A140480 -id:A140480 - cp's OEIS frontend

A141815 RMS numbers with non-unique RMS values.

627215, 876785, 1289441, 1815073, 2265353, 3132935, 3188809, 4390505, 4647985, 4730879, 6542705, 9026087, 11369969, 12705511, 15203889, 15857471, 17888153, 18253913, 18578719, 20871649, 21026655, 21930545, 22321663, 23630711

Offset: 1

Andrew Weimholt, Jul 07 2008

nonn

			627215 is an RMS Number with an RMS value of 160225.
876785 is an RMS Number with an RMS value of 160225.
Since these two RMS numbers have the same RMS value, their RMS values are non-unique and therefore they belong to the sequence.

Cf. A140480, A141812, A141813, A141814, A141816.

A141816 RMS values of the RMS numbers with non-unique RMS values: a(n) is the Root Mean Square of the divisors of A141815(n).

Original entry on oeis.org

160225, 160225, 456025, 456025, 801125, 801125, 801125, 801125, 1185665, 1185665, 1185665, 2280125, 4032145, 2280125, 4032145, 4005625, 6326125, 6456125, 6569225, 5226065, 4032145, 4005625, 4005625, 5928325, 6326125

Offset: 1

Andrew Weimholt, Jul 07 2008

nonn

			a(1)=160225, because A141815(1)=627215 and the Root Mean Square of the divisors of 627215 is 160225.

Cf. A140480, A141812, A141813, A141814, A141815.

A153876 a(n) = Sum_{i=2^(n-1)..2^n-1} sigma_0(i), sigma_0(i) number of divisors of n, n positive integer.

Original entry on oeis.org

1, 4, 11, 29, 68, 160, 364, 820, 1813, 3981, 8674, 18782, 40387, 86443, 184232, 391188, 827787, 1746443, 3674573, 7712561, 16151933, 33757505, 70422235, 146659055, 304947023, 633152157, 1312820598, 2718674046, 5623413203, 11618957217, 23982175093, 49452872529

Offset: 1

Ctibor O. Zizka, Jan 03 2009

This sequence tells how many binary numbers with n digits are there in the multiplication matrix [1,...,2^n -1]x[1,...,2^n -1]. In general, counting how many base-B numbers of length n are there in the multiplication matrix [1,...,B^n -1]x[1,...,B^n -1] gives a(n)= sum_{i=B^(n-1),(B^n)-1} sigma_0(i). Besides this motivation it is interesting to see the behavior of partial sums of sigma_0(i) on growing intervals : a(n)= sum_{i=f(n-1),f(n)} sigma_0(i).

Chai Wah Wu, Table of n, a(n) for n = 1..76

Cf. A000005, A006218, A095256, A140480, A346730.

a(n) = sum(i=2^(n-1), 2^n-1, numdiv(i)); \\ Michel Marcus, Oct 10 2021

from math import isqrt
def A153876(n): return ((t:=isqrt(b:=(1<Chai Wah Wu, Oct 23 2023

a(n) = A085831(n) - A085831(n-1)-1. - R. J. Mathar, Jan 05 2009

a(n) = Sum_{k>=1} k * A346730(n,k). - Alois P. Heinz, Aug 01 2021

a(14)-a(32) from Alois P. Heinz, Aug 01 2021

A327055 Numbers m such that the arithmetic mean and the quadratic mean (the root mean square) of the divisors of m are both integers.

