A140480 -id:A140480 - cp's OEIS frontend

A066885 a(n) = (prime(n)^2 + 1)/2.

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

Offset: 2

Enoch Haga, Jan 22 2002

a(n) is the average of the numbers from 1 to prime(n)^2. It's also the average of the primes in a prime(n) X prime(n) example of Haga's conjecture (see link below).
If a(n) is a square c^2, then prime(n) is an NSW prime (A088165) and a prime RMS number (A140480). - Ctibor O. Zizka, Aug 26 2008
The sequence starts with a(2) = (3^2 + 1)/2 = 5 since a(1) would be (2^2 + 1)/2 = 5/2. - Michael B. Porter, Dec 14 2009

Harry J. Smith, Table of n, a(n) for n = 2..1000
Carlos Rivera, Conjecture 26. The Calendar-like square Conjecture, The Prime Puzzles and Problems Connection.

Cf. A066883, A066886.
Cf. A088165, A140480.
Cf. A084921.
Partial sums of A124434.

[(NthPrime(n)^2+1)/2 : n in [2..50]]; // Wesley Ivan Hurt, Jun 23 2015

A066885:=n->(ithprime(n)^2+1)/2: seq(A066885(n), n=2..50); # Wesley Ivan Hurt, Jun 23 2015

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

A066885(n) = (prime(n)^2+1)/2 \\ Michael B. Porter, Dec 14 2009

{ for (n=2, 1000, write("b066885.txt", n, " ", (prime(n)^2 + 1)/2) ) } \\ Harry J. Smith, Apr 04 2010

a(n) = 1 + A084921(n). - R. J. Mathar, Sep 30 2011
a(n) mod 4 = 1. - Altug Alkan, Apr 08 2016
Product_{n>=2} (1 - 1/a(n)) = 2/3. - Amiram Eldar, Jun 03 2022

Edited by Dean Hickerson, Jun 08 2002

A145551 Numbers k such that product of divisors of k / sum of divisors of k is an integer.

Original entry on oeis.org

1, 6, 28, 30, 66, 84, 102, 120, 210, 270, 318, 330, 364, 420, 462, 496, 510, 546, 570, 642, 672, 690, 714, 840, 868, 870, 924, 930, 966, 1080, 1092, 1122, 1320, 1410, 1428, 1488, 1518, 1590, 1638, 1722, 1770, 1782, 1890, 1932, 2040, 2130, 2226, 2280, 2310

Offset: 1

Ctibor O. Zizka, Oct 13 2008

Numbers k such that A007955(k)/A000203(k) is an integer

Numbers k such that k^sigma_0(k) is a multiple of sigma_1(k)^2. - Chai Wah Wu, Mar 09 2016

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1100 from Paolo P. Lava)

Cf. A000203, A007955, A140480

A007955 := proc(n) local dvs,d ; dvs := numtheory[divisors](n) ; mul(d,d=dvs) ; end: A000203 := proc(n) local dvs,d ; dvs := numtheory[divisors](n) ; add(d,d=dvs) ; end: isA145551 := proc(n) RETURN(A007955(n) mod A000203(n) = 0) ; end: for n from 1 to 10000 do if isA145551(n) then printf("%d,",n) ; fi; od: # R. J. Mathar, Oct 14 2008

spQ[n_]:=Module[{ds=Divisors[n]},IntegerQ[(Times@@ds)/Total[ds]]]; Select[ Range[2500],spQ] (* Harvey P. Dale, Jun 26 2012 *)
Select[Range[2500], Divisible[#^(DivisorSigma[0, #]/2), DivisorSigma[1, #]] &] (* Amiram Eldar, Nov 08 2020 *)

from sympy import divisor_sigma
A145551_list = [n for n in range(1,10**3) if not n**divisor_sigma(n,0) % divisor_sigma(n,1)**2] # Chai Wah Wu, Mar 09 2016

90, 96, 108, 126, 132, 140 removed, extended by R. J. Mathar, Oct 14 2008

A144695 Numbers n such that sigma_1(n)/sigma_0(n) = c^2, c an integer.

