A003557 -id:A003557 - cp's OEIS frontend

A351260 Lexicographically earliest infinite sequence such that a(i) = a(j) => A003415(i) = A003415(j), A003557(i) = A003557(j) and A046523(i) = A046523(j), for all i, j >= 1.

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, 42, 2, 43, 2, 44, 45, 46, 47, 48, 2, 49, 50, 51, 2, 52, 2, 53, 54, 55, 47, 56, 2, 57, 58, 59, 2, 60, 41, 61, 62, 63, 2, 64, 37, 65, 66, 67, 68, 69, 2, 70, 71, 72, 2, 73, 2, 74, 56

Offset: 1

Antti Karttunen, Feb 06 2022

Restricted growth sequence transform of the triplet [A003415(n), A003557(n), A046523(n)].
For all i, j >= 1:
  A305800(i) = A305800(j) => a(i) = a(j),
  a(i) = a(j) => A294877(i) = A294877(j),
  a(i) = a(j) => A300249(i) = A300249(j),
  a(i) = a(j) => A344025(i) = A344025(j).

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

Cf. A003415, A003557, A046523.
Cf. also A305800, A294877, A300249, A344025.
Differs from A300235, A305895 and A327931 for the first time at n=105, where a(105) = 56, while A300235(105) = A305895(105) = A327931(105) = 75.
Differs from A300249 for the first time at n=425, where a(425) = 299, while A300249(425) = 198.

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) = {my(fac); if(n<1, 0, fac=factor(n); sum(i=1, matsize(fac)[1], n*fac[i, 2]/fac[i, 1]))}; \\ From A003415
A003557(n) = (n/factorback(factorint(n)[, 1]));
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ From A046523
Aux351260(n) = [A003415(n), A003557(n), A046523(n)];
v351260 = rgs_transform(vector(up_to,n,Aux351260(n)));
A351260(n) = v351260[n];

A353520 Lexicographically earliest infinite sequence such that a(i) = a(j) => A003415(i) = A003415(j), A003557(i) = A003557(j) and A053669(i) = A053669(j), for all i, j >= 1.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Apr 25 2022

nonn

Restricted growth sequence transform of the triplet [A003415(n), A003557(n), A053669(n)].
For all i, j:
  A305801(i) = A305801(j) => a(i) = a(j),
  a(i) = a(j) => A007814(i) = A007814(j),
  a(i) = a(j) => A344025(i) = A344025(j).

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

Cf. A003415, A003557, A007814, A053669.

Cf. also A305801, A328470, A344025, A351260.

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) = {my(fac); if(n<1, 0, fac=factor(n); sum(i=1, matsize(fac)[1], n*fac[i, 2]/fac[i, 1]))}; \\ From A003415
A003557(n) = (n/factorback(factorint(n)[, 1]));
A053669(n) = forprime(p=2, , if(n%p, return(p))); \\ From A053669
Aux353520(n) = [A003415(n), A003557(n), A053669(n)];
v353520 = rgs_transform(vector(up_to,n,Aux353520(n)));
A353520(n) = v353520[n];

A353571 Prime-shifted variant of A342001: a(n) = A349905(n) / A003557(A003961(n)).

Original entry on oeis.org

0, 1, 1, 2, 1, 8, 1, 3, 2, 10, 1, 13, 1, 14, 12, 4, 1, 11, 1, 17, 16, 16, 1, 18, 2, 20, 3, 25, 1, 71, 1, 5, 18, 22, 18, 16, 1, 26, 22, 24, 1, 103, 1, 29, 19, 32, 1, 23, 2, 13, 24, 37, 1, 14, 20, 36, 28, 34, 1, 106, 1, 40, 27, 6, 24, 119, 1, 41, 34, 131, 1, 21, 1, 44, 17, 49, 24, 151, 1, 31, 4, 46, 1, 158, 26, 50, 36

Offset: 1

Antti Karttunen, Apr 27 2022

nonn

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

Cf. A003415, A003557, A003961, A342001, A349905, A353572.