Original entry on oeis.org

1, 7, 41, 239, 287, 1673, 3055, 6665, 9545, 9799, 9855, 21385, 26095, 34697, 46655, 66815, 68593, 68985, 125255, 155287, 182665, 242879, 273265, 380511, 391345, 404055, 421655, 627215, 730145, 814463, 823537, 876785, 1069895, 1087009, 1166399, 1204281, 1256489

Offset: 1

Jaroslav Krizek, Oct 07 2019

nonn

Numbers m such that A000203(m) / A000005(m) and sqrt(A001157(m) / A000005(m)) are both integers.
Intersection of A003601 and A140480.
Sequence deviates from A140480 (RMS numbers); first deviation is at a(461), a(461) = 2226133343. Number A140480(461) = 2217231104 is the first RMS number that are not arithmetic (see A327056 for such numbers).
Corresponding values of A000203(a(n)) / A000005(a(n)): 1, 4, 21, 120, 84, 480, 504, 1056, 1512, 2520, 1110, 2016, 4158, ...
Corresponding values of sqrt(A001157(a(n)) / A000005(a(n))): 1, 5, 29, 169, 145, 845, 1105, 2405, 3445, 4901, 2665, 5525, ... (sequence deviates from A141812).

			Number 41 is a term because sigma(41) / tau(41) = 42 / 2 = 21 and sqrt((1^2 + 41^2)  / tau(41) ) = sqrt(1682 /  2) = 29.
Values of means of the first RMS number 2217231104 that is not in the sequence: 418652080/9 (noninteger) and 247511537 (integer).

Giovanni Resta, Table of n, a(n) for n = 1..7391 (terms < 10^13)

Cf. A000005, A000203, A001157, A140480, A141812, A158298, A158299, A327056.

[m: m in [1..10^6] | IsIntegral(SumOfDivisors(m) / NumberOfDivisors(m)) and IsIntegral(Sqrt(&+[d^2: d in Divisors(m)] / NumberOfDivisors(m)))]

aQ[n_] := IntegerQ[DivisorSigma[1, n]/(d = DivisorSigma[0, n])] && IntegerQ @ Sqrt[DivisorSigma[2, n]/d]; Select[Range[10^5], aQ] (* Amiram Eldar, Oct 07 2019 *)

A140972 Numbers k such that arithmetic mean of squares of first k Lucas numbers is an integer.

Original entry on oeis.org

1, 10, 12, 24, 36, 48, 60, 72, 96, 108, 120, 144, 168, 180, 192, 216, 240, 250, 288, 300, 324, 336, 360, 384, 432, 442, 480, 504, 540, 550, 552, 576, 600, 612, 624, 648, 660, 672, 684, 720, 768, 840, 864, 900, 960, 972, 1008, 1068, 1080, 1104, 1152, 1176, 1200

Offset: 1

Ctibor O. Zizka, Jul 27 2008

The root mean square RMS(L(0),...,L(k-1)) is firstly an integer for k = 36.

			k = 10: (L(0)^2+...+L(9)^2)/10 = 935.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harvey P. Dale)

Cf. A000032, A140480, A140971.

With[{nn=1200},Transpose[Select[Thread[{Range[nn],Accumulate[ LucasL[ Range[0,nn-1]]^2]}],IntegerQ[Last[#]/First[#]]&]][[1]]] (* Harvey P. Dale, Jul 15 2012 *)

list(lim) = {my(L1 = 2, L2 = 1, s = 5); print1(1, ", "); for(k = 3, lim, L3 = L1 + L2; s += L3^2; if(!(s % k), print1(k, ", ")); L1 = L2; L2 = L3);} \\ Amiram Eldar, Jul 04 2025

k such that (L(0)^2+ L(1)^2+ ... + L(k-1)^2) / k is an integer, where L(i) is the i-th Lucas number.

a(1) inserted and a(48) onwards added by R. J. Mathar, Aug 04 2008

A144711 Numbers n such that [sum_i=1..r (p(i)^2)]/r = c^2. p(i) prime divisors of n, c integer.

Original entry on oeis.org

2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, 25, 27, 29, 31, 32, 37, 41, 43, 47, 49, 53, 59, 61, 64, 67, 71, 73, 79, 81, 83, 89, 97, 101, 103, 107, 109, 113, 119, 121, 125, 127, 128, 131, 137, 139, 149, 151, 157, 161, 163, 167, 169, 173, 179, 181, 191, 193, 197, 199, 211

Offset: 1

Ctibor O. Zizka, Sep 19 2008

A005063(n)/A001221(n) = c^2.
Also numbers n such that the root mean square (quadratic mean) of the prime divisors of n is an integer.
These numbers are power of primes (p^k with k>=1) (A000961) or in A255580. - Daniel Lignon, Feb 26 2015

Daniel Lignon, Table of n, a(n) for n = 1..1000

Cf. A005063, A001221, A140480.