Original entry on oeis.org

1, 7, 17, 22, 30, 31, 71, 94, 97, 115, 119, 127, 138, 154, 164, 165, 199, 210, 214, 217, 241, 260, 265, 318, 337, 343, 374, 382, 449, 497, 510, 513, 517, 527, 577, 647, 658, 668, 679, 682, 705, 745, 759, 805, 862, 881, 889, 894, 930, 966, 967, 996, 1102, 1125

Offset: 1

Ctibor O. Zizka, Sep 19 2008

A000203(n)/A000005(n) = c^2. Generalized sigma-sequences are sequences of numbers n for which sigma_r(n)/sigma_s(n) = c^t . Sigma_i(n) denotes sum of i-th powers of divisors of n; c,r,s,t positive integers. This sequence has r=1,s=0,t=2, sequence A003601 has r=1,s=0,t=1, sequence {1,21,53,85,102,110,127,217,431,....} has r=1,s=0,t=3, sequence A020487 has r=2,s=1,t=1, sequence A020486 has r=2,s=0,t=1, sequence A140480 has r=2,s=0,t=2.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Divisor function

Cf. A000005, A000203, A140480, A003601, A020487, A020486.

A000005 := proc(n) numtheory[tau](n) ; end: A000203 := proc(n) numtheory[sigma](n) ; end: isA144695 := proc(n) local s ; s := A000005(n) ; if s <> 0 then issqr(A000203(n)/s) ; else false ; fi; end: for n from 1 to 5000 do if isA144695(n) then printf("%d,",n) ; fi; od: # R. J. Mathar, Sep 20 2008

Select[Range[1125], IntegerQ @ Sqrt[DivisorSigma[1, #]/DivisorSigma[0, #]] &] (* Amiram Eldar, Nov 20 2019 *)

isok(m) = my(f=factor(m), q=sigma(f)/numdiv(f)); issquare(q) && !frac(q); \\ Michel Marcus, Mar 15 2022

More terms from R. J. Mathar, Sep 20 2008

A141391 a(n) is the smallest unused number such that the RMS (Root Mean Square) of a(1) through a(n) is an integer.

Original entry on oeis.org

1, 7, 5, 11, 28, 14, 182, 70, 2, 66, 1518, 462, 1540, 616, 296, 600, 1950, 750, 10, 730, 2336, 876, 2436, 996, 3154, 1162, 5698, 210, 1554, 3234, 1638, 5382, 1872, 23088, 4368, 5934, 201, 4359, 77991, 7021, 13090, 4270, 12950, 74, 12802, 76466, 16954

Offset: 1

Andrew Weimholt, Jun 29 2008

nonn

Actual RMS values are given by A141392

			a(4) = 11. Sqrt(mean(1^2, 7^2, 5^2, k^2)) is an integer for k in {5,11,37}. We have already used 5 for a(3), so 11 is the smallest value available for a(4).

Giovanni Resta, Table of n, a(n) for n = 1..200

Cf. A141392, A141393, A141394, A140480.

s={1};Do[i=1;Until[!MemberQ[s,i]&&IntegerQ[RootMeanSquare[Append[s,i]]],i++];AppendTo[s,i],{n,46}];s (* James C. McMahon, Jul 24 2025 *)

A141813 Primitive RMS numbers: RMS numbers which are not the product of two smaller RMS numbers.

Original entry on oeis.org

1, 7, 41, 239, 3055, 6665, 9545, 9855, 26095, 34697, 155287, 380511, 421655, 627215, 814463, 823537, 1166399, 1204281, 1256489, 1289441, 1815073, 2265353, 2544697, 2627343, 3132935, 3188809, 3762639, 4647985, 4730879, 4963127, 4995569, 5054015, 5143945

Offset: 1

Andrew Weimholt, Jul 07 2008

nonn

RMS numbers (see A140480) are numbers such that the RMS (Root Mean Square) of their divisors is an integer. If A and B both appear in A140480 and GCD(A,B)=1, then A*B is also in A140480. This sequence contains only those RMS numbers that are not a product of smaller RMS numbers.

			The RMS Number 287 is not in the sequence because 287=7*41 and both 7 and 41 are RMS numbers.

Giovanni Resta, Table of n, a(n) for n = 1..2768 (terms < 10^13, first 666 terms from Donovan Johnson)

Cf. A140480, A141812, A141814, A141815, A141816.

A141814 RMS values of the Primitive RMS numbers: a(n) is the Root Mean Square of the divisors of A141813(n).

