A294895 -id:A294895 - cp's OEIS frontend

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

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)).

A294897 a(n) = Product_{d|n, gcd(d,n/d)>1} prime(A101296(gcd(d,n/d))-1).

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 4, 2, 1, 1, 4, 1, 1, 1, 12, 1, 4, 1, 4, 1, 1, 1, 16, 2, 1, 4, 4, 1, 1, 1, 36, 1, 1, 1, 80, 1, 1, 1, 16, 1, 1, 1, 4, 4, 1, 1, 144, 2, 4, 1, 4, 1, 16, 1, 16, 1, 1, 1, 16, 1, 1, 4, 252, 1, 1, 1, 4, 1, 1, 1, 1600, 1, 1, 4, 4, 1, 1, 1, 144, 12, 1, 1, 16, 1, 1, 1, 16, 1, 16, 1, 4, 1, 1, 1, 1296, 1, 4, 4, 80, 1, 1, 1, 16, 1

Offset: 1

Antti Karttunen, Nov 21 2017

nonn

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

Cf. A005117 (the positions of ones).
Cf. A101296, A294895.
Cf. also A292258 (A292259), A293515, A294875 for similar filter sequences.

up_to = 16384
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; };
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ This function from Charles R Greathouse IV, Aug 17 2011
v101296 = rgs_transform(vector(up_to, n, A046523(n)));
A101296(n) = v101296[n];
A294897(n) = { my(m=1); fordiv(n,d,if(gcd(d,n/d)>1, m *= prime(A101296(gcd(d,n/d))-1))); m; };

a(n) = Product_{d|n} A008578(A101296(gcd(d,n/d))).

For n >= 1, A001222(a(n)) = A048105(n).

A295879 Multiplicative with a(p) = 1, a(p^e) = prime(e-1) if e > 1.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 2, 1, 1, 1, 5, 1, 2, 1, 2, 1, 1, 1, 3, 2, 1, 3, 2, 1, 1, 1, 7, 1, 1, 1, 4, 1, 1, 1, 3, 1, 1, 1, 2, 2, 1, 1, 5, 2, 2, 1, 2, 1, 3, 1, 3, 1, 1, 1, 2, 1, 1, 2, 11, 1, 1, 1, 2, 1, 1, 1, 6, 1, 1, 2, 2, 1, 1, 1, 5, 5, 1, 1, 2, 1, 1, 1, 3, 1, 2, 1, 2, 1, 1, 1, 7, 1, 2, 2, 4, 1, 1, 1, 3, 1, 1, 1, 6, 1, 1, 1, 5, 1, 1, 1, 2, 2, 1, 1, 3, 2, 1, 1, 2, 3, 2, 1, 13

Offset: 1

Antti Karttunen, Nov 29 2017

This sequence can be used as a filter. It matches at least to the following sequences related to the counting of various non-unitary prime divisors:
For all i, j:
  a(i) = a(j) => A056170(i) = A056170(j), as A056170(n) = A001222(a(n)).
  a(i) = a(j) => A162641(i) = A162641(j).
  a(i) = a(j) => A295659(i) = A295659(j).
  a(i) = a(j) => A295662(i) = A295662(j).
  a(i) = a(j) => A295883(i) = A295883(j), as A295883(n) = A007949(a(n)).
  a(i) = a(j) => A295884(i) = A295884(j).
An encoding of the prime signature of A057521(n), the powerful part of n. - Peter Munn, Apr 06 2024

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences computed from exponents in factorization of n.

Cf. A003557, A008578, A057521, A064989, A181819.
Cf. also A293515, A294875, A294895, A294897, A295878, A351944.
Differs from A000688 for the first time at n=128, where a(128) = 13, while A000688(128) = 15.

Array[Apply[Times, FactorInteger[#] /. {p_, e_} /; p > 0 :> Which[p == 1, 1, e == 1, 1, True, Prime[e - 1]]] &, 128] (* Michael De Vlieger, Nov 29 2017 *)

a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i,2] == 1, 1, prime(f[i,2]-1)));} \\ Amiram Eldar, Nov 18 2022

a(1) = 1; for n>1, if n = Product prime(i)^e(i), then a(n) = Product A008578(e(i)).
a(n) = A064989(A181819(n)).
a(n) = A181819(A003557(n)). - Antti Karttunen, Apr 03 2022
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Product_{p prime} (1 + 1/p^2  + Sum_{k>=1} (prime(k+1)-prime(k))/p^(k+2)) = 2.208... . - Amiram Eldar, Nov 18 2022

cp's OEIS Frontend

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

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A294897 a(n) = Product_{d|n, gcd(d,n/d)>1} prime(A101296(gcd(d,n/d))-1).

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A295879 Multiplicative with a(p) = 1, a(p^e) = prime(e-1) if e > 1.

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