A294876 -id:A294876 - cp's OEIS frontend

A293514 a(n) = Product_{d|n, d>1} prime(A286561(n,d)), where A286561(n,d) gives the highest exponent of d dividing n.

1, 2, 2, 6, 2, 8, 2, 20, 6, 8, 2, 48, 2, 8, 8, 84, 2, 48, 2, 48, 8, 8, 2, 320, 6, 8, 20, 48, 2, 128, 2, 264, 8, 8, 8, 864, 2, 8, 8, 320, 2, 128, 2, 48, 48, 8, 2, 2688, 6, 48, 8, 48, 2, 320, 8, 320, 8, 8, 2, 3072, 2, 8, 48, 1560, 8, 128, 2, 48, 8, 128, 2, 11520, 2, 8, 48, 48, 8, 128, 2, 2688, 84, 8, 2, 3072, 8, 8, 8, 320

Offset: 1

Antti Karttunen, Nov 11 2017

nonn

			For n = 24, its divisors larger than one are: 2, 3, 4, 6, 8, 12, 24. Only 2 has valuation > 1, namely A286561(24,2) = 3 (as 2^3 divides 24), while the other six have valuation 1. Thus a(24) = prime(1)^6 * prime(3) = 64*5 = 320.
For n = 64, its divisors larger than one are: 2, 4, 8, 16, 32, 64. We see that 2^6 = 4^3 = 8^2 = 64, while valuation of the last three 16, 32 and 64 is 1. Thus a(64) = prime(1)^3 * prime(2) * prime(3) * prime(6) = 2^3 * 3 * 5 * 13 = 1560.

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences computed from exponents in factorization of n

Cf. A000040, A286561.

Cf. also A293515, A293524, A294876, A293442.

A293514(n) = { my(m=1); fordiv(n,d,if(d>1, m *= prime(valuation(n,d)))); m; };

a(n) = Product_{d|n, d>1} A000040(A286561(n,d)).
Other identities. For all n >= 1:
A001222(a(n)) = A032741(n).
A007814(a(n)) = A056595(n) [See A046951.]
1+A056239(a(n)) = A169594(n).
A064989(a(n)) = A293515(n).

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

A295666 a(n) = Product_{d|n, gcd(d,n/d) is prime} gcd(d,n/d).

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 4, 3, 1, 1, 4, 1, 1, 1, 4, 1, 9, 1, 4, 1, 1, 1, 16, 5, 1, 9, 4, 1, 1, 1, 4, 1, 1, 1, 36, 1, 1, 1, 16, 1, 1, 1, 4, 9, 1, 1, 16, 7, 25, 1, 4, 1, 81, 1, 16, 1, 1, 1, 16, 1, 1, 9, 4, 1, 1, 1, 4, 1, 1, 1, 144, 1, 1, 25, 4, 1, 1, 1, 16, 9, 1, 1, 16, 1, 1, 1, 16, 1, 81, 1, 4, 1, 1, 1, 16, 1, 49, 9, 100, 1, 1, 1, 16, 1

Offset: 1

Antti Karttunen, Nov 26 2017

nonn

			For n = 12, with divisors 1, 2, 3, 4, 6, 12, we select from the sequence gcd(1,12/1), gcd(2,12/2), gcd(3,12/3), gcd(4,12/4), gcd(6,12/6), gcd(12,12/12) = 1, 2, 1, 1, 2, 1 only those that are primes, namely the two 2's, and form their product, thus a(12) = 2*2 = 4.
For n = 100, with divisors 1, 2, 4, 5, 10, 20, 25, 50, 100, we select from the sequence gcd(1,100/1), gcd(2,100/2), gcd(4,100/4), gcd(5,100/5), gcd(10,100/10), gcd(20,100/20), gcd(25,100/25), gcd(50,100/50), gcd(100,100/100) = 1, 2, 1, 5, 10, 5, 1, 2, 1, only those that are primes, namely 2, 5, 5 and 2, thus a(100) = 2*5*5*2 = 100.

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

Cf. A003557, A056170, A010051, A294876, A294895, A295665.

a[n_]:=Product[GCD[i,n/i]^Boole[PrimeQ[GCD[i,n/i]]],{i,Divisors[n]}]; Array[a,105] (* Stefano Spezia, Feb 20 2024 *)

A295666(n) = { my(m=1,p); fordiv(n, d, p = gcd(d, n/d); if(isprime(p), m *= p)); m; };

a(n) = Product_{d|n} gcd(d,n/d)^A010051(gcd(d,n/d)).
a(n) = A295665(A294876(n)).
Other identities. For all n >= 1:
A001221(a(n)) = A056170(n) = A001221(A003557(n)).

A294895 a(n) = Product_{d|n, gcd(d,n/d)>1} prime(gcd(d,n/d)-1).

