A278908 -id:A278908 - cp's OEIS frontend

A368978 The number of bi-unitary divisors of n that are squares (A000290).

1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 2, 1, 2, 2, 1, 1, 1, 3, 1, 1, 1, 4, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 2, 2, 2, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 2, 4, 1, 1, 1, 2, 1, 1, 1, 4, 1, 1, 2, 2, 1, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1

Offset: 1

Amiram Eldar, Jan 11 2024

First differs from A007424, A278908, A307848, A323308, A358260 and A365549 at n = 32.

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

Cf. A000005, A000290, A005117, A013662, A062503, A222266, A286324, A368977.
Cf. A007424, A278908, A307848, A323308, A365549.
Similar sequences: A046951, A056624, A358260, A368980.

f[p_, e_] := If[OddQ[e], (e + 1)/2, 2*Floor[(e+2)/4]]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = vecprod(apply(x -> if(x%2, (x+1)/2, 2*((x+2)\4)), factor(n)[, 2]));

Multiplicative with a(p^e) = (e + 1)/2 if e is odd, and 2*floor((e+2)/4) if e is even.
a(n) >= 1, with equality if and only if n is squarefree (A005117).
a(n) <= A286324(n), with equality if and only if n is in A062503.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = zeta(4) * Product_{p prime} (1 + 1/p^2 - 1/p^4 + 1/p^5) = 1.58922450321701775833... .

A380398 The number of unitary divisors of n that are perfect powers (A001597).

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 2, 1, 2, 2, 1, 1, 1, 2, 1, 1, 1, 4, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 2, 2, 2, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 2, 2, 1, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1

Offset: 1

Amiram Eldar, Jan 23 2025

First differs from A368978 at n = 32, from A007424 and A369163 at n = 36, from A278908, A307848, A358260 and A365549 at n = 64, and from A323308 at n = 72.
a(n) depends only on the prime signature of n (A118914).
The record values are 2^k, for k = 0, 1, 2, ..., and they are attained at A061742(k).
The sum of unitary divisors of n that are perfect powers is A380400(n).

			a(4) = 2 since 4 have 2 unitary divisors that are perfect powers, 1 and 4 = 2^2.
a(72) = 3 since 72 have 3 unitary divisors that are perfect powers, 1, 8 = 2^3, and 9 = 3^2.

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

Cf. A001597, A005117, A008683, A061742, A077610, A091050, A118914, A323308, A380399, A380400.

Cf. A007424, A278908, A307848, A323308, A358260, A365549, A368978, A369163.

ppQ[n_] := n == 1 || GCD @@ FactorInteger[n][[;; , 2]] > 1; a[n_] := DivisorSum[n, 1 &, CoprimeQ[#, n/#] && ppQ[#] &]; Array[a, 100]

a(n) = sumdiv(n, d, gcd(d, n/d) == 1 && (d == 1 || ispower(d)));

a(n) = Sum_{d|n, gcd(d, n/d) == 1} [d in A001597], where [] is the Iverson bracket.
a(n) = A091050(n) - A380399(n).
a(n) = 1 if and only if n is squarefree (A005117).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 1 - Sum_{k>=2} mu(k)*(zeta(k)/zeta(k+1) - 1) = 1.49341326536904597349..., where mu is the Moebius function (A008683).

A383761 Irregular triangle read by rows in which the n-th row lists the exponential squarefree exponential divisors of n.

Original entry on oeis.org

1, 2, 3, 2, 4, 5, 6, 7, 2, 8, 3, 9, 10, 11, 6, 12, 13, 14, 15, 2, 4, 17, 6, 18, 19, 10, 20, 21, 22, 23, 6, 24, 5, 25, 26, 3, 27, 14, 28, 29, 30, 31, 2, 32, 33, 34, 35, 6, 12, 18, 36, 37, 38, 39, 10, 40, 41, 42, 43, 22, 44, 15, 45, 46, 47, 6, 12, 7, 49, 10, 50

Offset: 1

Amiram Eldar, May 09 2025

Differs from A322791, A361255 and A383760 at rows 16, 48, 80, 81, 112, 144, 162, ... .

An exponential squarefree exponential divisor (or e-squarefree e-divisor) d of a number n is a divisor d of n such that for every prime divisor p of n, the p-adic valuation of d is a squarefree divisor of the p-adic valuation of n.

			The first 10 rows are:
  1
  2
  3
  2, 4
  5
  6
  7
  2, 8
  3, 9
  10

Amiram Eldar, Table of n, a(n) for n = 1..15331 (first 10000 rows, flattened)
László Tóth, On certain arithmetic functions involving exponential divisors, II, Annales Univ. Sci. Budapest., Sect. Comp., Vol. 27 (2007), pp. 155-166; arXiv preprint, arXiv:0708.3557 [math.NT], 2007-2009.

Cf. A278908 (row lengths), A361174 (row sums).

