A066885 -id:A066885 - cp's OEIS frontend

A110494 Least k such that prime(n)^2 divides binomial(2k,k).

3, 5, 13, 25, 61, 85, 145, 181, 265, 421, 481, 685, 841, 925, 1105, 1405, 1741, 1861, 2245, 2521, 2665, 3121, 3445, 3961, 4705, 5101, 5305, 5725, 5941, 6385, 8065, 8581, 9385, 9661, 11101, 11401, 12325, 13285, 13945, 14965, 16021, 16381, 18241, 18625, 19405

Offset: 1

T. D. Noe, Jul 22 2005

For prime p > sqrt(2n), p^2 does not divide binomial(2n,n).

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000 (first 60 terms from T. D. Noe)

Cf. A110493 (largest prime p such that p^2 divides binomial(2n, n)).

t=Table[f=FactorInteger[Binomial[2n, n]]; s=Select[f, #[[2]]>1&]; If[s=={}, 0, s[[ -1, 1]]], {n, 100}]; Table[p=Prime[i]; First[Flatten[Position[t, p]]], {i, PrimePi[Max[t]]}]
lk[n_]:=Module[{k=1,c=Prime[n]^2},While[!Divisible[Binomial[2k,k],c], k=k+2]; k]; Array[lk,40] (* Harvey P. Dale, Oct 10 2012 *)

fv(n,p)=my(s); while(n\=p, s+=n); s
a(n)=my(p=prime(n),k=p^2\2+1); while(fv(2*k,p)-2*fv(k,p)<2,k++); k \\ Charles R Greathouse IV, Mar 27 2014

a(n)=prime(n)^2\2+1 \\ Charles R Greathouse IV, Mar 27 2014

a(n) = (prime(n)^2+1)/2 for n > 1.

a(n) = A066885(n), n > 1. - R. J. Mathar, Aug 18 2008

A368085 Square array read by ascending antidiagonals: row n is the trajectory of P under the 'Px+1' map, where P = n-th prime.

Original entry on oeis.org

2, 3, 5, 5, 10, 11, 7, 26, 5, 23, 11, 50, 13, 16, 47, 13, 122, 25, 66, 8, 95, 17, 170, 61, 5, 33, 4, 191, 19, 290, 85, 672, 1, 11, 2, 383, 23, 362, 145, 17, 336, 8, 56, 1, 767, 29, 530, 181, 29, 222, 168, 4, 28, 4, 1535, 31, 842, 265, 3440, 494, 111, 84, 2, 14, 2, 3071

Offset: 1

Paolo Xausa, Dec 11 2023

The 'Px+1 map' is defined as follows: if there exists p = smallest prime < P which divides x then x = x/p, otherwise x = P*x + 1.

			Array begins:
  [ 1]   2,   5,  11,    23,   47,   95, 191, 383,  767, ... = A153893
  [ 2]   3,  10,   5,    16,    8,    4,   2,   1,    4, ... = A033478
  [ 3]   5,  26,  13,    66,   33,   11,  56,  28,   14, ... = A057688
  [ 4]   7,  50,  25,     5,    1,    8,   4,   2,    1, ... = A368113
  [ 5]  11, 122,  61,   672,  336,  168,  84,  42,   21, ... = A368114
  [ 6]  13, 170,  85,    17,  222,  111,  37, 482,  241, ... = A057684
  [ 7]  17, 290, 145,    29,  494,  247,  19, 324,  162, ... = A368115
  [ 8]  19, 362, 181,  3440, 1720,  860, 430, 215,   43, ... = A057685
  [ 9]  23, 530, 265,    53, 1220,  610, 305,  61, 1404, ... = A057686
  [10]  29, 842, 421, 12210, 6105, 2035, 407,  37, 1074, ... = A057687
  ...    |    |    |
      A000040 | A066885 (from n = 2)
           A066872

Paolo Xausa, Table of n, a(n) for n = 1..11325 (antidiagonals 1..150, flattened).

Rows 1-10: A153893, A033478, A057688, A368113, A368114, A057684, A368115, A057685, A057686, A057687.
Columns 1-3: A000040, A066872, A066885 (from n = 2).
Main diagonal gives A368159.
Cf. A057689, A057690, A057691.

