A003557 -id:A003557 - cp's OEIS frontend

A346485 Möbius transform of A342001, where A342001(n) = A003415(n)/A003557(n).

0, 1, 1, 1, 1, 3, 1, 1, 1, 5, 1, 2, 1, 7, 6, 1, 1, 1, 1, 4, 8, 11, 1, 2, 1, 13, 1, 6, 1, 14, 1, 1, 12, 17, 10, 0, 1, 19, 14, 4, 1, 20, 1, 10, 4, 23, 1, 2, 1, 1, 18, 12, 1, 1, 14, 6, 20, 29, 1, 8, 1, 31, 6, 1, 16, 32, 1, 16, 24, 34, 1, 0, 1, 37, 2, 18, 16, 38, 1, 4, 1, 41, 1, 12, 20, 43, 30, 10, 1, 4, 18, 22, 32, 47

Offset: 1

Antti Karttunen, Aug 26 2021

nonn

Conjecture 1: After the initial zero, the positions of other zeros is given by A036785.

Conjecture 2: No negative terms. Checked up to n = 2^24.

Antti Karttunen, Table of n, a(n) for n = 1..10000
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A003415, A003557, A008683, A036785, A342001, A347232 [= a(A276086(n))].

Cf. also A300251, A300717, A347234, A347235, A347395.

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));
A346485(n) = sumdiv(n,d,moebius(n/d)*A342001(d));

a(n) = Sum_{d|n} A008683(n/d) * A342001(d).
Dirichlet g.f.: Product_{p prime} (1+p^(1-s)-p^(-s)) * Sum_{p prime} p^s/((p^s-1)*(p^s+p-1)). - Sebastian Karlsson, May 08 2022
Sum_{k=1..n} a(k) ~ c * A065464 * n^2 / 2, where c = Sum_{j>=2} (1/2 + (-1)^j * (Fibonacci(j) - 1/2))*PrimeZetaP(j) = 0.4526952873143153104685540856936425315834753528741817723313791528384... - Vaclav Kotesovec, Mar 04 2023

A294877 Lexicographically earliest such sequence a that a(i) = a(j) => A003557(i) = A003557(j) and A046523(i) = A046523(j), for all i, j.

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 5, 6, 4, 2, 7, 2, 4, 4, 8, 2, 9, 2, 7, 4, 4, 2, 10, 11, 4, 12, 7, 2, 13, 2, 14, 4, 4, 4, 15, 2, 4, 4, 10, 2, 13, 2, 7, 9, 4, 2, 16, 17, 18, 4, 7, 2, 19, 4, 10, 4, 4, 2, 20, 2, 4, 9, 21, 4, 13, 2, 7, 4, 13, 2, 22, 2, 4, 18, 7, 4, 13, 2, 16, 23, 4, 2, 20, 4, 4, 4, 10, 2, 24, 4, 7, 4, 4, 4, 25, 2, 26, 9, 27, 2, 13, 2, 10, 13, 4, 2, 28, 2, 13

Offset: 1

Antti Karttunen, Nov 11 2017

nonn

Restricted growth sequence transform of A291757, which means that this is the lexicographically least sequence a, such that for all i, j: a(i) = a(j) <=> A291757(i) = A291757(j) <=> A003557(i) = A003557(j) and A046523(i) = A046523(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.
Also the restricted growth sequence transform of A294876, Product_{d|n, d>1} prime(gcd(d,n/d)). (This was the original definition).
For all i, j:
  A295300(i) = A295300(j) => a(i) = a(j),
  A319347(i) = A319347(j) => a(i) = a(j),
  a(i) = a(j) => A055155(i) = A055155(j).

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

Cf. also A291751, A291757, A294876, A295300, A295888, A319347, A322021.

Cf. A000188, A055155, A294897, A295666, A322020 (a few of the matched sequences).

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; };
A294876(n) = { my(m=1); fordiv(n,d,if(d>1, m *= prime(gcd(d,n/d)))); m; };
v294877 = rgs_transform(vector(up_to,n,A294876(n)));
A294877(n) = v294877[n];

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
v294877 = rgs_transform(vector(up_to,n,[A003557(n),A046523(n)]));
A294877(n) = v294877[n]; \\ Antti Karttunen, Nov 28 2018

Name changed and comments added by Antti Karttunen, Nov 28 2018

A336550 Numbers k such that A007947(k) divides sigma(k) and A003557(k)-1 either divides A326143(k) [= A001065(k) - A007947(k)], or both are zero.

Original entry on oeis.org

6, 24, 28, 96, 120, 234, 384, 496, 936, 1536, 1638, 6144, 8128, 24576, 42588, 98304, 393216, 1089270, 1572864, 6291456, 25165824, 33550336, 100663296, 115048440, 402653184, 1185125760, 1610612736

Offset: 1

Antti Karttunen, Jul 28 2020

Numbers k such that gcd(sigma(k)-A007947(k), A007947(k)) == A007947(k) are those in A175200. These are equal to k such that gcd(A326143(k), A007947(k)) = gcd(sigma(k)-A007947(k)-k, A007947(k)) are equal to A007947(k).
Sequence is infinite because all numbers of the form 6*4^n (A002023) are present.
Question: Are there any odd terms?