A005063 := proc(n) add(p^2,p=numtheory[factorset](n)) ; end: A001221 := proc(n) nops(numtheory[factorset](n)) ; end: isA144711 := proc(n) local sofpr ; sofpr := A001221(n) ; if sofpr <> 0 then issqr(A005063(n)/sofpr) ; else false ; fi; end: for n from 1 to 500 do if isA144711(n) then printf("%d,",n) ; fi; od: # R. J. Mathar, Sep 20 2008

Select[Range[2,1000],IntegerQ[RootMeanSquare[Select[Divisors[#],PrimeQ]]]&] (* Daniel Lignon, Feb 26 2015 *)

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

A153022 Numbers n such that 1 plus the sum of squares of the first n primes is divisible by n+1.

Original entry on oeis.org

6, 10, 150, 238, 56824, 665460, 18468766, 193274008, 635705422, 790546908, 33256634229, 66874977806, 68066767730

Offset: 1

Ctibor O. Zizka, Dec 16 2008

No further term <200000. [R. J. Mathar, Jan 17 2009]
a(11) > pi(10^11). [Donovan Johnson, Mar 10 2010]
a(14) > 118*10^9. - Robert Price, Apr 10 2013

			1 plus the sum of squares of the first 6 primes: (1 + 2^2 + 3^2 + ... + 13^2)/7 = 54, thus 6 is an element of the sequence.

Cf. A111441, A045345, A140480

{n: n+1 | A024525(n) }. [R. J. Mathar, Jan 17 2009]

a(5) from R. J. Mathar, Jan 17 2009
a(6)-a(10) from Donovan Johnson, Mar 10 2010
a(11)-a(13) from Robert Price, Apr 10 2013

A224988 Even RMS numbers: even numbers n such that root mean square of divisors of n is an integer.

Original entry on oeis.org

2217231104, 6221622528, 9644780288, 12127073024, 15377570560, 15520617728, 22426778880, 25138541824, 34766068480, 43551357696, 49424655104, 56022543104, 67513462016, 84107119360, 84889511168, 90906475264, 107642993920, 156987452160, 174347951360, 175969792768

Offset: 1

Donovan Johnson, Apr 25 2013

nonn

Even numbers from A140480.
The first 20 terms are all divisible by 30976. 30976 = 2^8*11^2.
a(21) > 2*10^11.
All the 83 terms up to 10^13 are divisible by 30976. - Giovanni Resta, Oct 29 2019

			n = 2217231104 (even). sigma_2(n) = 6616291782395055852. n has 108 divisors. 6616291782395055852/108 = 247511537^2.

Giovanni Resta, Table of n, a(n) for n = 1..83 (terms < 10^13)
Donovan Johnson, 177 terms > 2*10^11

Cf. A000005, A001157, A140480.

forstep(n=2, 10^10, 2, s=sigma(n, 2); nd=numdiv(n); if(s%nd==0, if(issquare(s\nd), print(n))))

isok(n) = my(s=sigma(n, 2), nd=numdiv(n)); if(s%nd==0, issquare(s\nd), 0); \\ program adapted by Michel Marcus, Oct 29 2019

Even numbers n such that A001157(n)/A000005(n) is a square.

A247135 Composite numbers for which the root mean square of proper divisors is an integer.