Cf. A322791, A361255, A383760.

sqfDivQ[n_, d_] := SquareFreeQ[d] && Divisible[n, d];
expSqfDivQ[n_, d_] := Module[{f = FactorInteger[n]}, And @@ MapThread[sqfDivQ, {f[[;; , 2]], IntegerExponent[d, f[[;; , 1]]]}]]; expSqfDivs[1] = {1};
expSqfDivs[n_] := Module[{d = Rest[Divisors[n]]}, Select[d, expSqfDivQ[n, #] &]];
Table[expSqfDivs[n], {n, 1, 70}] // Flatten

A383959 The number of prime powers p^e having the property that e is a unitary divisor of the p-adic valuation of n.

Original entry on oeis.org

0, 1, 1, 2, 1, 2, 1, 2, 2, 2, 1, 3, 1, 2, 2, 2, 1, 3, 1, 3, 2, 2, 1, 3, 2, 2, 2, 3, 1, 3, 1, 2, 2, 2, 2, 4, 1, 2, 2, 3, 1, 3, 1, 3, 3, 2, 1, 3, 2, 3, 2, 3, 1, 3, 2, 3, 2, 2, 1, 4, 1, 2, 3, 4, 2, 3, 1, 3, 2, 3, 1, 4, 1, 2, 3, 3, 2, 3, 1, 3, 2, 2, 1, 4, 2, 2, 2

Offset: 1

Amiram Eldar, May 16 2025

First differs from A238949 at n = 64.	
First differs from A383960 at n = 256.		
Also, the number of prime powers p^e having the property that e is a squarefree divisor of the p-adic valuation of n.

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

Cf. A001221, A034444, A077761, A085548, A238949, A278908, A361255, A383863, A383960.

f[p_, e_] := 2^PrimeNu[e]; a[1] = 0; a[n_] := Plus @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = vecsum(apply(x -> 1 << omega(x), factor(n)[, 2]));

Additive with a(p^e) = A034444(e) = 2^A001221(e).

Sum_{k=1..n} a(k) ~ n*(log(log(n)) + B - C + D), where B is Mertens's constant (A077761), C = Sum_{p prime} 1/p^2 (A085548), and D = Sum_{p prime, e>=2} (1-1/p)*A034444(e)/p^e = 0.92341081050532387352... .

A349025 a(n) is multiplicative with a(p^e) = Sum_{d||e} p^(e-d), where d||e are the unitary divisors of e.

Original entry on oeis.org

1, 1, 1, 3, 1, 1, 1, 5, 4, 1, 1, 3, 1, 1, 1, 9, 1, 4, 1, 3, 1, 1, 1, 5, 6, 1, 10, 3, 1, 1, 1, 17, 1, 1, 1, 12, 1, 1, 1, 5, 1, 1, 1, 3, 4, 1, 1, 9, 8, 6, 1, 3, 1, 10, 1, 5, 1, 1, 1, 3, 1, 1, 4, 57, 1, 1, 1, 3, 1, 1, 1, 20, 1, 1, 6, 3, 1, 1, 1, 9, 28, 1, 1, 3, 1

Offset: 1

Amiram Eldar, Nov 06 2021

First differs from A348963 at n = 16.

A number k is an exponential unitary harmonic number (A349026) if and only if a(k) | k * A278908(k).

Amiram Eldar, Table of n, a(n) for n = 1..10000
Nicuşor Minculete, Contribuţii la studiul proprietăţilor analitice ale funcţiilor aritmetice - Utilizarea e-divizorilor, Ph.D. thesis, Academia Română, 2012. See section 4.3, pp. 90-94.

The unitary version of A348963.

Cf. A005117, A278908, A349026.

f[p_, e_] := DivisorSum[e, p^(e - #) &, CoprimeQ[#, e/#] &]; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = 1 if and only if n is squarefree (A005117).

A358659 Decimal expansion of the asymptotic mean of the ratio between the number of exponential unitary divisors and the number of exponential divisors.

Original entry on oeis.org

9, 8, 4, 8, 8, 3, 6, 4, 1, 8, 7, 7, 2, 2, 8, 2, 9, 4, 0, 9, 5, 3, 7, 0, 1, 3, 8, 0, 4, 8, 9, 6, 1, 1, 3, 7, 6, 4, 7, 3, 1, 6, 3, 2, 2, 2, 2, 7, 0, 5, 8, 1, 3, 4, 5, 5, 0, 0, 6, 3, 6, 2, 3, 5, 5, 0, 2, 2, 3, 9, 6, 8, 0, 6, 5, 9, 0, 8, 2, 3, 8, 0, 0, 8, 1, 8, 9, 3, 8, 0, 9, 5, 5, 7, 4, 0, 8, 7, 6, 9, 1, 3, 3, 4, 4

Offset: 0

Amiram Eldar, Nov 25 2022

			0.984883641877228294095370138048961137647316322227058...

