A348963 -id:A348963 - cp's OEIS frontend

A349019 Modified e-perfect numbers: numbers k such that A348963(k) | k.

1, 2, 3, 5, 6, 7, 10, 11, 12, 13, 14, 15, 17, 19, 21, 22, 23, 26, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 46, 47, 51, 53, 55, 57, 58, 59, 60, 61, 62, 65, 66, 67, 69, 70, 71, 73, 74, 77, 78, 79, 82, 83, 84, 85, 86, 87, 89, 91, 93, 94, 95, 97, 101

Offset: 1

Amiram Eldar, Nov 06 2021

nonn

First differs from A225354 at n = 25.
Not to be confused with modified exponential perfect numbers (A323757).
Sándor (2006) showed that the exponential harmonic numbers of type 2 (A348964) are terms in this sequence.
All the squarefree numbers are terms (A005117), since A348963(k) = 1 if k is squarefree.

			12 is a term since A348963(12) = 3 is a divisor of 12.

Amiram Eldar, Table of n, a(n) for n = 1..10000
József Sándor, On exponentially harmonic numbers, Scientia Magna, Vol. 2, No. 3 (2006), pp. 44-47.
József Sándor, Selected Chapters of Geomety, Analysis and Number Theory, 2005, pp. 141-145.

A005117, A348964 and A349020 are subsequences.

Cf. A225354, A323757, A348963.

f[p_, e_] := p^e/DivisorSum[e, p^(e - #) &]; modEPerfQ[1] = True; modEPerfQ[n_] := IntegerQ[Times @@ f @@@ FactorInteger[n]]; Select[Range[100], modEPerfQ]

A348964 Exponential harmonic (or e-harmonic) numbers of type 2: numbers k such that the harmonic mean of the exponential divisors of k is an integer.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 10, 11, 12, 13, 14, 15, 17, 18, 19, 21, 22, 23, 26, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 46, 47, 51, 53, 55, 57, 58, 59, 60, 61, 62, 65, 66, 67, 69, 70, 71, 73, 74, 75, 77, 78, 79, 82, 83, 84, 85, 86, 87, 89, 90, 91, 93, 94

Offset: 1

Amiram Eldar, Nov 05 2021

nonn

Sándor (2006) proved that all the squarefree numbers are e-harmonic of type 2.
Equivalently, numbers k such that A348963(k) | k * A049419(k).
Apparently, most exponential harmonic numbers of type 1 (A348961) are also terms of this sequence. Those that are not exponential harmonic numbers of type 2 are 1936, 5808, 9680, 13552, 17424, 29040, ...

			The squarefree numbers are trivial terms. If k is squarefree, then it has a single exponential divisor, k itself, and thus the harmonic mean of its exponential divisors is also k, which is an integer.
12 is a term since its exponential divisors are 6 and 12, and their harmonic mean, 8, is an integer.

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.
József Sándor, On exponentially harmonic numbers, Scientia Magna, Vol. 2, No. 3 (2006), pp. 44-47.
József Sándor, Selected Chapters of Geomety, Analysis and Number Theory, 2005, pp. 141-145.

A005117 and A348965 are subsequences.
Cf. A049419, A322791, A348961, A348963.
Similar sequences: A001599, A006086, A063947, A286325, A319745.

f[p_, e_] := p^e * DivisorSigma[0, e] / DivisorSum[e, p^(e-#) &]; ehQ[1] = True; ehQ[n_] := IntegerQ[Times @@ f @@@ FactorInteger[n]]; Select[Range[100], ehQ]

A349020 Modified e-perfect numbers (A349019) that are not squarefree.

Original entry on oeis.org

12, 36, 40, 60, 84, 120, 132, 150, 156, 180, 204, 208, 228, 252, 270, 276, 280, 348, 360, 372, 396, 420, 440, 444, 468, 492, 516, 520, 540, 544, 564, 600, 612, 624, 636, 660, 680, 684, 708, 732, 760, 780, 804, 828, 840, 852, 876, 920, 924, 948, 996, 1020, 1040

Offset: 1

Amiram Eldar, Nov 06 2021

nonn

Since all the squarefree numbers (A005117) are modified e-perfect numbers, these are the nontrivial terms of A349019.

			12 = 2^2 * 3 is a term since it is not squarefree and A348963(12) = 3 is a divisor of 12.

Amiram Eldar, Table of n, a(n) for n = 1..10000
József Sándor, On exponentially harmonic numbers, Scientia Magna, Vol. 2, No. 3 (2006), pp. 44-47.
József Sándor, Selected Chapters of Geomety, Analysis and Number Theory, 2005, pp. 141-145.

Intersection of A013929 and A349019.
A348965 is a subsequence.
Cf. A005117.

f[p_, e_] := p^e/DivisorSum[e, p^(e - #) &]; modEPerfQ[1] = True; modEPerfQ[n_] := IntegerQ[Times @@ f @@@ FactorInteger[n]]; Select[Range[1000], ! SquareFreeQ[#] && modEPerfQ[#] &]

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).

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A349019 Modified e-perfect numbers: numbers k such that A348963(k) | k.

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A348964 Exponential harmonic (or e-harmonic) numbers of type 2: numbers k such that the harmonic mean of the exponential divisors of k is an integer.

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A349020 Modified e-perfect numbers (A349019) that are not squarefree.

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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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