A338325 -id:A338325 - cp's OEIS frontend

A338326 The number of biquadratefree powerful numbers (A338325) between the consecutive squares n^2 and (n+1)^2.

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

Offset: 1

Amiram Eldar, Oct 22 2020

nonn

Dehkordi (1998) proved that for each k>=0 the sequence of numbers m such that a(m) = k has a positive asymptotic density.

			a(2) = 1 since there is one biquadratefree powerful number, 8 = 2^3, between 2^2 = 4 and 3^2 = 9.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Massoud H. Dehkordi, Asymptotic formulae for some arithmetic functions in number theory, Ph.D. thesis, Loughborough University, 1998.

Cf. A119241, A337736, A338325.

bqfpowQ[n_] := AllTrue[FactorInteger[n][[;; , 2]], MemberQ[{2, 3 }, #] &]; a[n_] := Count[Range[n^2 + 1, (n + 1)^2 - 1], _?bqfpowQ]; Array[a, 100]

A338327 a(n) is the least number k such that there are exactly n biquadratefree powerful numbers (A338325) between k^2 and (k+1)^2.

Original entry on oeis.org

1, 2, 14, 36, 234, 3510, 211297, 487425, 20136429

Offset: 0

Amiram Eldar, Oct 22 2020

a(n) is the least k such that A338326(k) = n.
Dehkordi (1998) proved that for each k>=0 the sequence of numbers m such that A338326(m) = k has a positive asymptotic density. Therefore, this sequence is infinite.
a(9) > 10^10. - Bert Dobbelaere, Oct 29 2020

			a(0) = 1 since there are no biquadratefree powerful numbers between 1^2 = 1 and 2^2 = 4.
a(1) = 2 since there is one biquadratefree powerful number, 8 = 2^3, between 2^2 = 4 and 3^2 = 8.
a(2) = 14 since there are 2 biquadratefree powerful numbers, 200 = 2^3 * 5^2 and 216 = 2^3 * 3^3, between 14^2 = 196 and 15^2 = 225.

Massoud H. Dehkordi, Asymptotic formulae for some arithmetic functions in number theory, Ph.D. thesis, Loughborough University, 1998.

Cf. A119242, A337737, A338325, A338326.

bqfpowQ[n_] := AllTrue[FactorInteger[n][[;; , 2]], MemberQ[{2, 3 }, #] &]; f[n_] := Count[Range[n^2 + 1, (n + 1)^2 - 1], _?bqfpowQ]; mx = 5; s = Table[0, {mx}]; c = 0; n = 1; While[c < mx, i = f[n] + 1; If[i <= mx && s[[i]] == 0, c++; s[[i]] = n]; n++]; s

a(8) from Bert Dobbelaere, Oct 29 2020

A338387 Numbers k such that there are no biquadratefree powerful numbers (A338325) between k^2 and (k+1)^2.

Original entry on oeis.org

1, 3, 4, 6, 7, 9, 12, 13, 15, 16, 17, 20, 21, 23, 24, 26, 27, 28, 29, 30, 32, 34, 35, 38, 39, 40, 41, 43, 44, 45, 47, 49, 50, 54, 56, 60, 61, 62, 63, 64, 66, 68, 69, 71, 74, 75, 76, 77, 79, 80, 81, 83, 84, 85, 86, 89, 90, 91, 95, 97, 99, 100, 101, 105, 106, 107

Offset: 1

Amiram Eldar, Oct 23 2020

nonn

Positions of 0's in A338326.

The asymptotic density of this sequence is 0.623265038... (Dehkordi, 1998).

			1 is a term since the two numbers between 1^2 = 1 and (1+1)^2 = 4, 2 and 3, are not biquadratefree powerful.
2 is not a term since there is a biquadratefree powerful number, 8 = 2^3, between 2^2 = 4 and (2+1)^2 = 9.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Massoud H. Dehkordi, Asymptotic formulae for some arithmetic functions in number theory, Ph.D. thesis, Loughborough University, 1998.