Index entries for sequences where odd perfect numbers must occur, if they exist at all

Intersection of A175200 and A336552.
Cf. A007947, A326143, A336551.
Cf. A000396, A002023, A326145 (subsequences).
Cf. also A336641 for a similar construction.

A007947(n) = factorback(factorint(n)[, 1]);
isA336550(n) = { my(r=A007947(n), s=sigma(n), u=((n/r)-1)); (!(s%r) && (gcd(u,(s-r-n))==u)); };

A347129 a(n) = A347130(n) / A003557(n), where A347130 is the Dirichlet convolution of the identity function with the arithmetic derivative of n.

Original entry on oeis.org

0, 1, 1, 3, 1, 10, 1, 6, 3, 14, 1, 24, 1, 18, 16, 10, 1, 21, 1, 36, 20, 26, 1, 44, 3, 30, 6, 48, 1, 124, 1, 15, 28, 38, 24, 45, 1, 42, 32, 68, 1, 164, 1, 72, 39, 50, 1, 70, 3, 27, 40, 84, 1, 36, 32, 92, 44, 62, 1, 276, 1, 66, 51, 21, 36, 244, 1, 108, 52, 236, 1, 78, 1, 78, 33, 120, 36, 284, 1, 110, 10, 86, 1, 372, 44

Offset: 1

Antti Karttunen, Aug 23 2021

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10080
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A003415, A003557, A347126 [= a(A276086(n))], A347130.

Cf. also A342001, A347127, A347128.

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]));
A347130(n) = sumdiv(n,d,d*A003415(n/d));
A347129(n) = (A347130(n) / A003557(n));

a(n) = A347130(n) / A003557(n).

A347235 Dirichlet convolution of Euler phi with A342001, where A342001(n) = A003415(n) / A003557(n).

Original entry on oeis.org

0, 1, 1, 3, 1, 8, 1, 7, 4, 12, 1, 21, 1, 16, 14, 15, 1, 27, 1, 33, 18, 24, 1, 47, 6, 28, 13, 45, 1, 87, 1, 31, 26, 36, 22, 69, 1, 40, 30, 75, 1, 119, 1, 69, 51, 48, 1, 99, 8, 63, 38, 81, 1, 84, 30, 103, 42, 60, 1, 219, 1, 64, 67, 63, 34, 183, 1, 105, 50, 183, 1, 153, 1, 76, 75, 117, 34, 215, 1, 159, 40, 84, 1, 303, 42

Offset: 1

Antti Karttunen, Aug 26 2021

nonn

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

Cf. A000010, A003415, A003557, A342001.

Cf. also A346485, A347131, A347233, A347234, A347395.

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));
A347235(n) = sumdiv(n,d,eulerphi(d)*A342001(n/d));

A347235(n) = sum(k=1,n,A342001(gcd(n,k))); \\ (Slow) - Antti Karttunen, Sep 02 2021

a(n) = Sum_{d|n} A000010(n/d) * A342001(d).

a(n) = Sum_{k=1..n} A342001(gcd(n,k)). - Antti Karttunen, Sep 02 2021

A291756 Compound filter: a(n) = P(A003557(n), A000010(n)), where P(n,k) is sequence A000027 used as a pairing function.

Original entry on oeis.org

1, 1, 2, 5, 7, 2, 16, 25, 31, 7, 46, 12, 67, 16, 29, 113, 121, 31, 154, 38, 67, 46, 232, 59, 281, 67, 334, 80, 379, 29, 436, 481, 191, 121, 277, 142, 631, 154, 277, 175, 781, 67, 862, 212, 328, 232, 1036, 261, 1135, 281, 497, 302, 1327, 334, 781, 355, 631, 379, 1654, 138, 1771, 436, 706, 1985, 1129, 191, 2146, 530, 947, 277, 2416, 607, 2557, 631, 951, 668, 1771

Offset: 1

Antti Karttunen, Sep 10 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16385

Cf. A000010, A000027, A003557, A291757, A291764.

A000010(n) = eulerphi(n);
A003557(n) = n/factorback(factor(n)[, 1]); \\ This function from Charles R Greathouse IV, Nov 17 2014
A291756(n) = (1/2)*(2 + ((A003557(n)+A000010(n))^2) - A003557(n) - 3*A000010(n));

a(n) = (1/2)*(2 + ((A003557(n)+A000010(n))^2) - A003557(n) - 3*A000010(n)).

A291757 a(n) = (1/2)(2 + ((A003557(n)+A046523(n))^2) - A003557(n) - 3A046523(n)).

