A140480 -id:A140480 - cp's OEIS frontend

A247142 Numbers for which the root mean square of proper divisors is an integer.

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 35, 37, 41, 43, 47, 49, 53, 55, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 145, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 215, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 307, 311, 313, 317, 331, 337, 341

Offset: 1

Daniel Lignon, Nov 20 2014

nonn

All the prime numbers (A000040) are in this sequence. But there are other numbers (A247135).

Daniel Lignon, Table of n, a(n) for n = 1..1249

Cf. A140480 (RMS of all divisors is an integer).

Select[Range[2,1000],IntegerQ[RootMeanSquare[Most[Divisors[#]]]]&]

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

A144857 Numbers k that divide Sum_{i=1..k} phi(i)^2, where phi(i) = totient function A000010.

Original entry on oeis.org

1, 2, 3, 6, 26, 190, 610, 2078, 2670, 7038, 16466, 89973, 150374, 157298, 163367, 419090, 640627, 879702, 3479689, 5618437, 11304721, 74106171, 471591726, 475915439, 1198344149, 2270643086, 3051266010

Offset: 1

Ctibor O. Zizka, Sep 23 2008

Does a number k exist such that RootMeanSquare(phi(1), ..., phi(k)) is an integer?

Cf. A000010, A057434, A048290, A140480.

lst = {}; s = 0; Do[ s = s + EulerPhi[n]^2; If[ Mod[s, n] == 0, AppendTo[lst, n]], {n, 10^9}]; lst (* Robert G. Wilson v, Oct 02 2008 *)

s=0;for(n=1,1e6,s+=eulerphi(n)^2;if(s%n==0,print1(n", "))) \\ Charles R Greathouse IV, Mar 05 2013

{k: k | A057434(k)}. - R. J. Mathar, Sep 29 2008

a(8)-a(11) from R. J. Mathar, Sep 29 2008
a(12)-a(24) from Robert G. Wilson v, Oct 02 2008
a(25)-a(27) from Donovan Johnson, Aug 21 2011

A144922 Numbers k such that k*sigma_2(k)/sigma_1(k) is an integer.

Original entry on oeis.org

1, 4, 6, 9, 12, 16, 18, 20, 24, 25, 28, 36, 44, 45, 48, 49, 50, 54, 60, 64, 72, 81, 90, 92, 96, 100, 108, 112, 117, 121, 132, 140, 144, 150, 153, 162, 168, 169, 180, 192, 196, 198, 200, 204, 216, 225, 228, 234, 240, 242, 252, 256, 270, 288, 289, 294, 300

Offset: 1

Ctibor O. Zizka, Sep 25 2008

Numbers k such that k*A001157(k)/A000203(k) is an integer. This sequence is connected closely with Ore divisor numbers (A001599) and RMS numbers (A140480).

This sequence is infinite. E.g., all the numbers of the form 3*2^m, for m >= 1, are terms, since 3*2^m * sigma_2(3*2^m) / sigma_1(3*2^m) = 5 * 2^(m-1) * (2^(m+1)+1) is an integer. - Amiram Eldar, Dec 25 2024

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

Cf. A000203, A001157, A001599, A140480.

A020487 is a subsequence.

Select[Range[300],IntegerQ[(#*DivisorSigma[2,#])/DivisorSigma[1,#]]&] (* Harvey P. Dale, Oct 28 2018 *)

is(k) = my(f = factor(k)); !((k*sigma(f, 2)) % sigma(f)); \\ Amiram Eldar, Dec 25 2024

More terms from Harvey P. Dale, Oct 28 2018

A145450 Numbers k such that k/A000005(k) is a square.

Original entry on oeis.org

1, 2, 36, 108, 128, 225, 288, 441, 450, 600, 864, 882, 1089, 1176, 1521, 1944, 2000, 2178, 2601, 2904, 3042, 3240, 3249, 4056, 4761, 5202, 6144, 6498, 6561, 6936, 7569, 8649, 8664, 9522, 10000, 11760, 12321, 12696, 13122, 15129, 15138, 16641, 17298

Offset: 1

Ctibor O. Zizka, Oct 10 2008

nonn

Subsequence of A033950.

			36/A000005(36) = 36/9 = 4 = 2^2, hence 36 is in the sequence.

Amiram Eldar, Table of n, a(n) for n = 1..5479 (terms below 10^9)

Cf. A000005 (sigma_0, number of divisors of n), A033950 (refactorable numbers), A140480 (RMS numbers).

