A069482 -id:A069482

A069487

Areas of Pythagorean triangles (A069482, A069484, A069486).

+20
4

30, 240, 840, 5544, 6864, 26520, 23256, 73416, 208104, 107880, 467976, 473304, 296184, 727560, 1494600, 2101344, 863760, 3138816, 2625864, 1492704, 5259504, 4248936, 7623384, 12845904, 7759224, 4244424

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CROSSREFS

Cf. A069482, A069484, A069486.

A272101

Square root of largest square dividing A069482(n).

+20
0

1, 4, 2, 6, 4, 2, 6, 2, 2, 2, 2, 2, 2, 6, 10, 4, 4, 16, 2, 12, 4, 18, 2, 4, 6, 2, 2, 12, 2, 4, 2, 2, 2, 24, 10, 2, 8, 2, 2, 8, 12, 2, 16, 2, 6, 2, 2, 30, 4, 2, 4, 8, 2, 2, 4, 2, 6, 2, 6, 2, 24, 20, 2, 4, 6, 36, 2, 6, 4, 6

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COMMENTS

a(n) is the square root of the square part of A069482(n).

FORMULA

a(n) = A000188(A069482(n)). - Michel Marcus, Apr 27 2016

CROSSREFS

Cf. A000188, A069482

A001248

Squares of primes.

+10
597

4, 9, 25, 49, 121, 169, 289, 361, 529, 841, 961, 1369, 1681, 1849, 2209, 2809, 3481, 3721, 4489, 5041, 5329, 6241, 6889, 7921, 9409, 10201, 10609, 11449, 11881, 12769, 16129, 17161, 18769, 19321, 22201, 22801, 24649, 26569, 27889, 29929, 32041, 32761, 36481

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CROSSREFS

Cf. A024450 (partial sums), A069482 (first differences).

A069484

a(n) = prime(n+1)^2 + prime(n)^2.

+10
23

13, 34, 74, 170, 290, 458, 650, 890, 1370, 1802, 2330, 3050, 3530, 4058, 5018, 6290, 7202, 8210, 9530, 10370, 11570, 13130, 14810, 17330, 19610, 20810, 22058, 23330, 24650, 28898, 33290, 35930, 38090, 41522, 45002

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COMMENTS

Together with A069482(n) and A069486(n) a Pythagorean triangle is formed with area = A069487(n).

CROSSREFS

Cf. A069485, A075892, A069482, A069486.

A056811

Number of primes not exceeding square root of n: primepi(sqrt(n)).

+10
10

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

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FORMULA

For k = 1, 2, ..., repeat k A069482(k) (that is, prime(k+1)^2 - prime(k)^2) times, and add 0 three times at the beginning (or begin the preceding by k = 0, with prime(0) set to 1). - Jean-Christophe Hervé, Oct 30 2013

CROSSREFS

Cf. A069482, A056813.

A078667

Integers that occur more than once as the difference of the squares of two consecutive primes.

+10
7

72, 120, 168, 312, 408, 552, 600, 768, 792, 912, 1032, 1848, 2472, 3048, 3192, 3288, 3528, 3720, 4008, 4920, 5160, 5208, 5808, 5928, 6072, 6480, 6792, 6840, 6888, 7080, 7152, 7248, 7512, 7728, 7800, 8520, 8760, 9072, 11400, 11880, 11928, 12792, 13200, 13320

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COMMENTS

1848 is the first integer that occurs exactly three times. The next few are 6888, 14280, 16008, 19152. 4920 is the first integer that occurs exactly four times. See A069482 for more details. - Richard R. Forberg, Feb 06 2015

A069486

a(n) = 2*prime(n)*prime(n+1).

+10
4

12, 30, 70, 154, 286, 442, 646, 874, 1334, 1798, 2294, 3034, 3526, 4042, 4982, 6254, 7198, 8174, 9514, 10366, 11534, 13114, 14774, 17266, 19594, 20806, 22042, 23326, 24634, 28702, 33274, 35894, 38086, 41422, 44998

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COMMENTS

together with A069482(n) and A069484(n) a Pythagorean triangle is formed with area = A069487(n).

CROSSREFS

Cf. A006094, A069482, A069484, A069487.

A257513

Square array A(row,col) = A083221(row+1,col) - A083221(row,col): the first differences of each column of array constructed from the sieve of Eratosthenes.

+10
4

1, 5, 2, 9, 16, 2, 13, 20, 24, 4, 17, 34, 42, 72, 2, 21, 38, 36, 66, 48, 4, 25, 52, 54, 96, 78, 120, 2, 29, 56, 48, 90, 60, 102, 72, 4, 33, 70, 66, 120, 90, 144, 114, 168, 6, 37, 74, 88, 158, 124, 194, 160, 230, 312, 2, 41, 88, 92, 138, 84, 150, 96, 162, 232, 120, 6, 45, 92, 114, 190, 140, 226, 176, 262, 360, 248, 408, 4

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CROSSREFS

Column 1: A001223, Column 2: A069482, Column 3: A109805, Column 4: A226502 (apart from the first term).

A056813

Largest non-unitary prime factor of LCM(1,...,n); that is, the largest prime which occurs to power > 1 in prime factorization of LCM(1,..,n).

+10
3

1, 1, 1, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7

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COMMENTS

For n>0, prime(n) appears {(prime(n+1))^2 - (prime(n))^2} times [from n=(prime(n))^2 to n=(prime(n+1))^2 - 1], that is, A000040(n) appears A069482(n) times (from n=A001248(n) to n=A084920(n+1)). - Lekraj Beedassy, Mar 31 2005

FORMULA

To get the sequence, repeat 1 three times, and then for any k >= 1, repeat A000040(k) A069482(k) times; or equivalently, for any k >= 1, repeat A008578(k) a number of times equal to A008578(k+1)^2 - A008578(k)^2. - Jean-Christophe Hervé, Oct 29 2013

CROSSREFS

Cf. A000040, A001248, A008578, A069482.

A069483

Largest prime factor of prime(n+1)^2 - prime(n)^2.

+10
3

5, 2, 3, 3, 3, 5, 3, 7, 13, 5, 17, 13, 7, 5, 5, 7, 5, 3, 23, 3, 19, 3, 43, 31, 11, 17, 7, 3, 37, 7, 43, 67, 23, 5, 5, 11, 5, 11, 17, 11, 5, 31, 3, 13, 11, 41, 31, 5, 19, 11, 59, 5, 41, 127, 13, 19, 5, 137, 31, 47, 5, 7, 103, 13, 7, 7, 167, 19, 29, 13, 89, 11, 37, 47, 127, 193, 131, 19

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FORMULA

a(n) = A006530(A069482(n)).

EXAMPLE

A069482(12) = A000040(13)^2 - A000040(12)^2 = 41^2 - 37^2 = 1681 - 1369 = 312 = 2*2*2*3*13, therefore a(12) = 13.

CROSSREFS

Cf. A006530, A069482, A069485.