Original entry on oeis.org

1, 2, 2, 12, 2, 16, 2, 59, 18, 16, 2, 80, 2, 16, 16, 261, 2, 94, 2, 80, 16, 16, 2, 355, 33, 16, 129, 80, 2, 436, 2, 1097, 16, 16, 16, 826, 2, 16, 16, 355, 2, 436, 2, 80, 94, 16, 2, 1493, 52, 125, 16, 80, 2, 505, 16, 355, 16, 16, 2, 1832, 2, 16, 94, 4497, 16, 436, 2, 80, 16, 436, 2, 3415, 2, 16, 125, 80, 16, 436, 2, 1493, 888, 16, 2, 1832, 16, 16, 16, 355, 2

Offset: 1

Antti Karttunen, Sep 10 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16385

Cf. A000027, A003557, A046523, A291750, A291752, A291756, A291758, A294877 (rgs-transform), A319347.

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
A291757(n) = (1/2)*(2 + ((A003557(n)+A046523(n))^2) - A003557(n) - 3*A046523(n));

a(n) = (1/2)*(2 + ((A003557(n)+A046523(n))^2) - A003557(n) - 3*A046523(n)).

Name changed by Antti Karttunen, Nov 28 2018

A295886 Filter-sequence combining A003557(n) and A023900(n).

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Dec 03 2017

nonn

For all i, j: a(i) = a(j) => A295887(i) = A295887(j).

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

Cf. A003557, A023900, A295876, A295887.

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); };
A023900(n) = sumdivmult(n, d, d*moebius(d)); \\ This function from Charles R Greathouse IV, Sep 09 2014
v295876 = rgs_transform(vector(up_to,n,A023900(n)))
A295876(n) = v295876[n];
Anotsubmitted6(n) = (1/2)*(2 + ((A003557(n)+A295876(n))^2) - A003557(n) - 3*A295876(n));
write_to_bfile(1,rgs_transform(vector(up_to,n,Anotsubmitted6(n))),"b295886.txt");

Restricted growth sequence transform of sequence a(n) = (1/2)*(2 + ((A003557(n) + A295876(n))^2) - A003557(n) - 3*A295876(n)).

A322320 a(n) = gcd(A003557(n), A173557(n)).

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Dec 05 2018

nonn

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

Cf. A000010, A003557, A173557, A322321, A322351, A322352.

Cf. also A066086, A322318.

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
A173557(n) = factorback(apply(p -> p-1, factor(n)[, 1]));
A322320(n) = gcd(A173557(n), A003557(n));

a(n) = gcd(A003557(n), A173557(n)) = gcd(A322351(n), A322352(n)).

a(n) = A000010(n) / A322321(n).

A348494 a(n) = A348492(n) / A003557(n), where A348492 is the GCD of the arithmetic derivative (A003415) and Pillai's arithmetical function (A018804).

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Oct 21 2021

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A003415, A003557, A018804, A342001, A347128, A348492, A348500 [= a(A276086(n))].

Cf. also A348496.

Array[GCD[Total@ GCD[#1, Range[#1]], #1 Total[#2/#1 & @@@ #2]]/Apply[Times, Map[#1^(#2 - 1) & @@ # &, #2]] & @@ {#, FactorInteger[#]} &, 105] (* Michael De Vlieger, Oct 21 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]));
A018804(n) = sumdiv(n, d, n*eulerphi(d)/d); \\ From A018804
A348492(n) = gcd(A003415(n), A018804(n));
A348494(n) = (A348492(n)/A003557(n));

a(n) = gcd(A342001(n), A347128(n)).

a(n) = A348492(n) / A003557(n), where A348492(n) = gcd(A003415(n), A018804(n)).

cp's OEIS Frontend

A346485 Möbius transform of A342001, where A342001(n) = A003415(n)/A003557(n).

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A294877 Lexicographically earliest such sequence a that a(i) = a(j) => A003557(i) = A003557(j) and A046523(i) = A046523(j), for all i, j.

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A336550 Numbers k such that A007947(k) divides sigma(k) and A003557(k)-1 either divides A326143(k) [= A001065(k) - A007947(k)], or both are zero.

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A347129 a(n) = A347130(n) / A003557(n), where A347130 is the Dirichlet convolution of the identity function with the arithmetic derivative of n.

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A347235 Dirichlet convolution of Euler phi with A342001, where A342001(n) = A003415(n) / A003557(n).

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A291756 Compound filter: a(n) = P(A003557(n), A000010(n)), where P(n,k) is sequence A000027 used as a pairing function.

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A291757 a(n) = (1/2)*(2 + ((A003557(n)+A046523(n))^2) - A003557(n) - 3*A046523(n)).

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A295886 Filter-sequence combining A003557(n) and A023900(n).

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A322320 a(n) = gcd(A003557(n), A173557(n)).

A291757 a(n) = (1/2)(2 + ((A003557(n)+A046523(n))^2) - A003557(n) - 3A046523(n)).