A372603 -id:A372603 - cp's OEIS frontend

A372601 The maximal exponent in the prime factorization of the largest exponentially odd divisor of n.

0, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 3, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 3, 1, 3, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 3, 3, 1, 1, 1, 1, 1, 1

Offset: 1

Amiram Eldar, May 07 2024

First differs from A331273 at n = 64.

Differs from A363332 at n = 1, 216, 432, 648, 864, 1000, ... .

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A051903, A109613, A350390, A368711.
Cf. A331273, A363332.
Similar sequences: A007424, A368781, A372602, A372603, A372604.

f[n_] := n - If[EvenQ[n], 1, 0]; a[n_] := f[Max[FactorInteger[n][[;; , 2]]]]; a[1] = 0; Array[a, 100]

s(n) = (n+1) \ 2 * 2 - 1;
a(n) = if(n>1, s(vecmax(factor(n)[,2])), 0);

a(n) = A051903(A350390(n)).
a(n) = A109613(A051903(n)-1) for n >= 2.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 1 + 2 * Sum_{i>=1} (1 - (1/zeta(2*i+1))) = 1.42929441950714075659... .

A372602 The maximal exponent in the prime factorization of the largest square dividing n.

Original entry on oeis.org

0, 0, 0, 2, 0, 0, 0, 2, 2, 0, 0, 2, 0, 0, 0, 4, 0, 2, 0, 2, 0, 0, 0, 2, 2, 0, 2, 2, 0, 0, 0, 4, 0, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0, 2, 2, 0, 0, 4, 2, 2, 0, 2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 2, 6, 0, 0, 0, 2, 0, 0, 0, 2, 0, 0, 2, 2, 0, 0, 0, 4, 4, 0, 0, 2, 0, 0, 0

Offset: 1

Amiram Eldar, May 07 2024

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A008833, A051903, A052928.

Similar sequences: A007424, A368781, A372601, A372603, A372604.

f[n_] := 2 * Floor[n/2]; a[n_] := f[Max[FactorInteger[n][[;; , 2]]]]; a[1] = 0; Array[a, 100]

s(n) = n \ 2 * 2;
a(n) = if(n>1, s(vecmax(factor(n)[,2])), 0);

a(n) = A051903(A008833(n)).
a(n) = A052928(A051903(n)).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 2 * Sum_{i>=1} (1 - (1/zeta(2*i))) = 0.98112786070359477197... .

A372604 The maximal exponent in the prime factorization of the largest divisor of n whose number of divisors is a power of 2.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 3, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 3, 1, 3, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 3, 3, 1, 1, 1, 1, 1, 1

Offset: 1

Amiram Eldar, May 07 2024

First differs from A331273 at n = 32.
Differs from A368247 at n = 1, 128, 216, 256, 384, 432, 512, ... .
All the terms are of the form 2^k-1 (A000225).

			4 has 3 divisors, 1, 2 and 4. The number of divisors of 4 is 3, which is not a power of 2. The number of divisors of 2 is 2, which is a power of 2. Therefore, A372379(4) = 2 and a(4) = A051903(2) = 1.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000225, A051903, A092323, A372379, A372466.
Cf. A331273, A368247.
Similar sequences: A007424, A368781, A372601, A372602, A372603.

f[n_] := 2^Floor[Log2[n + 1]] - 1; a[n_] := f[Max[FactorInteger[n][[;; , 2]]]]; a[1] = 0; Array[a, 100]

s(n) = 2^exponent(n+1) - 1;
a(n) = if(n>1, s(vecmax(factor(n)[,2])), 0);

a(n) = A051903(A372379(n)).
a(n) = A092323(A051903(n)+1).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 1 + Sum_{i>=1} 2^i * (1 - 1/zeta(2^(i+1)-1)) = 1.36955053734097783559... .

A375339 If n has exactly one non-unitary prime factor then a(n) is the exponent of the highest power of this prime that divides n, otherwise a(n) = 0.

Original entry on oeis.org

0, 0, 0, 2, 0, 0, 0, 3, 2, 0, 0, 2, 0, 0, 0, 4, 0, 2, 0, 2, 0, 0, 0, 3, 2, 0, 3, 2, 0, 0, 0, 5, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 2, 2, 0, 0, 4, 2, 2, 0, 2, 0, 3, 0, 3, 0, 0, 0, 2, 0, 0, 2, 6, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 2, 0, 0, 0, 4, 4, 0, 0, 2, 0, 0, 0, 3, 0, 2, 0, 2, 0, 0, 0, 5, 0, 2, 2, 0, 0, 0, 0, 3, 0

Offset: 1

Amiram Eldar, Aug 12 2024

First differs from A212172, A275812 and A372603 at n = 36.
If n = m * p^e, such that m is squarefree, p is a prime that does not divide m and e >= 2, then a(n) = e, otherwise a(n) = 0.
By definition all the positive terms are larger than 1.
The asymptotic density of 0's in this sequence is 1 - Sum_{p prime} (1/(p^2-1)) / zeta(2) = 1 - A059956 * A154945 = 0.66461069244308962639... .
The asymptotic density of the occurrences of k >= 2 in this sequence is Sum_{p prime} (1/(p^(k-1)*(p+1))) / zeta(2). E.g., 0.200755... (A271971) for k = 2, 0.0741777... for k = 3, and 0.0320652... for k = 4.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A005361, A048109, A051903, A060687, A082293, A190641, A359466, A375341 (the positive terms).
Cf. A013661, A059956, A154945, A271971, A375340.
Cf. A212172, A275812, A372603.

a[n_] := Module[{e = Select[FactorInteger[n][[;; , 2]], # > 1 &]}, If[Length[e] == 1, e[[1]], 0]]; Array[a, 100]

a(n) = {my(e = select(x -> x > 1, factor(n)[,2])); if(#e == 1, e[1], 0);}

a(n) = A051903(n) * A359466(n).
a(n) = A005361(n) * A359466(n).
a(A190641(n)) >= 2.
a(n) = 2 if and only if n is in A060687.
a(n) = 3 if and only if n is in A048109.
a(n) <= 3 if and only if n is in A082293.
Asymptotic mean:  = Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{p prime} (2*p-1)/((p-1)^2*(p+1)) / zeta(2) = A375340 / A013661 = 0.92105359989459565838... .
Asymptotic second raw moment:  = Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)^2 = Sum_{p prime} (4*p^2-3*p+1)/((p-1)^3*(p+1)) / zeta(2) = 3.04027120804428071157... .
The asymptotic second central moment, or variance, is  - ^2 = 2.19193147416548680815... and the asymptotic standard deviation is sqrt( - ^2) = 1.48051729951577627898... .

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A372601 The maximal exponent in the prime factorization of the largest exponentially odd divisor of n.

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A372602 The maximal exponent in the prime factorization of the largest square dividing n.

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A372604 The maximal exponent in the prime factorization of the largest divisor of n whose number of divisors is a power of 2.

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A375339 If n has exactly one non-unitary prime factor then a(n) is the exponent of the highest power of this prime that divides n, otherwise a(n) = 0.

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