A295666 -id:A295666 - cp's OEIS frontend

A296204 Numbers k such that Product_{d|k, gcd(d,k/d) is prime} gcd(d,k/d) = k; the fixed points of A295666.

1, 36, 100, 144, 196, 225, 324, 400, 441, 484, 676, 784, 1089, 1156, 1225, 1296, 1444, 1521, 1936, 2025, 2116, 2500, 2601, 2704, 3025, 3249, 3364, 3844, 3969, 4225, 4624, 4761, 5476, 5625, 5776, 5929, 6724, 7225, 7396, 7569, 8281, 8464, 8649, 8836, 9025, 9604, 9801, 10000, 11236, 12321, 13225, 13456, 13689, 13924, 14161

Offset: 1

Antti Karttunen, Dec 18 2017

nonn

Cf. A295666, A296205 (the square roots).

a295666[n_]:=Product[GCD[i,n/i]^Boole[PrimeQ[GCD[i,n/i]]],{i,Divisors[n]}]; Select[Range[14500], a295666[#]==# &] (* Stefano Spezia, Feb 20 2024 *)

A056170 Number of non-unitary prime divisors of n.

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 1, 2, 0, 0, 0, 1, 0

Offset: 1

Labos Elemer, Jul 27 2000

A prime factor of n is unitary iff its exponent is 1 in the prime factorization of n. (Of course for any prime p, GCD(p, n/p) is either 1 or p. For a unitary prime factor it must be 1.)
Number of squared primes dividing n. - Reinhard Zumkeller, May 18 2002
a(A005117(n)) = 0; a(A013929(n)) > 0; a(A190641(n)) = 1. - Reinhard Zumkeller, Dec 29 2012
First differences of A013940. - Jason Kimberley, Feb 01 2017
Number of exponents larger than 1 in the prime factorization of n. - Antti Karttunen, Nov 28 2017

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index entries for sequences computed from exponents in factorization of n.

Cf. A000188, A001221, A003557, A013940, A034444, A046660, A048105, A056169, A085548, A124010, A162641, A212177, A275812, A295659, A295666.

Cf. A057427(a(n)) = 1 - A008966(n).

a056170 = length . filter (> 1) . a124010_row
-- Reinhard Zumkeller, Dec 29 2012

A056170:=func;
[A056170(n):n in[1..105]];
// Jason Kimberley, Jan 22 2017

A056170 := n -> nops(select(t -> (t[2]>1), ifactors(n)[2]));
seq(A056170(n),n=1..100); # Robert Israel, Jun 03 2014

a[n_] := Count[FactorInteger[n], {, k /; k > 1}]; Table[a[n], {n, 105}]  (* Jean-François Alcover, Mar 23 2011 *)
Table[Count[FactorInteger[n][[All,2]],?(#>1&)],{n,110}] (* _Harvey P. Dale, Jul 08 2019 *)

a(n)=my(f=factor(n)[,2]); sum(i=1,#f,f[i]>1) \\ Charles R Greathouse IV, May 18 2015

from sympy import factorint
def a(n):
    f = factorint(n)
    return sum([1 for i in f if f[i]!=1]) # Indranil Ghosh, Apr 24 2017

Additive with a(p^e) = 0 if e = 1, 1 otherwise.
G.f.: Sum_{k>=1} x^(prime(k)^2)/(1 - x^(prime(k)^2)). - Ilya Gutkovskiy, Jan 01 2017
a(n) = log_2(A000005(A071773(n))). - observed by Velin Yanev, Aug 20 2017, confirmed by Antti Karttunen, Nov 28 2017
From Antti Karttunen, Nov 28 2017: (Start)
a(n) = A001221(n) - A056169(n).
a(n) = omega(A000188(n)) = omega(A003557(n)) = omega(A057521(n)) = omega(A295666(n)), where omega = A001221.
For all n >= 1 it holds that:
a(A003557(n)) = A295659(n).
a(n) >= A162641(n).
(End)
Dirichlet g.f.: primezeta(2s)*zeta(s). - Benedict W. J. Irwin, Jul 11 2018
Asymptotic mean: lim_{n->oo} (1/n) * Sum_{k=1..n} a(k) = Sum_{p prime} 1/p^2 = 0.452247... (A085548). - Amiram Eldar, Nov 01 2020
a(n) = A275812(n) - A046660(n). - Amiram Eldar, Jan 09 2024

Minor edits by Franklin T. Adams-Watters, Mar 23 2011

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

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).

A296205 Numbers k such that Product_{d|k^2, gcd(d,k^2/d) is prime} gcd(d,k^2/d) = k^2.

Original entry on oeis.org

1, 6, 10, 12, 14, 15, 18, 20, 21, 22, 26, 28, 33, 34, 35, 36, 38, 39, 44, 45, 46, 50, 51, 52, 55, 57, 58, 62, 63, 65, 68, 69, 74, 75, 76, 77, 82, 85, 86, 87, 91, 92, 93, 94, 95, 98, 99, 100, 106, 111, 115, 116, 117, 118, 119, 122, 123, 124, 129, 133, 134, 141, 142, 143, 145, 146, 147, 148, 153, 155, 158, 159, 161

Offset: 1

Antti Karttunen, Dec 18 2017

nonn

Except for a(1) = 1, these appear to be cubefree numbers with two distinct prime factors, or Heinz numbers of integer partitions with two distinct parts, none appearing more than twice. The enumeration of these partitions by sum is given by A307370. Equivalently, except for a(1) = 1, this sequence is the intersection of A004709 and A007774. - Gus Wiseman, Jul 03 2019

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000196, A295666, A296204.
Cf. A006881, A054753, A085986 (seem to be subsequences).
Cf. A004709, A007774, A056239, A112798, A118914, A307370, A325240.

filter:= proc(k) local d,r,v;
   r:= 1;
   for d in numtheory:-divisors(k^2) do
     v:= igcd(d,k^2/d);
     if isprime(v) then r:= r*v fi
   od;
   r = k^2
end proc:
select(filter, [$1..200]); # Robert Israel, Feb 20 2024

a(n) = A000196(A296204(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

A296204 Numbers k such that Product_{d|k, gcd(d,k/d) is prime} gcd(d,k/d) = k; the fixed points of A295666.

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A056170 Number of non-unitary prime divisors of 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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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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A296205 Numbers k such that Product_{d|k^2, gcd(d,k^2/d) is prime} gcd(d,k^2/d) = k^2.

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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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