A003557 -id:A003557 - cp's OEIS frontend

A322352 a(n) = max(A003557(n), A173557(n)).

1, 1, 2, 2, 4, 2, 6, 4, 3, 4, 10, 2, 12, 6, 8, 8, 16, 3, 18, 4, 12, 10, 22, 4, 5, 12, 9, 6, 28, 8, 30, 16, 20, 16, 24, 6, 36, 18, 24, 4, 40, 12, 42, 10, 8, 22, 46, 8, 7, 5, 32, 12, 52, 9, 40, 6, 36, 28, 58, 8, 60, 30, 12, 32, 48, 20, 66, 16, 44, 24, 70, 12, 72, 36, 8, 18, 60, 24, 78, 8, 27, 40, 82, 12, 64, 42, 56, 10, 88, 8

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, A322351, A322355.

a[n_] := If[n == 1, 1, Module[{f=FactorInteger[n]}, Max[ 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]));
A322352(n) = max(A003557(n), A173557(n));

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

a(n) = A000010(n) / A322351(n).

A323370 Lexicographically earliest sequence such that for all i, j, a(i) = a(j) => f(i) = f(j) where f(n) = [A000035(n), A003557(n), A023900(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, 52, 3, 58, 59, 60, 3, 61, 62, 63, 64, 65, 3, 66, 67, 68, 57, 69, 67, 70, 3, 71, 72, 73, 3, 74, 3, 75, 76

Offset: 1

Antti Karttunen, Jan 13 2019

nonn

Restricted growth sequence transform of function f, defined as f(n) = 0 when n is an odd prime, and f(n) = [A000035(n), A003557(n), A023900(n)] for all other numbers.
For all i, j:
  A305801(i) = A305801(j) => a(i) = a(j),
  a(i) = a(j) => A323367(i) = A323367(j),
  a(i) = a(j) => A323371(i) = A323371(j).

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

Cf. A000035, A003557, A023900, A092248, A295886, A305801, A319340, A322587, A323367, A323371.

Differs from A323405 for the first time at n=78, where a(78) = 52, while A323405(78) = 58.

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
Aux323370(n) = if((n>2)&&isprime(n),0,[(n%2), A003557(n), A023900(n)]);
v323370 = rgs_transform(vector(up_to, n, Aux323370(n)));
A323370(n) = v323370[n];

A336551 a(n) = A003557(n) - 1.

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 0, 3, 2, 0, 0, 1, 0, 0, 0, 7, 0, 2, 0, 1, 0, 0, 0, 3, 4, 0, 8, 1, 0, 0, 0, 15, 0, 0, 0, 5, 0, 0, 0, 3, 0, 0, 0, 1, 2, 0, 0, 7, 6, 4, 0, 1, 0, 8, 0, 3, 0, 0, 0, 1, 0, 0, 2, 31, 0, 0, 0, 1, 0, 0, 0, 11, 0, 0, 4, 1, 0, 0, 0, 7, 26, 0, 0, 1, 0, 0, 0, 3, 0, 2, 0, 1, 0, 0, 0, 15, 0, 6, 2, 9, 0, 0, 0, 3, 0

Offset: 1

Antti Karttunen, Jul 28 2020

nonn

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

Cf. A003557, A007947, A066503.

A336551(n) = { my(f=factor(n)); for(i=1, #f~, f[i, 2] = f[i, 2]-1); (factorback(f)-1); };

a(n) = A066503(n) / A007947(n).

A347128 a(n) = A018804(n) / A003557(n), where A018804 is Pillai's arithmetical function.

Original entry on oeis.org

1, 3, 5, 4, 9, 15, 13, 5, 7, 27, 21, 20, 25, 39, 45, 6, 33, 21, 37, 36, 65, 63, 45, 25, 13, 75, 9, 52, 57, 135, 61, 7, 105, 99, 117, 28, 73, 111, 125, 45, 81, 195, 85, 84, 63, 135, 93, 30, 19, 39, 165, 100, 105, 27, 189, 65, 185, 171, 117, 180, 121, 183, 91, 8, 225, 315, 133, 132, 225, 351, 141, 35, 145, 219, 65, 148

Offset: 1

Antti Karttunen, Aug 23 2021

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

Cf. A003557, A018804, A348494 [= gcd(a(n), A342001(n))], A348496 [= gcd(a(n), A347129(n))].

