); };
A344592(n) = A328572(A108951(n));

A003557(n) = { my(f=factor(n)); for (i=1, #f~, f[i, 2] = f[i, 2]-1); factorback(f); }; \\ From A003557
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A344592(n) = A003557(A276086(A108951(n)));

a(n) = A328572(A108951(n)) = A003557(A324886(n)) = A003557(A276086(A108951(n))).
a(n) = A329047(n) / A342920(n).
a(n) = A085731(A324886(n)) = gcd(A324886(n), A329047(n)) = A324886(n) / A346091(n). - Antti Karttunen, Jul 09 2021

A347395 Dirichlet convolution of Liouville's lambda (A008836) with A342001, where A342001(n) = A003415(n)/A003557(n).

Original entry on oeis.org

0, 1, 1, 1, 1, 3, 1, 2, 1, 5, 1, 3, 1, 7, 6, 2, 1, 2, 1, 5, 8, 11, 1, 5, 1, 13, 2, 7, 1, 14, 1, 3, 12, 17, 10, 2, 1, 19, 14, 9, 1, 20, 1, 11, 5, 23, 1, 5, 1, 2, 18, 13, 1, 4, 14, 13, 20, 29, 1, 14, 1, 31, 7, 3, 16, 32, 1, 17, 24, 34, 1, 3, 1, 37, 3, 19, 16, 38, 1, 9, 2, 41, 1, 20, 20, 43, 30, 21, 1, 9, 18, 23, 32

Offset: 1

Antti Karttunen, Sep 02 2021

It seems that all the terms after the initial zero are strictly positive. Checked up to n = 2^24. Compare to A346485.

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A003415, A003557, A008836, A342001, A347396 [= a(A276086(n))].

Cf. also A346485, A347235.

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]));
A342001(n) = (A003415(n) / A003557(n));
A008836(n) = ((-1)^bigomega(n));
A347395(n) = sumdiv(n,d,A008836(n/d)*A342001(d));

a(n) = Sum_{d|n} A008836(n/d) * A342001(d).

Sum_{k=1..n} a(k) ~ c * A065464 * Pi^4 * n^2 / 180, where c = Sum_{j>=2} (1/2 + (-1)^j * (Fibonacci(j) - 1/2))*PrimeZetaP(j) = 0.4526952873143153104685540856936425315834753528741817723313791528384... - Vaclav Kotesovec, Mar 04 2023

A300717 Möbius transform of A003557, n divided by its largest squarefree divisor.

Original entry on oeis.org

1, 0, 0, 1, 0, 0, 0, 2, 2, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 6, 0, 0, 0, 0, 8, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 16, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 18, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 6

Offset: 1

Antti Karttunen, Mar 11 2018

Multiplicative because A003557 is. - Andrew Howroyd, Jul 27 2018

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

Cf. A000010, A008683, A003557, A300718, A300719, A003958.

with(numtheory): A003557 := n -> n/ilcm(op(numtheory[factorset](n))):
seq(add(mobius(d)*A003557(n/d), d in divisors(n)), n=1..100); # Ridouane Oudra, Nov 18 2019

Table[DivisorSum[n, MoebiusMu[#] EulerPhi[#] EulerPhi[n/#] &], {n, 108}] (* Michael De Vlieger, Nov 18 2019 *)
f[p_, e_] := If[e == 1, 0, (p - 1)*p^(e - 2)]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Dec 06 2022 *)

A003557(n) = { my(f=factor(n)); for (i=1, #f~, f[i, 2] = max(0,f[i, 2]-1)); factorback(f); }; \\ From A003557
A300717(n) = sumdiv(n,d,moebius(n/d)*A003557(d));

a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i,2] == 1, 0, (f[i,1] - 1)*f[i,1]^(f[i,2] - 2)));} \\ Amiram Eldar, Dec 06 2022

a(n) = Sum_{d|n} A008683(n/d)*A003557(d).
a(n) = A000010(n) - A300718(n).
a(n) = A003557(n) - A300719(n).
Multiplicative with a(p) = 0 and a(p^e) = (p-1)*p^(e-2) for prime p and e>1. - Werner Schulte, Sep 27 2018
Dirichlet convolution with A003958 equals A000010. - Werner Schulte, Sep 28 2018
a(n) = Sum_{d|n} mu(d)*phi(d)*phi(n/d). - Ridouane Oudra, Nov 18 2019
Dirichlet convolution of A000010 and A097945. - R. J. Mathar, Jun 02 2020
From Richard L. Ollerton, May 07 2021: (Start)
a(n) = Sum_{k=1..n} phi(gcd(n,k))*mu(gcd(n,k)).
a(n) = Sum_{k=1..n} phi(gcd(n,k))*mu(n/gcd(n,k)). (End)

cp's OEIS Frontend

A342001 Arithmetic derivative without its inherited divisor; the arithmetic derivative of n divided by A003557(n), which is a common divisor of both n and A003415(n).

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A328572 Primorial base expansion of n converted into its prime product form, but with 1 subtracted from all nonzero digits: a(n) = A003557(A276086(n)).

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A291751 Lexicographically earliest such sequence a that a(i) = a(j) => A003557(i) = A003557(j) and A048250(i) = A048250(j), for all i, j.

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A291750 Compound filter: a(n) = P(A003557(n), A048250(n)), where P(n,k) is sequence A000027 used as a pairing function.

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A295300 Lexicographically earliest sequence such that a(i) = a(j) => f(i) = f(j), where f(n) = [A003557(n), A046523(n), A048250(n)].

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A344025 Lexicographically earliest infinite sequence such that a(i) = a(j) => A003415(i) = A003415(j) and A003557(i) = A003557(j), for all i, j >= 1.

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A295887 Filter sequence combining A003557(n) and A173557(n); the restricted growth sequence transform of A291756.

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A344592 a(n) = A003557(A276086(A108951(n))).

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