A069482 -id:A069482 - cp's OEIS frontend

A069487 Areas of Pythagorean triangles (A069482, A069484, A069486).

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

Offset: 1

Reinhard Zumkeller, Mar 29 2002

nonn

			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

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.

Cf. A069482, A069484, A069486.

Cf. A000040, A030078, A127917.

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

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

A272101 Square root of largest square dividing A069482(n).

Original entry on oeis.org

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

Offset: 1

Benedict W. J. Irwin, Apr 20 2016

nonn

Analogous to A001223 with 2-norm.

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

			sqrt(5)=1*sqrt(5), a(n)=1.
sqrt(16)=4*sqrt(1), a(n)=4.
sqrt(24)=2*sqrt(6), a(n)=2.

Cf. A000188, A069482

Cf. A001223, A068361.

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

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

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

A001248 Squares of primes.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane

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

Ray Chandler, Table of n, a(n) for n = 1..10000 (first 5000 terms from N. J. A. Sloane)
Nicholas John Bizzell-Browning, LIE scales: Composing with scales of linear intervallic expansion, Ph. D. Thesis, Brunel Univ. (UK, 2024). See p. 143.
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

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).

a001248 n = a001248_list !! (n-1)
a001248_list = map (^ 2) a000040_list -- Reinhard Zumkeller, Sep 23 2011

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

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

Prime[Range[30]]^2 (* Zak Seidov, Dec 07 2011 *)
Select[Range[40000], DivisorSigma[0, #] == 3 &] (* Carlos Eduardo Olivieri, Jun 01 2015 *)

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

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

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

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

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)

A069484 a(n) = prime(n+1)^2 + prime(n)^2.

Original entry on oeis.org

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

Offset: 1

Reinhard Zumkeller, Mar 29 2002

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

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.

Cf. A069485, A075892, A069482, A069486.

Cf. A001248, A000040, A048851.

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

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 *)

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

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

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.

A056811 Number of primes not exceeding square root of n: primepi(sqrt(n)).

Original entry on oeis.org

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

Offset: 1

Labos Elemer, Aug 28 2000

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

			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.

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

Cf. A000196, A000720, A003418, A056170.

Cf. A069482, A056813.

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

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

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

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

A078667 Integers that occur more than once as the difference of the squares of two consecutive primes.

Original entry on oeis.org

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

Offset: 1

Jon Perry, Dec 15 2002

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

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

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A090783, A090784, A090785, A091878.

N:= 20000: # for terms <= N
V:= Vector(N):
p:= 3:
while 4*(p-1) <= N do
  q:= p; p:= nextprime(p);
  v:= p^2 - q^2;
  if v > N then next fi;
  V[v]:= V[v]+1
od:
select(v -> V[v] > 1, 2*[$1..N/2]); # Robert Israel, Aug 22 2025

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 *)

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]",")))

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

A069486 a(n) = 2prime(n)prime(n+1).

Original entry on oeis.org

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

Offset: 1

Reinhard Zumkeller, Mar 29 2002

nonn

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

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

Cf. A006094, A069482, A069484, A069487.

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

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.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, May 01 2015

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), ...

			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
...

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.

(define (A257513 n) (A257513bi (A002260 n) (A004736 n)))
(define (A257513bi row col) (- (A083221bi (+ 1 row) col) (A083221bi row col))) ;; A083221bi given in A083221.

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

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).

Original entry on oeis.org

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

Offset: 1

Labos Elemer, Aug 28 2000

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

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

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

Cf. A054041, A056170.

Cf. A000040, A001248, A008578, A069482.

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 *)

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

Corrected offset by Jean-Christophe Hervé, Oct 29 2013

A069483 Largest prime factor of prime(n+1)^2 - prime(n)^2.

Original entry on oeis.org

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

Offset: 1

Reinhard Zumkeller, Mar 29 2002, Aug 05 2007

nonn

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

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

Cf. A006530, A069482, A069485.

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

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

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

cp's OEIS Frontend

A069487 Areas of Pythagorean triangles (A069482, A069484, A069486).

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A272101 Square root of largest square dividing A069482(n).

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A001248 Squares of primes.

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A069484 a(n) = prime(n+1)^2 + prime(n)^2.

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A056811 Number of primes not exceeding square root of n: primepi(sqrt(n)).

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A078667 Integers that occur more than once as the difference of the squares of two consecutive primes.

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A069486 a(n) = 2*prime(n)*prime(n+1).

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A069486 a(n) = 2prime(n)prime(n+1).