Cf. A336175, A338325, A338326, A338327, A338388, A338389, A338390, A338391, A338392.

bqfpowQ[n_] := AllTrue[FactorInteger[n][[;; , 2]], MemberQ[{2, 3}, #] &]; Select[Range[100], !AnyTrue[Range[#^2 + 1, (# + 1)^2 - 1], bqfpowQ] &]

A338388 Numbers k such that there is a single biquadratefree powerful number (A338325) between k^2 and (k+1)^2.

Original entry on oeis.org

2, 5, 8, 10, 11, 18, 19, 22, 25, 33, 37, 42, 46, 48, 51, 52, 53, 55, 57, 58, 59, 65, 70, 73, 78, 87, 88, 92, 94, 96, 102, 103, 104, 109, 111, 114, 115, 116, 119, 121, 122, 135, 144, 145, 149, 150, 155, 157, 164, 165, 166, 176, 181, 182, 183, 185, 190, 191, 195

Offset: 1

Amiram Eldar, Oct 23 2020

nonn

Positions of 1's in A338326.

The asymptotic density of this sequence is 0.308276695... (Dehkordi, 1998).

			2 is a term since there is a single biquadratefree powerful number, 8 = 2^3, between 2^2 = 4 and (2+1)^2 = 9.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Massoud H. Dehkordi, Asymptotic formulae for some arithmetic functions in number theory, Ph.D. thesis, Loughborough University, 1998.

Cf. A336176, A338325, A338326, A338327, A338387, A338389, A338390, A338391, A338392.

bqfpowQ[n_] := AllTrue[FactorInteger[n][[;; , 2]], MemberQ[{2, 3}, #] &]; Select[Range[200], Count[Range[#^2 + 1, (# + 1)^2 - 1], _?bqfpowQ] == 1 &]

A338389 Numbers k such that there are exactly two biquadratefree powerful numbers (A338325) between k^2 and (k+1)^2.

Original entry on oeis.org

14, 31, 67, 72, 82, 93, 98, 110, 132, 140, 156, 172, 189, 192, 223, 240, 257, 281, 285, 322, 347, 368, 379, 407, 410, 414, 426, 441, 455, 468, 472, 481, 488, 514, 515, 517, 524, 525, 537, 551, 555, 574, 579, 602, 613, 664, 680, 693, 702, 703, 737, 743, 749, 755

Offset: 1

Amiram Eldar, Oct 23 2020

nonn

Positions of 2's in A338326.

The asymptotic density of this sequence is 0.058757863... (Dehkordi, 1998).

			14 is a term since there are exactly two biquadratefree powerful numbers, 200 = 2*3 * 5^2 and 216 = 2^3 * 3^3, between 14^2 = 196 and (14+1)^2 = 225.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Massoud H. Dehkordi, Asymptotic formulae for some arithmetic functions in number theory, Ph.D. thesis, Loughborough University, 1998.

Cf. A336177, A338325, A338326, A338327, A338387, A338388, A338390, A338391, A338392.

bqfpowQ[n_] := AllTrue[FactorInteger[n][[;; , 2]], MemberQ[{2, 3}, #] &]; Select[Range[800], Count[Range[#^2 + 1, (# + 1)^2 - 1], _?bqfpowQ] == 2 &]

A338390 Numbers k such that there are exactly three biquadratefree powerful numbers (A338325) between k^2 and (k+1)^2.

Original entry on oeis.org

36, 161, 364, 659, 722, 771, 896, 911, 981, 987, 1241, 1359, 1486, 1575, 1822, 2042, 2090, 2435, 2537, 2582, 2733, 2870, 2873, 2967, 2983, 3012, 3101, 3108, 3198, 3222, 3278, 3419, 3465, 3544, 3668, 3855, 3860, 3934, 4024, 4092, 4188, 4426, 4437, 4494, 4511, 4522

Offset: 1

Amiram Eldar, Oct 23 2020

nonn

Positions of 3's in A338326.