Select[Range[20000],IntegerQ[Sqrt[#/DivisorSigma[0,#]]]&] (* Harvey P. Dale, May 21 2012 *)

{m=18000; for(n=1, m, s=sigma(n,0); if(n%s==0&&issquare(n/s), print1(n,",")))}

Edited and extended by Klaus Brockhaus, Oct 15 2008

A147591 RootMeanSquare(digits of n) is an integer.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 17, 22, 33, 44, 55, 66, 71, 77, 88, 99, 111, 115, 151, 157, 175, 222, 333, 444, 511, 517, 555, 571, 666, 715, 751, 777, 888, 999, 1111, 1135, 1153, 1177, 1315, 1339, 1351, 1393, 1513, 1531, 1557, 1575, 1717, 1755, 1771, 1933, 2000

Offset: 1

Ctibor O. Zizka, Nov 08 2008

			n=1236, RMS(digits of 1351)=sqrt((1^2+3^2+5^2+1^2)/4)=sqrt(36/4)=sqrt(9)=3.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A140480

Select[Range[2000],IntegerQ[RootMeanSquare[IntegerDigits[#]]]&] (* Harvey P. Dale, Sep 02 2017 *)

A147980 Given a set of positive integers A={1,2,...,n-1,n}, n>=2. Take subsets of A of the form {1,...,n} so only subsets containing numbers 1 and n are allowed. Then a(1)=1 and a(n) is the number of subsets where arithmetic mean of the subset is an integer.

Original entry on oeis.org

1, 0, 2, 0, 4, 4, 8, 12, 28, 44, 84, 156, 288, 540, 1020, 1904, 3616, 6860, 13024, 24836, 47448, 90772, 174072, 334348, 643112, 1238928, 2389956, 4615916, 8925808, 17278680, 33482196, 64944060, 126083448, 244989096, 476416560, 927167752, 1805691728, 3519062820

Offset: 1

Ctibor O. Zizka, Nov 18 2008

nonn

For n odd the value of the arithmetic mean for each possible subset equals (n+1)/2. For n even this value is n/2 or (n+2)/2. If looking after RootMeanSquare for the subset we obtain a sequence [1,0,0,0,0,0,2,...]. We see for example for n=7, A={1,2,3,4,5,6,7} and the only 2 subsets with an integer RootMeanSquare are {1,7}, {1,5,7}. Interestingly the value of RootMeanSquare is 5 for both subsets. So the sequence A140480 RMS numbers is a subsequence of it as a set of divisors of n is clearly a subset of n of the form {1,...,n}.

			n=5, A={1,2,3,4,5}. Subsets of A starting with 1 and ending with 5 are : {1,5}, {1,2,5}, {1,3,5}, {1,4,5}, {1,2,3,5}, {1,2,4,5}, {1,3,4,5}, {1,2,3,4,5}. Arithmetic mean of the subset is an integer for subsets : {1,5}, {1,3,5}, {1,2,4,5}, {1,2,3,4,5}. Thus a(5) = 4. The value of the arithmetic mean is 3 for all 4 subsets.

Alois P. Heinz, Table of n, a(n) for n = 1..100
Eric Weisstein's World of Mathematics, Arithmetic mean

Cf. A140480.

b:= proc(i,s,c) option remember; `if` (i=1, `if` (irem (s, c)=0, 1, 0), b(i-1, s, c)+ b(i-1, s+i, c+1)) end: a:= n-> `if` (n=1, 1, b (n-1, n+1, 2)): seq (a(n), n=1..40);  # Alois P. Heinz, May 06 2010

b[i_, s_, c_] := b[i, s, c] = If[i==1, If[Mod[s, c]==0, 1, 0], b[i-1, s, c] + b[i-1, s+i, c+1]];
a[n_] := If[n==1, 1, b[n-1, n+1, 2]];
Array[a, 40] (* Jean-François Alcover, Nov 20 2020, after Alois P. Heinz *)

More terms from Alois P. Heinz, May 06 2010

A152215 Numbers k such that sigma_2(k)/(sigma_1(k)*sigma_0(k)) = c, c an integer.

Original entry on oeis.org

1, 4, 25, 100, 121, 256, 289, 484, 529, 841, 1156, 1600, 1681, 2116, 2209, 2809, 3025, 3364, 3481, 5041, 6400, 6724, 6889, 7225, 7921, 8836, 10201, 11236, 11449, 12100, 12769, 13225, 13924, 17161, 18225, 18496, 18769, 20164, 21025, 22201, 27556, 27889, 28900

Offset: 1

Ctibor O. Zizka, Nov 29 2008

k : A001157(k)/(A000203(k)*A000005(k)) = c, c an integer.