Cf. also A173557, A347127.

f[p_, e_] := (e*(p - 1)/p + 1)*p^e; A347128[n_] := (Times @@ (f @@@ FactorInteger[n]))/(n/Times @@ (First[Transpose[FactorInteger[n]]]));Table[A347128[n], {n, 1, 76}] (* Robert P. P. McKone, Aug 23 2021, after Amiram Eldar *)

A347128(n) = { my(f=factor(n)); prod(i=1, #f~, ((f[i, 1]-1)*f[i, 2] + f[i, 1])); };

Multiplicative with a(p^e) = ((p-1)*e + p).

a(n) = A018804(n) / A003557(n).

A349340 Dirichlet inverse of A003557, where A003557 is multiplicative with a(p^e) = p^(e-1).

Original entry on oeis.org

1, -1, -1, -1, -1, 1, -1, -1, -2, 1, -1, 1, -1, 1, 1, -1, -1, 2, -1, 1, 1, 1, -1, 1, -4, 1, -4, 1, -1, -1, -1, -1, 1, 1, 1, 2, -1, 1, 1, 1, -1, -1, -1, 1, 2, 1, -1, 1, -6, 4, 1, 1, -1, 4, 1, 1, 1, 1, -1, -1, -1, 1, 2, -1, 1, -1, -1, 1, 1, -1, -1, 2, -1, 1, 4, 1, 1, -1, -1, 1, -8, 1, -1, -1, 1, 1, 1, 1, -1, -2, 1

Offset: 1

Antti Karttunen, Nov 18 2021

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

Cf. A003557, A076479, A326297 (absolute values).

Cf. also A325126, A349350, A349619.

f[p_, e_] := -(p - 1)^(e - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 18 2021 *)

A003557(n) = (n/factorback(factorint(n)[, 1]));
memoA349340 = Map();
A349340(n) = if(1==n,1,my(v); if(mapisdefined(memoA349340,n,&v), v, v = -sumdiv(n,d,if(dA003557(n/d)*A349340(d),0)); mapput(memoA349340,n,v); (v)));

A349340(n) = { my(f=factor(n)); prod(i=1, #f~, -((f[i,1]-1)^(f[i,2]-1))); };

Multiplicative with a(p^e) = -((p-1)^(e-1)).
a(n) = A076479(n) * A326297(n).
a(1) = 1; a(n) = -Sum_{d|n, d < n} A003557(n/d) * a(d).

A349396 Dirichlet convolution of A342001 ({arithmetic derivative of n}/A003557(n)) with A055615 (Dirichlet inverse of n).

Original entry on oeis.org

0, 1, 1, 0, 1, 0, 1, -1, -1, 0, 1, -2, 1, 0, 0, -2, 1, -6, 1, -2, 0, 0, 1, -2, -3, 0, -3, -2, 1, 0, 1, -3, 0, 0, 0, 2, 1, 0, 0, -2, 1, 0, 1, -2, -6, 0, 1, -2, -5, -20, 0, -2, 1, -6, 0, -2, 0, 0, 1, 0, 1, 0, -6, -4, 0, 0, 1, -2, 0, 0, 1, 8, 1, 0, -20, -2, 0, 0, 1, -2, -5, 0, 1, 0, 0, 0, 0, -2, 1, 0, 0, -2, 0, 0, 0

Offset: 1

Antti Karttunen, Nov 18 2021

sign

Dirichlet convolution of this sequence with A000010 (Euler phi) is A346485.

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

Cf. A003415, A003557, A055615, A342001.
Cf. A346485, A347234,  A347235, A347395, A347954, A347959, A347961, A347963 for Dirichlet convolutions of A342001 with other sequences.
Cf. also A349394.

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));
A055615(n) = (n*moebius(n));
A349396(n) = sumdiv(n,d,A342001(n/d)*A055615(d));

a(n) = Sum_{d|n} A055615(d) * A342001(n/d).

A353572 Shifted variant of A342002: a(n) = A353571(A276086(n)), where A353571(x) = A003415(A003961(x)) / A003557(A003961(x)) and A276086 is the primorial base exp-function.

Original entry on oeis.org

0, 1, 1, 8, 2, 11, 1, 10, 12, 71, 19, 92, 2, 13, 17, 86, 24, 107, 3, 16, 22, 101, 29, 122, 4, 19, 27, 116, 34, 137, 1, 14, 16, 103, 27, 136, 18, 131, 167, 886, 244, 1117, 29, 164, 222, 1051, 299, 1282, 40, 197, 277, 1216, 354, 1447, 51, 230, 332, 1381, 409, 1612, 2, 17, 21, 118, 32, 151, 25, 152, 202, 991, 279

Offset: 0

Antti Karttunen, Apr 27 2022

Cf. A003415, A003557, A003961, A276154, A342002, A349905, A353571, A353573 [= gcd(a(n), A342002(n))], A353574.