The asymptotic density of this sequence is 0.008234579... (Dehkordi, 1998).

			36 is a term since there are exactly three biquadratefree powerful numbers, 1323 = 3^3 * 7^2, 1331 = 11^3 and 1352 = 2^3 * 13^2, between 36^2 = 1296 and (36+1)^2 = 1369.

Amiram Eldar, Table of n, a(n) for n = 1..6500
Massoud H. Dehkordi, Asymptotic formulae for some arithmetic functions in number theory, Ph.D. thesis, Loughborough University, 1998.

Cf. A336178, A338325, A338326, A338327, A338387, A338388, A338389, A338391, A338392.

bqfpowQ[n_] := AllTrue[FactorInteger[n][[;; , 2]], MemberQ[{2, 3}, #] &]; Select[Range[1000], Count[Range[#^2 + 1, (# + 1)^2 - 1], _?bqfpowQ] == 3 &]

A338391 Numbers k such that there are exactly four biquadratefree powerful numbers (A338325) between k^2 and (k+1)^2.

Original entry on oeis.org

234, 675, 2426, 8075, 8391, 9093, 9548, 10214, 10340, 11213, 13816, 14523, 14970, 15593, 17329, 17803, 20649, 22483, 23020, 23128, 24842, 25971, 26318, 26557, 28241, 28677, 29124, 29837, 31058, 31338, 31732, 31907, 32490, 35676, 35765, 36302, 37599, 41077, 42577

Offset: 1

Amiram Eldar, Oct 23 2020

nonn

Positions of 4's in A338326.

The asymptotic density of this sequence is 0.000089634... (Dehkordi, 1998).

			234 is a term since there are exactly four biquadratefree powerful numbers, 54872 = 2^3 * 19^3, 54925 = 5^2 * 13^3, 55112 = 2^3 * 83^2 and 55125 = 3^2 * 5^3 * 7^2, between 234^2 = 54756 and (234+1)^2 = 55225.

Amiram Eldar, Table of n, a(n) for n = 1..600
Massoud H. Dehkordi, Asymptotic formulae for some arithmetic functions in number theory, Ph.D. thesis, Loughborough University, 1998.

Cf. A338325, A338326, A338327, A338387, A338388, A338389, A338390, A338392.

bqfpowQ[n_] := AllTrue[FactorInteger[n][[;; , 2]], MemberQ[{2, 3}, #] &]; Select[Range[10^4], Count[Range[#^2 + 1, (# + 1)^2 - 1], _?bqfpowQ] == 4 &]

A338392 Numbers k such that there are exactly five biquadratefree powerful numbers (A338325) between k^2 and (k+1)^2.

Original entry on oeis.org

3510, 3611, 16871, 25076, 26910, 35810, 50501, 83107, 101287, 104686, 111836, 152924, 153433, 217983, 239163, 247301, 262413, 266282, 277635, 294453, 298950, 340228, 344510, 362830, 369877, 385336, 475063, 524827, 536793, 537713, 539293, 567062, 568609, 614283

Offset: 1

Amiram Eldar, Oct 23 2020

nonn

Positions of 5's in A338326.

The asymptotic density of this sequence is 0.000011113... (Dehkordi, 1998).

			3510 is a term since there are exactly five biquadratefree powerful numbers, 12320648 = 2^3 * 17^2 * 73^2, 12321000 = 2^3 * 3^2 * 5^3 * 37^2, 12324500 = 2^2 * 5^3 * 157^2, 12325975 = 5^2 * 79^3 and 12326391 = 3^3 * 7^3 * 11^3, between 3510^2 = 12320100 and (3510+1)^2 = 12327121.

