A069482 -id:A069482

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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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

LINKS

Marius A. Burtea, Table of n, a(n) for n = 1..5000

César Aguilera, Two Prime Number Objects and The Velucchi Numbers, hal-02909691 [math.NT], 2020.

FORMULA

a(n) = A030078(n+1)*A000040(n) - A000040(n+1)*A030078(n).

a(n) = A000040(n+1)^3*A000040(n) - A000040(n+1)*A000040(n)^3.

a(n) = A000040(n)*A127917(n+1) - A127917(n)*A000040(n+1). - César Aguilera, Sep 18 2019

EXAMPLE

prime(2)^3 * prime(1) - prime(1)^3 * prime(2) = 3^3 * 2 - 2^3 * 3 = 54 - 24 = 30 that is the area of the Pythagorean triangle (5, 12, 13), so a(1) = 30. - Bernard Schott, Sep 23 2019

PROG

(Magma) [NthPrime(n+1)^3*NthPrime(n)-NthPrime(n+1)*(NthPrime(n)^3):n in [1..26]]; // Marius A. Burtea, Sep 19 2019

CROSSREFS

Cf. A069482, A069484, A069486.

Cf. A000040, A030078, A127917.

KEYWORD

nonn

AUTHOR

Reinhard Zumkeller, Mar 29 2002

STATUS

approved

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

Analogous to A001223 with 2-norm.

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

LINKS

Table of n, a(n) for n=1..70.

FORMULA

Conjectures: (Start)

a(A068361(n)) = A001223(A068361(n)).

a(A068361(n)) = 2 for n>1.

These are the only a(n)=A001223(n).

(End)

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

EXAMPLE

sqrt(5)=1*sqrt(5), a(n)=1.

sqrt(16)=4*sqrt(1), a(n)=4.

sqrt(24)=2*sqrt(6), a(n)=2.

MATHEMATICA

Table[Sqrt[Prime[n+1]^2-Prime[n]^2], {n, 1, 100}]/.Sqrt[_]->1

PROG

(PARI) a(n) = my(d=prime(n+1)^2 - prime(n)^2); sqrtint(d/core(d)); \\ Michel Marcus, Apr 27 2016

CROSSREFS

Cf. A000188, A069482

Cf. A001223, A068361.

KEYWORD

nonn

AUTHOR

Benedict W. J. Irwin, Apr 20 2016

STATUS

approved

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

Also 4, together with numbers n such that Sum_{d|n}(-1)^d = -A048272(n) = -3. - Benoit Cloitre, Apr 14 2002

Also, all solutions to the equation sigma(x) + phi(x) = 2x + 1. - Farideh Firoozbakht, Feb 02 2005

Unique numbers having 3 divisors (1, their square root, themselves). - Alexandre Wajnberg, Jan 15 2006

Smallest (or first) new number deleted at the n-th step in an Eratosthenes sieve. - Lekraj Beedassy, Aug 17 2006

Subsequence of semiprimes A001358. - Lekraj Beedassy, Sep 06 2006

Integers having only 1 factor other than 1 and the number itself. Every number in the sequence is a multiple of 1 factor other than 1 and the number itself. 4 : 2 is the only factor other than 1 and 4; 9 : 3 is the only factor other than 1 and 9; and so on. - Rachit Agrawal (rachit_agrawal(AT)daiict.ac.in), Oct 23 2007

The n-th number with p divisors is equal to the n-th prime raised to power p-1, where p is prime. - Omar E. Pol, May 06 2008

There are 2 Abelian groups of order p^2 (C_p^2 and C_p x C_p) and no non-Abelian group. - Franz Vrabec, Sep 11 2008

Also numbers n such that phi(n) = n - sqrt(n). - Michel Lagneau, May 25 2012

For n > 1, n is the sum of numbers from A006254(n-1) to A168565(n-1). - Vicente Izquierdo Gomez, Dec 01 2012

A078898(a(n)) = 2. - Reinhard Zumkeller, Apr 06 2015

Let r(n) = (a(n) - 1)/(a(n) + 1); then Product_{n>=1} r(n) = (3/5) * (4/5) * (12/13) * (24/25) * (60/61) * ... = 2/5. - Dimitris Valianatos, Feb 26 2019

Numbers k such that A051709(k) = 1. - Jianing Song, Jun 27 2021

LINKS

Ray Chandler, Table of n, a(n) for n = 1..10000 (first 5000 terms from N. J. A. Sloane)

R. P. Boas and N. J. A. Sloane, Correspondence, 1974

Marius Coman, On the special relation between the numbers of the form 505+ 1008k and the squares of primes, 2015.