Original entry on oeis.org

1, 5, 29, 169, 1105, 2405, 3445, 2665, 9425, 12325, 55205, 101065, 124501, 160225, 204425, 239425, 292825, 226525, 446165, 456025, 456025, 801125, 637325, 493025, 801125, 801125, 706225, 1185665, 1185665, 1759925, 1770305, 1291225, 1313845, 1185665, 1743625

Offset: 1

Andrew Weimholt, Jul 07 2008, Jul 09 2008

nonn

			a(5)=1105, because A141813(5)=3055, with divisors 1,5,13,47,65,235,611,3055 and RMS(1,5,13,47,65,235,611,3055) = 1105.

Giovanni Resta, Table of n, a(n) for n = 1..2768 (terms 1..187 from Andrew Weimholt, terms 188..666 from Donovan Johnson)

Cf. A140480, A141812, A141813, A141815, A141816.

A153111 Solutions of the Pell-like equation 1 + 6AA = 7BB, with A, B integers.

Original entry on oeis.org

1, 25, 649, 16849, 437425, 11356201, 294823801, 7654062625, 198710804449, 5158826853049, 133930787374825, 3477041644892401, 90269151979827601, 2343520909830625225, 60841274503616428249, 1579529616184196509249, 41006928746285492812225

Offset: 1

Ctibor O. Zizka, Dec 18 2008

B is of the form B(i) = 26*B(i-1) - B(i-2) for B(0) = 1, B(1) = 25 (this sequence).
A is of the form A(i) = 26*A(i-1) - A(i-2) for A(0) = 1, A(1) = 27.
In general a Pell-like equation of the form 1 + X*A*A = (X + 1)*B*B has the solution A(i) = (4*X + 2)*A(i-1) - A(i-2), for A(0) = 1 and A(1) = (4*X + 3), and B(i) = (4*X + 2)*B(i-1) - B(i-2) for B(0) = 1 and B(1) = (4*X + 1).
Examples in the OEIS:
  X = 1 gives A002315 for A(i) and A001653 for B(i);
  X = 2 gives A054320 for A(i) and A072256 for B(i);
  X = 3 gives A028230 for A(i) and A001570 for B(i);
  X = 4 gives A049629 for A(i) and A007805 for B(i);
  X = 5 gives A133283 for A(i) and A157014 for B(i);
  X = 6 gives A157461 for A(i) and this sequence for B(i).
Positive values of x (or y) satisfying x^2 - 26*x*y + y^2 + 24 = 0. - Colin Barker, Feb 20 2014

Vincenzo Librandi, Table of n, a(n) for n = 1..200
Luigi Cimmino, Algebraic relations for recursive sequences, arXiv:math/0510417 [math.NT], 2005-2008.
Jeroen Demeyer, Diophantine sets of polynomials over number fields, arXiv:0807.1970 [math.NT], 2008.
Franz Lemmermeyer, Conics - a Poor Man's Elliptic Curves, arXiv:math/0311306 [math.NT], 2003.
Index entries for linear recurrences with constant coefficients, signature (26,-1).

Cf. A002315, A001653, A054320, A072256, A001078, A028230, A001570, A049629, A007805, A133283, A140480.

Cf. similar sequences listed in A238379.

I:=[1,25]; [n le 2 select I[n] else 26*Self(n-1)-Self(n-2): n in [1..20]]; // Vincenzo Librandi, Feb 22 2014

CoefficientList[Series[(1 - x)/(x^2 - 26 x + 1), {x, 0, 40}], x] (* Vincenzo Librandi, Feb 22 2014 *)
LinearRecurrence[{26, -1}, {1, 25}, 20] (* Jean-François Alcover, Jan 07 2019 *)

Vec(-x*(x-1)/(x^2-26*x+1) + O(x^100)) \\ Colin Barker, Feb 20 2014

a(n) = 26*a(n-1) - a(n-2). - Colin Barker, Feb 20 2014
G.f.: -x*(x - 1) / (x^2 - 26*x + 1). - Colin Barker, Feb 20 2014
a(n) = (1/14)*(7 - sqrt(42))*(1 + (13 + 2*sqrt(42))^(2*n - 1))/(13 + 2*sqrt(42))^(n - 1). - Bruno Berselli, Feb 25 2014
E.g.f.: (1/7)*(7*cosh(2*sqrt(42)*x) - sqrt(42)*sinh(2*sqrt(42)*x))*exp(13*x) - 1. - Franck Maminirina Ramaharo, Jan 07 2019

More terms from Philippe Deléham, Sep 19 2009; corrected by N. J. A. Sloane, Sep 20 2009

Additional term from Colin Barker, Feb 20 2014

A141392 a(n) = RMS( A141391(1) through A141391(n) ).