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

Cf. A000005, A000203, A001157, A003601, A094471, A140480, A003601, A020486, A020487

Select[Range[50000],IntegerQ[DivisorSigma[2,#]/(DivisorSigma[1,#] DivisorSigma[ 0,#])]&] (* Harvey P. Dale, Feb 12 2013 *)

isok(k) = denominator(sigma(k,2)/(sigma(k, 1)*sigma(k,0))) == 1; \\ Michel Marcus, Jul 15 2019

More terms from Harvey P. Dale, Feb 12 2013

A152218 Numbers k such that sigma_2(k)*sigma_1(k)/sigma_0(k) is a perfect square.

Original entry on oeis.org

1, 4, 529, 2116, 2583, 3249, 3346, 6150, 10332, 12474, 12792, 12996, 28224, 38240, 59245, 85905, 91035, 103607, 142560, 176382, 212949, 236980, 249744, 343620, 360096, 364140, 379050, 414428, 450840, 751530, 787710, 788424, 851796, 1059474, 1132096, 1366407

Offset: 1

Ctibor O. Zizka, Nov 29 2008

nonn

Amiram Eldar, Table of n, a(n) for n = 1..500 (terms 1..200 from Donovan Johnson)

Cf. A000005, A000290, A000203, A001157, A140480, A144695.

fQ[n_] := IntegerQ[ Sqrt[ DivisorSigma[2, n] DivisorSigma[1, n]/DivisorSigma[0, n]]]; k = 1; lst = {}; While[k < 1132096, If[ fQ@k, AppendTo[lst, k]; Print@k]; k++ ]; lst (* Robert G. Wilson v, Sep 10 2010 *)
Select[Range[137*10^4],IntegerQ[Sqrt[(DivisorSigma[2,#]DivisorSigma[ 1,#])/ DivisorSigma[ 0,#]]]&] (* Harvey P. Dale, Jun 18 2018 *)

isok(k) = {my(f = factor(k)); issquare(sigma(f, 2) * sigma(f) / numdiv(f));} \\ Amiram Eldar, Feb 01 2025

{k: A001157(k)*A000203(k)/A000005(k) in A000290}.

Correct definition recovered by Jack Brennen
12 more terms from R. J. Mathar, Aug 25 2010
More terms from Robert G. Wilson v, Sep 10 2010

A158287 Composite RMS numbers: composite numbers c such that root mean square of divisors of c is an integer.

Original entry on oeis.org

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, 1289441

Offset: 1

Jaroslav Krizek, Mar 15 2009

nonn

a(n) = composite number c (A002808), iff sqrt(sigma_2(c)/tau(c)) = sqrt(A001157(c)/A000005(c)) = k, for k = natural numbers (A000027). Prime RMS numbers (NSW primes) in A088165.

16 of the first 1654 terms are even (the smallest is 2217231104). The first 16 even terms are all divisible by 30976. - Donovan Johnson, Apr 17 2013

			a(1) = 287, sqrt(A001157(287)/A000005(287)) = sqrt(84100/4) = 145, number 145 is an integer.

Giovanni Resta, Table of n, a(n) for n = 1..7424 (terms < 10^13, first 1654 terms from Donovan Johnson)

Cf. A001157, A000005, A088165, A140480.

Select[Range[13*10^5],CompositeQ[#]&&IntegerQ[RootMeanSquare[Divisors[ #]]]&] (* Harvey P. Dale, Sep 23 2022 *)

cp's OEIS Frontend

A247142 Numbers for which the root mean square of proper divisors is an integer.

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

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A144857 Numbers k that divide Sum_{i=1..k} phi(i)^2, where phi(i) = totient function A000010.

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A144922 Numbers k such that k*sigma_2(k)/sigma_1(k) is an integer.

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A145450 Numbers k such that k/A000005(k) is a square.

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A147591 RootMeanSquare(digits of n) is an integer.

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A147980 Given a set of positive integers A={1,2,...,n-1,n}, n>=2. Take subsets of A of the form {1,...,n} so only subsets containing numbers 1 and n are allowed. Then a(1)=1 and a(n) is the number of subsets where arithmetic mean of the subset is an integer.

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Maple

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A152215 Numbers k such that sigma_2(k)/(sigma_1(k)*sigma_0(k)) = c, c an integer.

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