Brady Haran and Matt Parker, Squaring Primes, Numberphile video (2018).

Eric Weisstein's World of Mathematics, Prime Power.

OEIS Wiki, Index entries for number of divisors

Index to sequences related to prime signature

FORMULA

n such that A062799(n) = 2. - Benoit Cloitre, Apr 06 2002

A000005(a(n)^(k-1)) = A005408(k) for all k>0. - Reinhard Zumkeller, Mar 04 2007

a(n) = A000040(n)^(3-1)=A000040(n)^2, where 3 is the number of divisors of a(n). - Omar E. Pol, May 06 2008

A000005(a(n)) = 3 or A002033(a(n)) = 2. - Juri-Stepan Gerasimov, Oct 10 2009

A033273(a(n)) = 3. - Juri-Stepan Gerasimov, Dec 07 2009

For n > 2: (a(n) + 17) mod 12 = 6. - Reinhard Zumkeller, May 12 2010

A192134(A095874(a(n))) = A005722(n) + 1. - Reinhard Zumkeller, Jun 26 2011

For n > 2: a(n) = 1 (mod 24). - Zak Seidov, Dec 07 2011

A211110(a(n)) = 2. - Reinhard Zumkeller, Apr 02 2012

a(n) = A087112(n,n). - Reinhard Zumkeller, Nov 25 2012

a(n) = prime(n)^2. - Jon E. Schoenfield, Mar 29 2015

Product_{n>=1} a(n)/(a(n)-1) = Pi^2/6. - Daniel Suteu, Feb 06 2017

Sum_{n>=1} 1/a(n) = P(2) = 0.4522474200... (A085548). - Amiram Eldar, Jul 27 2020

From Amiram Eldar, Jan 23 2021: (Start)

Product_{n>=1} (1 + 1/a(n)) = zeta(2)/zeta(4) = 15/Pi^2 (A082020).

Product_{n>=1} (1 - 1/a(n)) = 1/zeta(2) = 6/Pi^2 (A059956). (End)

MAPLE

A001248:=n->ithprime(n)^2; seq(A001248(k), k=1..50); # Wesley Ivan Hurt, Oct 11 2013

MATHEMATICA

Prime[Range[30]]^2 (* Zak Seidov, Dec 07 2011 *)

Select[Range[40000], DivisorSigma[0, #] == 3 &] (* Carlos Eduardo Olivieri, Jun 01 2015 *)

PROG

(PARI) forprime(p=2, 1e3, print1(p^2", ")) \\ Charles R Greathouse IV, Jun 10 2011

(PARI) A001248(n)=prime(n)^2 \\ M. F. Hasler, Sep 16 2012

(Haskell)

a001248 n = a001248_list !! (n-1)

a001248_list = map (^ 2) a000040_list -- Reinhard Zumkeller, Sep 23 2011

(Magma) [p^2: p in PrimesUpTo(300)]; // Vincenzo Librandi, Mar 27 2014

(SageMath) [n^2 for n in prime_range(1, 301)] # G. C. Greubel, May 02 2024

(Python)

from sympy import prime

def A001248(n): return prime(n)**2 # Chai Wah Wu, Aug 09 2024

CROSSREFS

Cf. A000005, A000040, A001358, A001694, A002033, A005408, A005722, A006254.

Cf. A008864, A033273, A049001, A048272, A051709, A059956, A060800, A062799.

Cf. A078898, A082020, A085548, A087112, A095874, A168565, A192134, A211110.

Cf. A258599.

Subsequence of A000430, A001358, and of A190641.

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

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane

STATUS

approved

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

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

LINKS

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Janyarak Tongsomporn, Saeree Wananiyaku, and Jörn Steuding, Sums of consecutive prime squares, Integers (2022) Vol. 22, #A9.

FORMULA

a(n) = A001248(n+1) + A001248(n) = A000040(n+1)^2 + A000040(n)^2.

a(n) = A048851(n+1).

a(n) = 2 * A075892(n) for n > 1.