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]));
A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); }; \\ From A003961
A353571(n) = { my(s=A003961(n)); (A003415(s)/A003557(s)); };

a(n) = A342001(A003961(n)) = A349905(n) / A003557(A003961(n)).

For all n >= 1, a(n) >= A342001(n).

A317935 Numerators of rational valued sequence whose Dirichlet convolution with itself yields A003557, n divided by largest squarefree divisor of n.

Original entry on oeis.org

1, 1, 1, 7, 1, 1, 1, 25, 11, 1, 1, 7, 1, 1, 1, 363, 1, 11, 1, 7, 1, 1, 1, 25, 19, 1, 61, 7, 1, 1, 1, 1335, 1, 1, 1, 77, 1, 1, 1, 25, 1, 1, 1, 7, 11, 1, 1, 363, 27, 19, 1, 7, 1, 61, 1, 25, 1, 1, 1, 7, 1, 1, 11, 9923, 1, 1, 1, 7, 1, 1, 1, 275, 1, 1, 19, 7, 1, 1, 1, 363, 1363, 1, 1, 7, 1, 1, 1, 25, 1, 11, 1, 7, 1, 1, 1, 1335, 1, 27, 11, 133, 1, 1, 1, 25, 1

Offset: 1

Antti Karttunen, Aug 12 2018

Multiplicative because A003557 is.

No negative terms among the first 2^20 terms. Is the sequence nonnegative?

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

Cf. A003557, A046644 (denominators).

Cf. also A300717, A300719, A318317.

A003557(n) = { my(f=factor(n)); for (i=1, #f~, f[i, 2]--); factorback(f); }; \\ From A003557
A317935aux(n) = if(1==n,n,(A003557(n)-sumdiv(n,d,if((d>1)&&(dA317935aux(d)*A317935aux(n/d),0)))/2);
A317935(n) = numerator(A317935aux(n));

a(n) = numerator of f(n), where f(1) = 1, f(n) = (1/2) * (A003557(n) - Sum_{d|n, d>1, d 1.

A322321 a(n) = lcm(A003557(n), A173557(n)).

Original entry on oeis.org

1, 1, 2, 2, 4, 2, 6, 4, 6, 4, 10, 2, 12, 6, 8, 8, 16, 6, 18, 4, 12, 10, 22, 4, 20, 12, 18, 6, 28, 8, 30, 16, 20, 16, 24, 6, 36, 18, 24, 4, 40, 12, 42, 10, 24, 22, 46, 8, 42, 20, 32, 12, 52, 18, 40, 12, 36, 28, 58, 8, 60, 30, 12, 32, 48, 20, 66, 16, 44, 24, 70, 12, 72, 36, 40, 18, 60, 24, 78, 8, 54, 40, 82, 12, 64, 42, 56, 20

Offset: 1

Antti Karttunen, Dec 05 2018

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences related to lcm's

Cf. A000010, A003557, A173557, A322320, A322351, A322352.

Cf. also A322319, A322359.

a[n_] := If[n == 1, 1, Module[{f=FactorInteger[n]}, LCM[ 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]));
A322321(n) = lcm(A003557(n), A173557(n));

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

a(n) = A000010(n) / A322320(n).

A323368 Lexicographically earliest sequence such that a(i) = a(j) => A000035(i) = A000035(j) and 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, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 15, 23, 24, 25, 26, 27, 28, 29, 21, 30, 31, 32, 31, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 29, 31, 43, 44, 45, 46, 47, 48, 49, 46, 50, 51, 52, 53, 54, 55, 39, 56, 57, 58, 59, 60, 61, 62, 59, 46, 63, 64, 65, 66, 67, 62, 68, 51, 69, 70, 71, 58, 72, 73, 74, 75, 76, 77, 78, 79, 54, 80, 59, 75, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90

Offset: 1

Antti Karttunen, Jan 12 2019

nonn

For all i, j:
  a(i) = a(j) => A007814(i) = A007814(j),
  a(i) = a(j) => A291751(i) = A291751(j),
  a(i) = a(j) => A296089(i) = A296089(j),
  a(i) = a(j) => A323238(i) = A323238(j).