Original entry on oeis.org

1, 5, 5, 7, 14, 14, 70, 70, 66, 66, 462, 462, 616, 616, 600, 600, 750, 750, 730, 730, 876, 876, 996, 996, 1162, 1162, 1582, 1554, 1554, 1638, 1638, 1872, 1872, 4368, 4368, 4419, 4359, 4359, 13209, 13090, 13090, 12950, 12950, 12802, 12802, 16954, 16954

Offset: 1

Andrew Weimholt, Jun 29 2008

nonn

			a(4) = 7 because first 4 terms of A141391 are 1,7,5,11 and sqrt(mean(1^2, 7^2, 5^2, 11^2)) = 7.

Giovanni Resta, Table of n, a(n) for n = 1..200

Cf. A141391, A141393, A141394, A140480.

lim=46;a141391={1};Do[i=1;Until[!MemberQ[a141391,i]&&IntegerQ[RootMeanSquare[Append[a141391,i]]],i++];AppendTo[a141391,i],{n,lim}];a[n_]:=RootMeanSquare[a141391[[1;;n]]];Array[a,lim] (* James C. McMahon, Jul 24 2025 *)

A141393 a(n) is the smallest number, larger than the previous, such that the RMS (Root Mean Square) of a(0) through a(n) is an integer.

Original entry on oeis.org

1, 7, 25, 33, 84, 294, 462, 750, 1155, 1705, 2431, 3017, 18130, 19684, 22052, 28996, 36907, 45925, 61957, 71309, 168889, 471799, 9998261, 11975939, 21577709, 29764925, 853783375, 1388314375, 1438304875, 2710683875, 2741707205, 62802867395, 72342543455

Offset: 0

Andrew Weimholt, Jun 29 2008

nonn

Actual RMS values are given by A141394.

Giovanni Resta, Table of n, a(n) for n = 0..100

Cf. A141391, A141392, A141394, A140480.

a(22)-a(32) from Giovanni Resta, Jan 22 2014

A141394 a(n) = RMS( A141393(0) through A141393(n) ).

Original entry on oeis.org

1, 5, 15, 21, 42, 126, 210, 330, 495, 715, 1001, 1295, 5180, 7252, 9028, 11356, 14195, 17535, 22211, 26887, 45241, 109871, 2087549, 3186259, 5326355, 7832875, 164490375, 308102375, 403718875, 634415375, 794973005, 11129622070, 16694433105, 22566503645

Offset: 0

Andrew Weimholt, Jun 29 2008

nonn

			a(3) = 21 because first 4 terms of A141393 are 1,7,25,33 and sqrt(mean(1^2, 7^2, 25^2, 33^2)) = 21.

Giovanni Resta, Table of n, a(n) for n = 0..100

Cf. A141391, A141392, A141393, A140480.

a(22)-a(33) from Giovanni Resta, Jan 22 2014

cp's OEIS Frontend

A066885 a(n) = (prime(n)^2 + 1)/2.

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A145551 Numbers k such that product of divisors of k / sum of divisors of k is an integer.

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A144695 Numbers n such that sigma_1(n)/sigma_0(n) = c^2, c an integer.

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A141391 a(n) is the smallest unused number such that the RMS (Root Mean Square) of a(1) through a(n) is an integer.

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A141813 Primitive RMS numbers: RMS numbers which are not the product of two smaller RMS numbers.

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A141814 RMS values of the Primitive RMS numbers: a(n) is the Root Mean Square of the divisors of A141813(n).

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A153111 Solutions of the Pell-like equation 1 + 6*A*A = 7*B*B, with A, B integers.

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A141392 a(n) = RMS( A141391(1) through A141391(n) ).

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A153111 Solutions of the Pell-like equation 1 + 6AA = 7BB, with A, B integers.