Px1[p_,n_]:=Catch[For[i=1,iA368085list[dmax_]:=With[{a=Reverse[Table[NestList[Px1[Prime[n],#]&,Prime[n],dmax-n],{n,dmax}]]},Array[Diagonal[a,#]&,dmax,1-dmax]];
A368085list[15] (* Generates 15 antidiagonals *)

A066886 Sum of the elements in any transversal of a prime(n) X prime(n) array containing the numbers from 1 to prime(n)^2 in standard order.

Original entry on oeis.org

5, 15, 65, 175, 671, 1105, 2465, 3439, 6095, 12209, 14911, 25345, 34481, 39775, 51935, 74465, 102719, 113521, 150415, 178991, 194545, 246559, 285935, 352529, 456385, 515201, 546415, 612575, 647569, 721505, 1024255, 1124111, 1285745

Offset: 1

Enoch Haga, Jan 22 2002

a(n) is the sum of the primes in a prime(n) X prime(n) example of Haga's conjecture (see link below).

Robert Israel, Table of n, a(n) for n = 1..10000
Carlos Rivera, The prime puzzles & problems connection, conjecture 26, The Prime Puzzles and Problems Connection.

Cf. A005097, A006003, A006254, A066883, A066885.

map(t -> t*(t^2+1)/2, [seq(ithprime(i),i=1..100)]); # Robert Israel, Apr 04 2018

a[n_] := Prime[n] (Prime[n]^2 + 1)/2; Table[a[n], {n, 50}]

apply(x->(x*(x^2+1)/2), primes(100)) \\ Michel Marcus, Apr 04 2018

a(n) = prime(n)*(prime(n)^2+1)/2, where prime(n) is the n-th prime.
a(n) = A006003(prime(n)). - Michel Marcus, Apr 04 2018
a(n) = A006254(n-1)^4 - A005097(n-1)^4, for n>1. - Dimitris Valianatos, Apr 10 2018

Edited by Dean Hickerson, Jun 08 2002

A066883 Number of primes in the interval [p(n), p(n)^2] minus p(n), where p(n) is the n-th prime.

Original entry on oeis.org

0, 0, 2, 5, 15, 21, 38, 46, 68, 108, 121, 171, 210, 227, 268, 341, 412, 441, 524, 585, 612, 711, 781, 888, 1042, 1126, 1165, 1247, 1286, 1381, 1720, 1814, 1972, 2018, 2306, 2361, 2536, 2715, 2838, 3029, 3217, 3290, 3635, 3709, 3848, 3920, 4370, 4836

Offset: 1

Enoch Haga, Jan 26 2002

Haga's conjecture (see link below) is that if the integers from 1 to p^2 (p prime) are put in a p by p square in standard order, then there's a transversal consisting of primes; i.e., a set of p primes containing exactly one number in each row and column. E.g., for p=5 the primes 5, 7, 11, 19, 23 work. Since p is needed for the p-th column, primes less than p can't be used. a(n) is the number of primes available minus the number needed for the transversal.

Paulo Ribenboim, The New Book of Prime Number Records, 3rd ed., 1995, Springer, pp. 397-398

Harry J. Smith, Table of n, a(n) for n = 1..1000
Carlos Rivera, The prime puzzles & problems connection, conjecture 26

Cf. A066885, A066886, A054272.

20 for Y=1 to 140 30 A=nxtprm(A):B=A^2 40 for X=A to B 50 if X=prmdiv(X) then C=C+1 60 next X 70 print A; C; C-A; "-"; 80 C=0 90 next Y

a[n_] := PrimePi[(p=Prime[n])^2]-PrimePi[p-1]-p

{ for (n=1, 1000, a=primepi((p=prime(n))^2) - primepi(p - 1) - p; write("b066883.txt", n, " ", a) ) } \\ Harry J. Smith, Apr 04 2010

a(n) = A054272(n)-A000040(n).

Edited by Dean Hickerson, Jun 08 2002

A155185 Primes in A155175.

Original entry on oeis.org

5, 13, 113, 1741, 5101, 8581, 9941, 21841, 26681, 47741, 82013, 481181, 501001, 1009621, 2356621, 2542513, 3279361, 3723721, 4277813, 7757861, 8124481, 13204661, 25311613, 30772013, 44170601, 48619661, 51521401, 52541501, 54236113, 60731221, 72902813

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jan 21 2009

nonn

Hypotenuse C (prime numbers only) of primitive Pythagorean triangles such that perimeters are Averages of twin prime pairs, q=p+1, a=q^2-p^2, c=q^2+p^2, b=2*p*q, ar=a*b/2; s=a+b+c, s-+1 are primes. p=1,q=2,a=3,b=4,c=5=prime,s=12-+1primes, ...