Nicuşor Minculete and László Tóth, Exponential unitary divisors, Annales Univ. Sci. Budapest., Sect. Comp. Vol. 35 (2011), pp. 205-216.

Cf. A049419, A278908.

Similar sequences: A307869, A308042, A308043.

r[n_] := 2^PrimeNu[n]/DivisorSigma[0, n]; $MaxExtraPrecision = 500; m = 500; f[x_] := Log[1 + Sum[x^e*(r[e] - r[e - 1]), {e, 4, m}]]; c = Rest[CoefficientList[Series[f[x], {x, 0, m}], x]*Range[0, m]]; RealDigits[Exp[f[1/2] + NSum[Indexed[c, k]*(PrimeZetaP[k] - 1/2^k)/k, {k, 2, m}, NSumTerms -> m, WorkingPrecision -> m]], 10, 120][[1]]

Equals lim_{m->oo} (1/m) Sum_{k=1..m} A278908(k)/A049419(k).

Equals Product_{p prime} (1 + Sum_{e >= 4} (r(e) - r(e-1))/p^e), where r(e) = A278908(e)/A049419(e).

A384557 The number of exponential unitary (or e-unitary) divisors of n that are exponentially odd numbers (A268335).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1

Offset: 1

Amiram Eldar, Jun 03 2025

First differs from A359411 at n = 2097152 = 2^21: a(2097152) = 4, while A359411(2097152) = 2.
First differs from A368979 at n = 512 = 2^9: a(512) = 2, while A368979(512) = 3.
First differs from A367516 at n = 128 = 2^7: a(128) = 2, while A367516(128) = 1.
First differs from A382291 at n = 128 = 2^7: a(128) = 2, while A382291(128) = 4.
First differs from A368168 at n = 64 = 2^6: a(64) = 2, while A368168(64) = 1.
The sum of these divisors is A384559(n), and the largest of them is A331737(n).
The number of exponential unitary (or e-unitary) divisors of n is A278908(n) and the number of divisors of n that are exponentially odd numbers is A322483(n).
All the terms are powers of 2. The first term that is greater than 2 is a(32768) = 4.

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

Cf. A068068, A138302, A268335, A278908, A322483, A331737, A359411, A367516, A368168, A368979, A382291, A384559.

f[p_, e_] := 2^PrimeNu[e/2^IntegerExponent[e, 2]]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = vecprod(apply(x -> 2^omega(x >> valuation(x, 2)) , factor(n)[, 2]));

Multiplicative with a(p^e) = A068068(e).
a(n) >= 1, with equality if and only if n is in A138302.
a(n) <= A278908(n), with equality if and only if n is an exponentially odd number.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Product_{p prime} (1 + Sum_{k>=2} (d(k) - d(k-1))/p^k) = 1.13551542615965557947..., where d(k) is the number of odd unitary divisors of k (A068068).

A338782 The largest e-squarefree e-divisor of n.

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Nov 08 2020

An exponential squarefree exponential divisor (e-squarefree e-divisor) d = Product p_i^f_i of n = Product p_i^e_i has f_i | e_i and f_i is squarefree for all i.
The largest of the A278908(n) e-squarefree e-divisors of n.
a(n) = n if and only if n is an exponentially squarefree number (A209061).

Amiram Eldar, Table of n, a(n) for n = 1..10000
László Tóth, On certain arithmetic functions involving exponential divisors, II, Annales Univ. Sci. Budapest., Sect. Comp., Vol. 27 (2007), pp. 155-166, arXiv preprint, arXiv:0708.3557 [math.NT], 2007-2009.

Cf. A005117, A007947, A209061, A278908.

rad[n_] := Times @@ (First@# & /@ FactorInteger[n]); f[p_, e_] := p^rad[e]; a[1]=1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

Multiplicative with a(p^e) = p^rad(e), where rad(k) is the largest squarefree number dividing k (A007947).

Sum_{n<=x} a(n) = (1/2) * c * x^2, where c = Product_{p prime} Sum{k>=4} (p^rad(k) - p^(1+rad(k-1)))/p^(2*k) = 0.9646498658... (Tóth, 2007).

cp's OEIS Frontend

A368978 The number of bi-unitary divisors of n that are squares (A000290).

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A380398 The number of unitary divisors of n that are perfect powers (A001597).

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A383761 Irregular triangle read by rows in which the n-th row lists the exponential squarefree exponential divisors of n.

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A383959 The number of prime powers p^e having the property that e is a unitary divisor of the p-adic valuation of n.

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A349025 a(n) is multiplicative with a(p^e) = Sum_{d||e} p^(e-d), where d||e are the unitary divisors of e.

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A358659 Decimal expansion of the asymptotic mean of the ratio between the number of exponential unitary divisors and the number of exponential divisors.

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A384557 The number of exponential unitary (or e-unitary) divisors of n that are exponentially odd numbers (A268335).

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A338782 The largest e-squarefree e-divisor of n.

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