Massoud H. Dehkordi, Asymptotic formulae for some arithmetic functions in number theory, Ph.D. thesis, Loughborough University, 1998.

Cf. A338325, A338326, A338327, A338387, A338388, A338389, A338390, A338391.

bqfpowQ[n_] := AllTrue[FactorInteger[n][[;; , 2]], MemberQ[{2, 3}, #] &]; Select[Range[25000], Count[Range[#^2 + 1, (# + 1)^2 - 1], _?bqfpowQ] == 5 &]

A330595 Decimal expansion of Product_{primes p} (1 + 1/p^2 + 1/p^3).

Original entry on oeis.org

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

Offset: 1

Vaclav Kotesovec, Dec 19 2019

			1.748932997843245303033906997685114802259883493595480897273662144084849...

Cf. A065464, A065473, A065476, A160889, A328017, A330594, A330596, A338325.

Do[Print[N[Exp[-Sum[q = Expand[(-p^2 - p^3)^j]; Sum[PrimeZetaP[Exponent[q[[k]], p]] * Coefficient[q[[k]], p^Exponent[q[[k]], p]], {k, 1, Length[q]}]/j, {j, 1, t}]], 110]], {t, 20, 200, 20}]

prodeulerrat(1 + 1/p^2 + 1/p^3) \\ Vaclav Kotesovec, Sep 19 2020

Equals Sum_{n>=1} 1/A338325(n). - Amiram Eldar, Oct 26 2020

A375073 Numbers whose prime factorization exponents include at least one 2, at least one 3 and no other exponents.

Original entry on oeis.org

72, 108, 200, 392, 500, 675, 968, 1125, 1323, 1352, 1372, 1800, 2312, 2700, 2888, 3087, 3267, 3528, 4232, 4500, 4563, 5292, 5324, 5400, 6125, 6728, 7688, 7803, 8575, 8712, 8788, 9000, 9747, 9800, 10584, 10952, 11979, 12168, 12348, 13068, 13448, 13500, 14283, 14792

Offset: 1

Amiram Eldar, Jul 29 2024

Numbers k such that the set of distinct prime factorization exponents of k (row k of A136568) is {2, 3}.

Number k such that A051904(k) = 2 and A051903(k) = 3.

Equals A338325 \ (A062503 UNION A062838).
Subsequence of A001694 and A046100.
A143610 is a subsequence.
Cf. A051903, A051904, A136568.
Cf. A082020, A157289, A330595.

Select[Range[15000], Union[FactorInteger[#][[;; , 2]]] == {2, 3} &]

PARI
```
is(k) = Set(factor(k)[,2]) == [2, 3];
```

Sum_{n>=1} 1/a(n) = Product_{p prime} (1 + 1/p^2 + 1/p^3) - 15/Pi^2 - zeta(3)/zeta(6) + 1 = A330595 - A082020 - A157289 + 1 = 0.047550294197921818806... .

cp's OEIS Frontend

A338326 The number of biquadratefree powerful numbers (A338325) between the consecutive squares n^2 and (n+1)^2.

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A338327 a(n) is the least number k such that there are exactly n biquadratefree powerful numbers (A338325) between k^2 and (k+1)^2.

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A338387 Numbers k such that there are no biquadratefree powerful numbers (A338325) between k^2 and (k+1)^2.

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A338388 Numbers k such that there is a single biquadratefree powerful number (A338325) between k^2 and (k+1)^2.

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A338389 Numbers k such that there are exactly two biquadratefree powerful numbers (A338325) between k^2 and (k+1)^2.

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A338390 Numbers k such that there are exactly three biquadratefree powerful numbers (A338325) between k^2 and (k+1)^2.

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A338391 Numbers k such that there are exactly four biquadratefree powerful numbers (A338325) between k^2 and (k+1)^2.

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A338392 Numbers k such that there are exactly five biquadratefree powerful numbers (A338325) between k^2 and (k+1)^2.

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