Cf. A020882, A020886, A020884, A020883, A024364, A024406, A155171, A155173, A155174, A155175, A155176, A155177, A155178, A155180, A088483, A001844, A096891, A066885, A099776, A110494, A081589

lst={};Do[p=n;q=p+1;a=q^2-p^2;c=q^2+p^2;b=2*p*q;ar=a*b/2;s=a+b+c;If[PrimeQ[s-1]&&PrimeQ[s+1],If[PrimeQ[c],AppendTo[lst,c]]],{n,8!}];lst (* corrected by Ray Chandler, Feb 11 2020 *)

Sequence corrected by Ray Chandler, Feb 11 2020

A140384 Consecutive triples (x=(p^2-1)/2,2*p,y=z=(p^2+1)/2), p prime, representing isosceles triangles with sides (x,y,z).

Original entry on oeis.org

4, 6, 5, 12, 10, 13, 24, 14, 25, 60, 22, 61, 84, 26, 85, 144, 34, 145, 180, 38, 181, 264, 46, 265, 420, 58, 421, 480, 62, 481, 684, 74, 685, 840, 82, 841, 924, 86, 925, 1104, 94, 1105, 1404, 106, 1405, 1740, 118, 1741, 1860, 122, 1861, 2244, 134, 2245, 2520, 142

Offset: 1

Juri-Stepan Gerasimov, Jun 13 2008

Or: Triples A084921(i), A100484(i), A066885(i), i=2,3... in compound order.

Edited by R. J. Mathar, Jun 16 2008 and Jun 17 2008

A143822 Primes p such that sigma_0((p*p + 1)/2) = 4.

Original entry on oeis.org

13, 17, 23, 31, 37, 53, 67, 89, 97, 103, 109, 113, 127, 137, 149, 151, 163, 167, 179, 197, 211, 223, 227, 229, 241, 263, 269, 277, 281, 283, 311, 331, 347, 359, 367, 373, 383, 389, 397, 419, 431, 433, 439, 479, 491, 503, 509, 541, 547, 587, 601, 617, 619, 653

Offset: 1

Ctibor O. Zizka, Sep 02 2008

A048161 are primes p such that sigma_0((p*p+1)/2)= 2. Primes p such that sigma_0((p*p+1)/2)= 3 gives all RMS numbers (A140480) with 2 divisors (prime RMS numbers, prime NSW numbers (A088165)) and all RMS numbers with 4 divisors as those are a multiple of two nonequal RMS prime numbers. In general we look after primes p such that sigma_0((p*p+1)/2) equals some given integer k. RMS numbers n=p_1*...*p_t have k=2^t divisors (p_i prime, t integer >=1) and sigma_2(p_1*...*p_t)=(2^t)* (q_1^r_1 *...* q_t^r_t), q_j prime, r_t integer >=1.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000005 (sigma_0), A048161, A088165, A140480, A002315.

A066885 := proc(n) local p; p :=ithprime(n) ; (p^2+1)/2 ; end: A000005 := proc(n) numtheory[tau](n) ; end: for n from 2 to 300 do if A000005(A066885(n)) = 4 then printf("%d,",ithprime(n)) ; fi; od: # R. J. Mathar, Sep 04 2008

Select[Range[650], PrimeQ[#] && DivisorSigma[0, (#^2 + 1)/2] == 4 &] (* Amiram Eldar, Mar 11 2020 *)
Select[Prime[Range[150]],DivisorSigma[0,(#^2+1)/2]==4&] (* Harvey P. Dale, Sep 22 2022 *)

97 inserted and extended by R. J. Mathar, Sep 04 2008

A155186 Primes in A155171.