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

Cf. A000035, A003557, A048250, A291750, A291751, A323238, A323369.
Differs from A296089 for the first time at n=103, where a(103)=88, while A296089(103)=56.
Cf. also A323366.

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); };
A048250(n) = factorback(apply(p -> p+1,factor(n)[,1]));
v323368 = rgs_transform(vector(up_to, n, [(n%2), A003557(n), A048250(n)]));
A323368(n) = v323368[n];

A323372 Lexicographically earliest sequence such that for all i, j, a(i) = a(j) => A003557(i) = A003557(j) and A323363(i) = A323363(j).

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, 29, 65, 66, 67, 68, 69, 58, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 79

Offset: 1

Antti Karttunen, Jan 13 2019

nonn

Restricted growth sequence transform of ordered pair [A003557(n), A323363(n)].
For all i, j:
  a(i) = a(j) => A291751(i) = A291751(j),
  a(i) = a(j) => A323364(i) = A323364(j).

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

Cf. A001615, A003557, A291750, A291751, A323363, A323364.

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; };
DirInverse(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = -sumdiv(n, d, if(dA001615(n) = (n * sumdivmult(n, d, issquarefree(d)/d)); \\ From A001615
v323363 = DirInverse(vector(up_to,n,A001615(n)));
A323363(n) = v323363[n];
A003557(n) = { my(f=factor(n)); for(i=1, #f~, f[i, 2] = f[i, 2]-1); factorback(f); };
v323372 = rgs_transform(vector(up_to, n, [A003557(n), A323363(n)]));
A323372(n) = v323372[n];

A323405 Lexicographically earliest sequence such that for all i, j, a(i) = a(j) => f(i) = f(j) where f(n) = [A003557(n), A023900(n), A063994(n)] for all other numbers, except f(n) = 0 for odd primes.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Jan 13 2019

nonn

For all i, j:
  A305801(i) = A305801(j) => a(i) = a(j),
  a(i) = a(j) => A323371(i) = A323371(j),
  a(i) = a(j) => A247074(i) = A247074(j).

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

Cf. A000010, A003557, A023900, A063994, A247074, A305801, A323371, A323404.
Differs from A323370 for the first time at n=78, where a(78) = 58, while A323370(78) = 52.
Cf. also A323374.

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] = max(0,f[i, 2]-1)); factorback(f); };
A023900(n) = sumdivmult(n, d, d*moebius(d)); \\ From A023900
A063994(n) = { my(f=factor(n)[, 1]); prod(i=1, #f, gcd(f[i]-1, n-1)); }; \\ From A063994
Aux323405(n) = if((n>2)&&isprime(n),0,[A003557(n), A023900(n), A063994(n)]);
v323405 = rgs_transform(vector(up_to, n, Aux323405(n)));
A323405(n) = v323405[n];

A335341 Sum of divisors of A003557(n).

Original entry on oeis.org

1, 1, 1, 3, 1, 1, 1, 7, 4, 1, 1, 3, 1, 1, 1, 15, 1, 4, 1, 3, 1, 1, 1, 7, 6, 1, 13, 3, 1, 1, 1, 31, 1, 1, 1, 12, 1, 1, 1, 7, 1, 1, 1, 3, 4, 1, 1, 15, 8, 6, 1, 3, 1, 13, 1, 7, 1, 1, 1, 3, 1, 1, 4, 63, 1, 1, 1, 3, 1, 1, 1, 28, 1, 1, 6, 3, 1, 1, 1

Offset: 1

R. J. Mathar, Jun 02 2020

The sum of the divisors d of n such that n/d is a coreful divisor of n (a coreful divisor of n is a divisor with the same squarefree kernel as n). The number of these divisors is A005361(n). - Amiram Eldar, Jun 30 2023

Antti Karttunen, Table of n, a(n) for n = 1..16383
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A000203, A003557, A005361 (number of divisors of A003557), A336567.

Cf. A047994, A173557.

