A331308 -id:A331308 - cp's OEIS frontend

A331592 a(n) is the smaller of the number of terms in the factorizations of n into (1) powers of distinct primes and (2) powers of squarefree numbers with distinct exponents that are powers of 2.

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

Offset: 1

Antti Karttunen and Peter Munn, Jan 21 2020

nonn

See A329332 for a description of the relationship between the two factorizations. From this relationship we get the formula a(n) = min(A001221(n), A001221(A225546(n))).
The result depends only on the prime signature of n.
k first appears at A191555(k).

			The factorization of 6 into powers of distinct primes is 6 = 2^1 * 3^1 = 2 * 3, which has 2 terms. Its factorization into powers of squarefree numbers with distinct exponents that are powers of 2 is 6 = 6^(2^0) = 6^1, which has 1 term. So a(6) is min(2,1) = 1.
The factorization of 40 into powers of distinct primes is 40 = 2^3 * 5^1 = 8 * 5, which has 2 terms. Its factorization into powers of squarefree numbers with distinct exponents that are powers of 2 is 40 = 10^(2^0) * 2^(2^1) = 10^1 * 2^2 = 10 * 4, which has 2 terms. So a(40) is min(2,2) = 2.

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

Cf. A000961, A001221, A191555, A293442, A329332.
Sequences with related definitions: A331308, A331591, A331593.
A003961, A225546 are used to express relationship between terms of this sequence.
Differs from = A071625 for the first time at n=216, where a(216) = 2, while A071625(216) = 1.

A331592(n) = min(omega(n), A331591(n)); \\ Uses also code from A331591.

a(n) = min(A001221(n), A331591(n)) = min(A001221(n), A001221(A293442(n))).
a(A225546(n)) = a(n).
a(A003961(n)) = a(n).
a(n^2) = a(n).

A331309 a(n) = A000005(A225546(n)), where A000005 gives the number of divisors of n.

Original entry on oeis.org

1, 2, 3, 2, 5, 4, 9, 4, 3, 6, 17, 6, 33, 10, 7, 2, 65, 6, 129, 10, 11, 18, 257, 8, 5, 34, 9, 18, 513, 8, 1025, 4, 19, 66, 13, 4, 2049, 130, 35, 12, 4097, 12, 8193, 34, 15, 258, 16385, 6, 9, 10, 67, 66, 32769, 12, 21, 20, 131, 514, 65537, 14, 131073, 1026, 27, 4, 37, 20, 262145, 130, 259, 14, 524289, 8, 1048577, 2050, 15

Offset: 1

Antti Karttunen, Jan 21 2020

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A000005, A048675, A225546, A331287, A331308, A331591.

A048675(n) = { my(f = factor(n)); sum(k=1, #f~, f[k, 2]*2^primepi(f[k, 1]))/2; };
A331309(n) = if(1==n,1,my(f=factor(n),u=#binary(vecmax(f[, 2])),prods=vector(u,x,1),m=1,e); for(i=1,u,for(k=1,#f~, if(bitand(f[k,2],m),prods[i] *= f[k,1])); m<<=1); prod(i=1,u,1+A048675(prods[i])));

a(n) = A000005(A225546(n)).

A331301 Lexicographically earliest infinite sequence such that a(i) = a(j) => f(i) = f(j), where f(n) = min(n, A225546(n)) for all other n, except for odd primes p, f(p) = 0.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Jan 21 2020

nonn

For all i, j:
  A305801(i) = A305801(j) => a(i) = a(j),
  a(i) = a(j) => A064179(i) = A064179(j),
  a(i) = a(j) => A064547(i) = A064547(j),
  a(i) = a(j) => A302777(i) = A302777(j),
  a(i) = a(j) => A331308(i) = A331308(j),
  a(i) = a(j) => A331287(i) = A331287(j),
  a(i) = a(j) => A331592(i) = A331592(j).

Antti Karttunen, Table of n, a(n) for n = 1..100000

Cf. A019565, A225546, A305801, A331174, A331288.

Cf. also A064179, A064547, A302777, A331308, A331287, A331592.

up_to = 10000;
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; };
Aux331301(n) = if((n%2)&&isprime(n),0,A331288(n)); \\ Needs also code from A331288.
v331301 = rgs_transform(vector(up_to, n, Aux331301(n)));
A331301(n) = v331301[n];

cp's OEIS Frontend

A331592 a(n) is the smaller of the number of terms in the factorizations of n into (1) powers of distinct primes and (2) powers of squarefree numbers with distinct exponents that are powers of 2.

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A331309 a(n) = A000005(A225546(n)), where A000005 gives the number of divisors of n.

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A331301 Lexicographically earliest infinite sequence such that a(i) = a(j) => f(i) = f(j), where f(n) = min(n, A225546(n)) for all other n, except for odd primes p, f(p) = 0.

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