Original entry on oeis.org

2, 7, 29, 101, 107, 197, 227, 457, 647, 829, 1549, 1627, 2221, 2309, 2347, 2521, 2677, 2801, 3181, 3299, 3529, 3541, 3557, 3739, 3769, 4231, 4549, 4871, 4987, 5651, 5827, 5881, 6037, 6079, 6637, 6827, 7517, 7639, 7937, 9787, 11621, 12041, 12329, 13009

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jan 21 2009

nonn

Numbers p (prime numbers only) of primitive Pythagorean triangles such that perimeters are Averages of twin prime pairs, q=p+1, a=q^2-p^2, c=q^2+p^2, b=2*p*q, s=a+b+c, s-+1 are primes.

Cf. A020882, A020886, A020884, A020883, A024364, A024406, A155171, A155173, A155174, A155175, A155176, A155177, A155178, A155180, A088483, A001844, A096891, A066885, A099776, A110494, A081589, A155185, A019389, A062090, A050150

lst={};Do[p=n;q=p+1;a=q^2-p^2;c=q^2+p^2;b=2*p*q;ar=a*b/2;s=a+b+c;If[PrimeQ[s-1]&&PrimeQ[s+1],If[PrimeQ[p],AppendTo[lst,p]]],{n,8!}];lst

A370760 Table read by rows: row n is the unique primitive Pythagorean triple (a,b,c) such that a = prime(n).

Original entry on oeis.org

3, 4, 5, 5, 12, 13, 7, 24, 25, 11, 60, 61, 13, 84, 85, 17, 144, 145, 19, 180, 181, 23, 264, 265, 29, 420, 421, 31, 480, 481, 37, 684, 685, 41, 840, 841, 43, 924, 925, 47, 1104, 1105, 53, 1404, 1405, 59, 1740, 1741, 61, 1860, 1861, 67, 2244

Offset: 2

Miguel-Ángel Pérez García-Ortega, Mar 01 2024

See Corolario 5.2.3 of the reference.

			Table begins:
  n=2:   3,   4,   5;
  n=3:   5,  12,  13;
  n=4:   7,  24,  25;
  n=5:  11,  60,  61;
  n=6:  13,  84,  85;
  ...

Miguel Ángel Pérez García-Ortega, José Manuel Sánchez Muñoz and José Miguel Blanco Casado, El Libro de las Ternas Pitagóricas, Preprint 2024.

Miguel-Ángel Pérez García-Ortega, Capítulo 5.

Cf. A000040, A065091 (short leg), A216244 (long leg), A066885 (hypotenuse), A005097 (inradius).

Apply[Join, Map[{#,(#^2-1)/2,(#^2+1)/2}&,Prime[Range[2,31]]]]

Row n = (a, b, c) = (p, ( p^2 - 1 ) / 2, ( p^2 + 1 ) / 2), where p = prime(n) = A000040(n).

A140392 Triples of height (a prime p), base length x and side length y=z of isosceles triangles.

Original entry on oeis.org

3, 8, 5, 5, 24, 13, 7, 48, 25, 11, 120, 61, 13, 168, 85, 17, 288, 145, 19, 360, 181, 23, 528, 265, 29, 840, 421, 31, 960, 481, 37, 1368, 685, 41, 1680, 841, 43, 1848, 925, 47, 2208, 1105, 53, 2808, 1405, 59, 3480, 1741, 61, 3720, 1861, 67, 4488, 2245, 71, 5040, 2521

Offset: 1

Juri-Stepan Gerasimov, Jun 13 2008

Or: consecutive triples of p=A000040(j), x=2*A084920(j), y=z= A066885(j), j>=2.

The area of the triangles is half the product of height and base length, p*x/2=A127918(j).

			Contains (p,x,y) = (3,8,5), (5,24,13), (7,48,25), (11,120,61), ...

Edited and extended by R. J. Mathar, Jun 17 2008

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A110494 Least k such that prime(n)^2 divides binomial(2k,k).

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A368085 Square array read by ascending antidiagonals: row n is the trajectory of P under the 'Px+1' map, where P = n-th prime.

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A066886 Sum of the elements in any transversal of a prime(n) X prime(n) array containing the numbers from 1 to prime(n)^2 in standard order.

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A066883 Number of primes in the interval [p(n), p(n)^2] minus p(n), where p(n) is the n-th prime.

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A155185 Primes in A155175.

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A140384 Consecutive triples (x=(p^2-1)/2,2*p,y=z=(p^2+1)/2), p prime, representing isosceles triangles with sides (x,y,z).

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A143822 Primes p such that sigma_0((p*p + 1)/2) = 4.

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A155186 Primes in A155171.

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