MAPLE

seq(ithprime(n)^2+ithprime(n+1)^2, n = 1 .. 100); # Stefano Spezia, Dec 21 2018

MATHEMATICA

Table[Prime[n]^2+Prime[n+1]^2, {n, 5!}] (* Vladimir Joseph Stephan Orlovsky, Apr 12 2010 *)

Total[#^2]&/@Partition[Prime[Range[50]], 2, 1] (* Harvey P. Dale, May 26 2012 *)

PROG

(PARI) v=primes(101); vector(#v-1, i, v[i]^2+v[i+1]^2) \\ Charles R Greathouse IV, Aug 21 2011

(Python)

from sympy import prime

for n in range(1, 101): print(n, prime(n)**2+prime(n+1)**2) # Stefano Spezia, Dec 21 2018

CROSSREFS

Cf. A069485, A075892, A069482, A069486.

Cf. A001248, A000040, A048851.

KEYWORD

nonn,easy

AUTHOR

Reinhard Zumkeller, Mar 29 2002

STATUS

approved

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,9

COMMENTS

Number of primes among factors of LCM(1,...,n) whose exponent is > 1, i.e., number of non-unitary prime factors of LCM(1,...,n).

Number of positive integers <= n with exactly 3 divisors.

Number of squared primes not exceeding n. - Wesley Ivan Hurt, May 24 2013

Maximum number of composite numbers not exceeding n that are all coprime to each other. - Yifan Xie, Jul 07 2024

LINKS

T. D. Noe, Table of n, a(n) for n = 1..10000

FORMULA

a(n) = A056170(A003418(n)) = A000720(A000196(n)).

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

G.f.: (1/(1 - x)) * Sum_{k>=1} x^(prime(k)^2). - Ilya Gutkovskiy, Sep 14 2019

a(n) ~ 2*n^(1/2)/log(n), by the prime number theorem. - Harry Richman, Jan 19 2022

EXAMPLE

If n=169,...,288 = p()^2,...,p(7)^2-1, then only the first 6 primes have exponents larger than 1, resulting in powers: 128, 81, 125, 49, 121, 169. So a(n)=6 for as much as 288-169+1 = 120 values of n.

MATHEMATICA

Table[PrimePi[Sqrt[n]], {n, 100}] (* T. D. Noe, Mar 13 2013 *)

PROG

(PARI) a(n) = primepi(sqrt(n)); \\ Michel Marcus, Apr 11 2016

(Python)

from math import isqrt

from sympy import primepi

def a(n): return primepi(isqrt(n))

print([a(n) for n in range(1, 88)]) # Michael S. Branicky, Jan 19 2022

CROSSREFS

Cf. A000196, A000720, A003418, A056170.

Cf. A069482, A056813.

KEYWORD

nonn,easy

AUTHOR

Labos Elemer, Aug 28 2000

STATUS

approved

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

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

LINKS

Table of n, a(n) for n=1..44.

EXAMPLE

120 = 31^2 - 29^2 = 17^2 - 13^2.

MATHEMATICA

Sort@ DeleteDuplicates@ Flatten@ Select[Gather[NextPrime[#]^2 - #^2 & /@ Prime@ Range@ 1200], Length@ # > 1 &] (* Michael De Vlieger, Mar 18 2015 *)

Select[Tally[Differences[Prime[Range[1000]]^2]], #[[2]]>1&][[;; , 1]]//Sort (* Harvey P. Dale, Nov 16 2023 *)

PROG

(PARI) pv(v)=vecsort(vecextract(v, concat("1..", vc-1))) op=2; v=vector(5000); vc=1; forprime (p=3, 5000, v[vc]=p^2-op^2; vc++; op=p) v=pv(v) for (i=2, length(v), if (v[i]==v[i-1], print1(v[i]", ")))

CROSSREFS

Cf. A090783, A090784, A090785, A091878.

KEYWORD

nonn

AUTHOR

Jon Perry, Dec 15 2002

EXTENSIONS

Duplicate terms removed, as suggested by Richard R. Forberg, by Jon E. Schoenfield, Mar 15 2015

STATUS

approved

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

a(n) = 2*A006094(n);

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

LINKS

Table of n, a(n) for n=1..35.

MATHEMATICA

2Times@@#&/@Partition[Prime[Range[40]], 2, 1] (* Harvey P. Dale, Dec 17 2012 *)

CROSSREFS

Cf. A006094, A069482, A069484, A069487.

KEYWORD

nonn

AUTHOR

Reinhard Zumkeller, Mar 29 2002

STATUS

approved

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

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

The array is read by downwards antidiagonals as A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), ...