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
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A353571(n) = { my(s=A003961(n)); (A003415(s)/A003557(s)); };
A353572(n) = A353571(A276086(n));

a(n) = A353571(A276086(n)).
a(n) = A342002(A276154(n)).
For all n >= 0, a(n) >= A342002(n).

A371088 Numbers k such that A003557(k) divides A276086(k), where A003557(k) is k divided by its largest squarefree divisor, and A276086 is the primorial base exp-function.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 9, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 25, 26, 29, 30, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 45, 46, 47, 49, 50, 51, 53, 55, 57, 58, 59, 61, 62, 63, 65, 66, 67, 69, 70, 71, 73, 74, 75, 77, 78, 79, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97, 98, 99, 101, 102, 103, 105, 106, 107, 109, 110, 111

Offset: 1

Antti Karttunen, Mar 12 2024

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences related to primorial base

Union of A005117 and A371086.
Cf. A003557, A276086, A371087 (characteristic function), A371089 (complement).
Cf. A358221 (subsequence).

PARI
```
\\ See A371087.
```

A319347 Filter sequence combining A000035(n) (parity of n), A003557(n), and A046523(n) (prime signature of n).

Original entry on oeis.org

1, 2, 3, 4, 3, 5, 3, 6, 7, 5, 3, 8, 3, 5, 9, 10, 3, 11, 3, 8, 9, 5, 3, 12, 13, 5, 14, 8, 3, 15, 3, 16, 9, 5, 9, 17, 3, 5, 9, 12, 3, 15, 3, 8, 18, 5, 3, 19, 20, 21, 9, 8, 3, 22, 9, 12, 9, 5, 3, 23, 3, 5, 18, 24, 9, 15, 3, 8, 9, 15, 3, 25, 3, 5, 26, 8, 9, 15, 3, 19, 27, 5, 3, 23, 9, 5, 9, 12, 3, 28, 9, 8, 9, 5, 9, 29, 3, 30, 18, 31, 3, 15, 3, 12, 32

Offset: 1

Antti Karttunen, Sep 24 2018

nonn

Restricted growth sequence transform of triple [A000035(n), A003557(n), A046523(n)], or equally, of triple [A007814(n), A003557(n), A046523(n)], or equally, of ordered pair [A000035(n), A291757(n)].

For all i, j: A305801(i) = A305801(j) => a(i) = a(j) => A305891(i) = A305891(j).

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

Cf. A000035, A003557, A007814, A046523, A291757, A305801, A305891.

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); };
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p=0); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };
v319347 = rgs_transform(vector(up_to,n,[A003557(n),(n%2),A046523(n)]));
A319347(n) = v319347[n];

A323163 Greatest common divisor of product (1+(p^e)) and product p^(e-1), where p ranges over prime factors of n, with e corresponding exponent; a(n) = gcd(A034448(n), A003557(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, 2, 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, 6, 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, 10, 1, 1, 1, 2, 1

Offset: 1

Antti Karttunen, Jan 09 2019

nonn

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

Cf. A003557, A034448, A322318, A323159.

Differs from A062760 for the first time at n=36, where a(36) = 2, while A062760(36) = 1.

A003557(n) = { my(f=factor(n)); for(i=1, #f~, f[i, 2] = f[i, 2]-1); factorback(f); };
A034448(n) = { my(f=factorint(n)); prod(k=1, #f~, 1+(f[k, 1]^f[k, 2])); }; \\ After code in A034448
A323163(n) = gcd(A003557(n), A034448(n));

a(n) = gcd(A003557(n), A034448(n)).

cp's OEIS Frontend

A322352 a(n) = max(A003557(n), A173557(n)).

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A323370 Lexicographically earliest sequence such that for all i, j, a(i) = a(j) => f(i) = f(j) where f(n) = [A000035(n), A003557(n), A023900(n)] for all other numbers, except f(n) = 0 for odd primes.

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A336551 a(n) = A003557(n) - 1.

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A347128 a(n) = A018804(n) / A003557(n), where A018804 is Pillai's arithmetical function.

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A349340 Dirichlet inverse of A003557, where A003557 is multiplicative with a(p^e) = p^(e-1).

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A349396 Dirichlet convolution of A342001 ({arithmetic derivative of n}/A003557(n)) with A055615 (Dirichlet inverse of n).

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A353572 Shifted variant of A342002: a(n) = A353571(A276086(n)), where A353571(x) = A003415(A003961(x)) / A003557(A003961(x)) and A276086 is the primorial base exp-function.

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A371088 Numbers k such that A003557(k) divides A276086(k), where A003557(k) is k divided by its largest squarefree divisor, and A276086 is the primorial base exp-function.

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A319347 Filter sequence combining A000035(n) (parity of n), A003557(n), and A046523(n) (prime signature of n).

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