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 4, 3, 1, 1, 4, 1, 1, 1, 20, 1, 9, 1, 4, 1, 1, 1, 16, 7, 1, 9, 4, 1, 1, 1, 100, 1, 1, 1, 396, 1, 1, 1, 16, 1, 1, 1, 4, 9, 1, 1, 400, 13, 49, 1, 4, 1, 81, 1, 16, 1, 1, 1, 16, 1, 1, 9, 1700, 1, 1, 1, 4, 1, 1, 1, 17424, 1, 1, 49, 4, 1, 1, 1, 400, 171, 1, 1, 16, 1, 1, 1, 16, 1, 81, 1, 4, 1, 1, 1, 10000, 1, 169, 9, 4508, 1, 1, 1, 16, 1

Offset: 1

Antti Karttunen, Nov 21 2017

nonn

For all i, j: a(i) = a(j) => A294897(i) = A294897(j).

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

Cf. A005117 (the positions of ones).

Cf. A294876, A294897.

A294895[n_] := Times @@ Prime[Select[Map[GCD[#, n/#] &, Divisors[n]], #>1 &] - 1];
Array[A294895, 100] (* Paolo Xausa, Feb 22 2024 *)

A294895(n) = { my(m=1); fordiv(n,d,if(gcd(d,n/d)>1, m *= prime(gcd(d,n/d)-1))); m; };

a(n) = Product_{d|n} A008578(gcd(d,n/d)).
a(n) = A064989(A294876(n)).
For n >= 1, A001222(a(n)) = A048105(n).
For n > 1, 1+A061395(a(n)) = A000188(n).

A295665 Fully multiplicative with a(prime(m)) = prime(k) when m = prime(k), and a(prime(m)) = 1 when m is not a prime.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Nov 26 2017

The number of applications to reach 1 is A322027(n). Positions of first appearances are A076610. - Gus Wiseman, Jan 17 2020

			For n = 360 = 2^3 * 3^2 * 5 = prime(1)^3 * prime(2)^2 * prime(3), 1 is not a prime, but 2 and 3 are, thus a(360) = 2^2 * 3 = 12.

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences computed from indices in prime factorization

Cf. A000040, A000720, A010051, A295666.
Cf. also A003963, A257538.
Positions of 1's are A320628.
Positions of terms > 1 are A331386.
Primes of prime index are A006450.
Primes of nonprime index are A007821.
Products of primes of prime index are A076610.
Products of primes of nonprime index are A320628.
The number of prime prime indices is A257994.
The number of nonprime prime indices is A330944.
Numbers whose prime indices are not all prime are A330945.
Cf. A001222, A007097, A018252, A056239, A112798, A320633, A322027, A330947, A330948.

Table[Times@@Cases[FactorInteger[n],{p_?(PrimeQ[PrimePi[#]]&),k_}:>PrimePi[p]^k],{n,40}] (* Gus Wiseman, Jan 17 2020 *)

(definec (A295665 n) (if (= 1 n) 1 (let ((k (A055396 n))) (* (if (zero? (A010051 k)) 1 k) (A295665 (A032742 n))))))

Multiplicative with a(p^e) = A000720(p)^(e*A010051(A000720(p))).
a(1) = 1; for n > 1, if A055396(n) is a prime, then a(n) = A055396(n) * a(A032742(n)), otherwise a(n) = a(A032742(n)).
Other identities. For all n >= 1:
a(A006450(n)) = A000040(n).
a(A007097(n)) = A007097(n-1).
a(A294876(n)) = A295666(n).

A322020 a(n) = Product_{d|n, gcd(d,n/d) is a prime power} gcd(d,n/d).

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 4, 3, 1, 1, 4, 1, 1, 1, 16, 1, 9, 1, 4, 1, 1, 1, 16, 5, 1, 9, 4, 1, 1, 1, 64, 1, 1, 1, 36, 1, 1, 1, 16, 1, 1, 1, 4, 9, 1, 1, 256, 7, 25, 1, 4, 1, 81, 1, 16, 1, 1, 1, 16, 1, 1, 9, 512, 1, 1, 1, 4, 1, 1, 1, 144, 1, 1, 25, 4, 1, 1, 1, 256, 81, 1, 1, 16, 1, 1, 1, 16, 1, 81, 1, 4, 1, 1, 1, 4096, 1, 49, 9, 100, 1, 1, 1, 16, 1

Offset: 1

Antti Karttunen, Nov 28 2018

nonn

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

Cf. A069513, A294876, A295666.

A322020(n) = { my(m=1,p); fordiv(n, d, p = gcd(d, n/d); if(isprimepower(p), m *= p)); m; };

a(n) = Product_{d|n} gcd(d,n/d)^A069513(gcd(d,n/d)).

cp's OEIS Frontend

A293514 a(n) = Product_{d|n, d>1} prime(A286561(n,d)), where A286561(n,d) gives the highest exponent of d dividing 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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A295666 a(n) = Product_{d|n, gcd(d,n/d) is prime} gcd(d,n/d).

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A294895 a(n) = Product_{d|n, gcd(d,n/d)>1} prime(gcd(d,n/d)-1).

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A295665 Fully multiplicative with a(prime(m)) = prime(k) when m = prime(k), and a(prime(m)) = 1 when m is not a prime.

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A322020 a(n) = Product_{d|n, gcd(d,n/d) is a prime power} gcd(d,n/d).

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