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..3321; the first 81 antidiagonals of the array

Index entries for sequences generated by sieves

FORMULA

A(row,col) = A083221(row+1,col) - A083221(row,col).

EXAMPLE

The top left corner of the array:

1, 5, 9, 13, 17, 21, 25, 29, 33, 37, 41, 45, 49, 53, 57, 61

2, 16, 20, 34, 38, 52, 56, 70, 74, 88, 92, 106, 110, 124, 128, 142

2, 24, 42, 36, 54, 48, 66, 88, 92, 114, 132, 126, 144, 138, 156, 178

4, 72, 66, 96, 90, 120, 158, 138, 190, 192, 186, 216, 254, 306, 300, 324

2, 48, 78, 60, 90, 124, 84, 140, 126, 108, 138, 172, 184, 144, 200, 186

4, 120, 102, 144, 194, 150, 226, 216, 198, 240, 290, 314, 270, 346, 336, 318

2, 72, 114, 160, 96, 176, 150, 120, 162, 208, 220, 156, 236, 210, 180, 260

4, 168, 230, 162, 262, 240, 210, 264, 326, 350, 282, 382, 360, 330, 430, 408

6, 312, 232, 360, 338, 304, 374, 456, 492, 412, 540, 518, 484, 612, 590, 672

2, 120, 248, 198, 144, 210, 280, 292, 180, 308, 258, 204, 332, 282, 352, 426

6, 408, 370, 320, 406, 504, 540, 428, 588, 550, 500, 660, 622, 720, 830, 730

4, 312, 246, 336, 434, 458, 318, 490, 432, 366, 538, 480, 578, 684, 552, 486

2, 168, 258, 352, 364, 204, 380, 306, 228, 404, 330, 424, 522, 366, 288, 378

4, 360, 470, 494, 330, 526, 456, 378, 574, 504, 614, 732, 576, 498, 600, 522

6, 600, 636, 460, 684, 614, 532, 756, 686, 816, 958, 794, 712, 830, 748, 866

...

PROG

(Scheme)

(define (A257513 n) (A257513bi (A002260 n) (A004736 n)))

(define (A257513bi row col) (- (A083221bi (+ 1 row) col) (A083221bi row col))) ;; A083221bi given in A083221.

CROSSREFS

Transpose: A257514.

Row 1: A016813.

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

Cf. A083221, A257251.

KEYWORD

nonn,tabl,look

AUTHOR

Antti Karttunen, May 01 2015

STATUS

approved

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,4

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

a(n) is the largest prime factor of A045948(n). [Matthew Vandermast, Oct 29 2008]

Alternative definition: a(n) = largest prime <= sqrt(n) (considering 1 as prime for this occasion, see A008578 for the 19th century definition of primes). - Jean-Christophe Hervé, Oct 29 2013

LINKS

Jean-Christophe Hervé, Table of n, a(n) for n = 1..10000

FORMULA

a(n) = prime(w) if prime(w)^2 <= n < prime(w+1)^2.

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

EXAMPLE

The j-th prime appears at the position of its square, at n = prime(j)^2.

MATHEMATICA

Table[f = Transpose[FactorInteger[LCM @@ Range[n]]]; pos = Position[f[[2]], _?(# > 1 &)]; If[pos == {}, 1, f[[1, pos[[-1]]]][[1]]], {n, 100}] (* T. D. Noe, Oct 30 2013 *)

CROSSREFS

Cf. A054041, A056170.

Cf. A000040, A001248, A008578, A069482.

KEYWORD

nonn,easy

AUTHOR

Labos Elemer, Aug 28 2000

EXTENSIONS

Corrected offset by Jean-Christophe Hervé, Oct 29 2013

STATUS

approved

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

LINKS

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

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.

MATHEMATICA

FactorInteger[#[[2]]-#[[1]]][[-1, 1]]&/@Partition[Prime[Range[80]]^2, 2, 1] (* Harvey P. Dale, Jan 17 2016 *)

PROG

(PARI) a(n) = my(f=factor(prime(n+1)^2 - prime(n)^2)); f[#f~, 1]; \\ Michel Marcus, Nov 12 2023

CROSSREFS

Cf. A006530, A069482, A069485.

KEYWORD

nonn

AUTHOR

Reinhard Zumkeller, Mar 29 2002, Aug 05 2007

STATUS

approved

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