A335341 := proc(n)
    local a,pe,p,e ;
    a := 1;
    for pe in ifactors(n)[2] do
        p := op(1,pe) ;
        e := op(2,pe) ;
        if e > 1 then
            a := a*(p^e-1)/(p-1) ;
        end if;
    end do:
    a ;
end proc:

f[p_, e_] := (p^e-1)/(p-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 26 2020 *)

a(n) = sigma(n/factorback(factor(n)[, 1])); \\ Michel Marcus, Jun 02 2020

a(n) = A000203(A003557(n)).
Multiplicative with a(p^1)=1 and a(p^e) = (p^e-1)/(p-1) if e>1.
A057723(n) = A007947(n)*a(n).
a(n) = 1 iff n in A005117.
a(n) = A336567(n) + A003557(n). - Antti Karttunen, Jul 28 2020
Dirichlet g.f.: zeta(s-1) * zeta(s) * Product_{p prime} (1 - 1/p^(s-1) + 1/p^(2*s-1)). - Amiram Eldar, Sep 09 2023
a(n) = A047994(n)/A173557(n). - Ridouane Oudra, Oct 30 2023

A341998 Arithmetic derivative of n divided by its largest squarefree divisor: a(n) = A003557(A003415(n)).

Original entry on oeis.org

1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 8, 1, 3, 4, 16, 1, 1, 1, 4, 1, 1, 1, 2, 1, 1, 9, 16, 1, 1, 1, 8, 1, 1, 2, 2, 1, 1, 8, 2, 1, 1, 1, 8, 1, 5, 1, 8, 1, 3, 2, 4, 1, 27, 8, 2, 1, 1, 1, 2, 1, 1, 1, 32, 3, 1, 1, 12, 1, 1, 1, 2, 1, 1, 1, 8, 3, 1, 1, 8, 18, 1, 1, 2, 1, 3, 16, 2, 1, 1, 2, 16, 1, 7, 4, 8, 1, 1, 5, 2, 1, 1, 1, 2, 1

Offset: 2

Antti Karttunen, Feb 28 2021

nonn

Antti Karttunen, Table of n, a(n) for n = 2..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 2..65539

Cf. A003415, A003557, A341994, A341997, A342001.

Cf. A328393 (positions of ones), A328303 (after its two initial terms, gives the positions of terms > 1).

A003415(n) = {my(fac); if(n<1, 0, fac=factor(n); sum(i=1, matsize(fac)[1], n*fac[i, 2]/fac[i, 1]))}; \\ From A003415
A003557(n) = { my(f=factor(n)); for (i=1, #f~, f[i, 2] = f[i, 2]-1); factorback(f); }; \\ From A003557
A341998(n) = if(n<=1,1,A003557(A003415(n)));

a(n) = A003557(A003415(n)).

cp's OEIS Frontend

A351260 Lexicographically earliest infinite sequence such that a(i) = a(j) => A003415(i) = A003415(j), A003557(i) = A003557(j) and A046523(i) = A046523(j), for all i, j >= 1.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

A353520 Lexicographically earliest infinite sequence such that a(i) = a(j) => A003415(i) = A003415(j), A003557(i) = A003557(j) and A053669(i) = A053669(j), for all i, j >= 1.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

A353571 Prime-shifted variant of A342001: a(n) = A349905(n) / A003557(A003961(n)).

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

A317935 Numerators of rational valued sequence whose Dirichlet convolution with itself yields A003557, n divided by largest squarefree divisor of n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Formula

A322321 a(n) = lcm(A003557(n), A173557(n)).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A323368 Lexicographically earliest sequence such that a(i) = a(j) => A000035(i) = A000035(j) and A003557(i) = A003557(j) and A048250(i) = A048250(j), for all i, j.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

A323372 Lexicographically earliest sequence such that for all i, j, a(i) = a(j) => A003557(i) = A003557(j) and A323363(i) = A323363(j).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

A323405 Lexicographically earliest sequence such that for all i, j, a(i) = a(j) => f(i) = f(j) where f(n) = [A003557(n), A023900(n), A063994(n)] for all other numbers, except f(n) = 0 for odd primes.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

A335341 Sum of divisors of A003557(n).

Views

Author

Keywords

Comments