Original entry on oeis.org

35, 49, 55, 145, 215, 341, 545, 589, 1189, 1681, 1769, 2449, 2641, 3005, 3131, 3599, 4681, 6931, 7601, 9899, 10469, 11215, 15871, 17639, 19511, 21691, 23711, 28345, 28369, 35429, 36521, 36811, 39059, 44609, 57121, 68189, 68759, 75349, 79921, 84419, 85801

Offset: 1

Daniel Lignon, Nov 20 2014

nonn

Of course, for all prime numbers the mean square of proper divisors is an integer.

			35 is a term because it is not a prime, its proper divisors are {1, 5, 7} and sqrt((1^2 + 5^2 + 7^2)/3) = 5, an integer. - _Colin Barker_, Nov 20 2014

Amiram Eldar, Table of n, a(n) for n = 1..5000 (terms 1..100 from Daniel Lignon)

Cf. A247142 (numbers for which the root mean square of proper divisors is an integer), A140480 (numbers for which the root mean square of all divisors is an integer) and A023884 (numbers for which the arithmetic mean of proper divisors is an integer).

Select[Range[2, 120000], (IntegerQ[RootMeanSquare[Most[Divisors[#]]]] && Not[PrimeQ[#]]) &]

s=[]; for(n=2, 120000, if(!isprime(n) && issquare((sigma(n, 2)-n^2)/(sigma(n, 0)-1)), s=concat(s, n))); s \\ Colin Barker, Nov 20 2014

A247136 Numbers for which the root mean square of nontrivial divisors is an integer.

Original entry on oeis.org

4, 9, 25, 49, 119, 121, 161, 169, 289, 343, 361, 369, 527, 529, 711, 721, 833, 841, 959, 961, 1081, 1127, 1241, 1369, 1681, 1695, 1767, 1849, 2047, 2209, 2809, 3281, 3335, 3481, 3553, 3713, 3721, 4207, 4489, 4633, 4681, 5041, 5047, 5215, 5329, 6241, 6713, 6887

Offset: 1

Daniel Lignon, Nov 20 2014

nonn

All the squares of prime numbers (A001248) have this property but there are other numbers (A247137): 119,161,343,369,527,711,721,833,959,1081...

			119 is a term because the nontrivial divisors of 119 are [7,17] and sqrt((7^2+17^2)/2)= 13 : it's an integer.

Daniel Lignon and Charles R Greathouse IV, Table of n, a(n) for n = 1..10000 (first 316 terms from Lignon)

Cf. A140480 (numbers for which the root mean square of all divisors is an integer), A247136 (numbers for which the root mean square of proper divisors is an integer) and A023886 (numbers for which the arithmetic mean of nontrivial divisors is an integer).

Select[Range[2,100000],(Not[PrimeQ[#]] && IntegerQ[ RootMeanSquare[ Rest[ Most[ Divisors[#]]]]])&]

integralRMS(v)=my(t=norml2(v)/#v); denominator(t)==1 && issquare(t)
is(n)=my(d=divisors(n)); #d>2 && integralRMS(d[2..#d-1]) \\ Charles R Greathouse IV, Nov 20 2014

Trivially a(n) << n^2 log^2 n. - Charles R Greathouse IV, Nov 20 2014

cp's OEIS Frontend

A141815 RMS numbers with non-unique RMS values.

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A141816 RMS values of the RMS numbers with non-unique RMS values: a(n) is the Root Mean Square of the divisors of A141815(n).

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A153876 a(n) = Sum_{i=2^(n-1)..2^n-1} sigma_0(i), sigma_0(i) number of divisors of n, n positive integer.

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A327055 Numbers m such that the arithmetic mean and the quadratic mean (the root mean square) of the divisors of m are both integers.

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Magma

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A140972 Numbers k such that arithmetic mean of squares of first k Lucas numbers is an integer.

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Mathematica

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A144711 Numbers n such that [sum_i=1..r (p(i)^2)]/r = c^2. p(i) prime divisors of n, c integer.

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Maple

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A153022 Numbers n such that 1 plus the sum of squares of the first n primes is divisible by n+1.

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A224988 Even RMS numbers: even numbers n such that root mean square of divisors of n is an integer.

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PARI

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