A107924 -id:A107924 - cp's OEIS frontend

A107925 Odd numbers k such that k^2 is an arithmetic number.

1, 7, 13, 19, 31, 37, 43, 61, 67, 73, 79, 91, 97, 103, 109, 121, 127, 133, 139, 151, 157, 163, 181, 193, 199, 211, 217, 223, 229, 241, 247, 259, 271, 277, 283, 301, 307, 313, 331, 337, 349, 367, 373, 379, 397, 403, 409, 421, 427, 433, 439, 457, 463, 469, 481

Offset: 1

Zak Seidov, Jun 10 2005

Also, odd numbers k such that the arithmetic and geometric means of the divisors of k^2 are both integer.
Even numbers with this property are more rare and given by A107924.
Contains A002476 as a subsequence.
From Jianing Song, Apr 25 2022: (Start)
For p prime, p^(k-1) is a term in A003601 if and only if (p^k-1)/(p-1) is divisible by k. So p^e is a term here if and only if k | (p^k-1)/(p-1) for k = 2*e+1. (Note that p cannot be equal to 2 if k | (p^k-1)/(p-1).)
If a,b are both here, gcd(a,b) = 1, then a*b is also here. If a is A107924 and b is here, gcd(a,b) = 1, then a*b is also in A107924.
Let r >= 1, p_1, p_2, ..., p_r be distinct primes, k_1, k_2, ..., k_r be odd numbers such that Product_{i=1..r} (p_i)^(k_i) is an arithmetic number. Then there exists a number i in 1..r such that (p_i)^(k_i) is an arithmetic number. See my link for a proof. (End)

Amiram Eldar, Table of n, a(n) for n = 1..10000
Jianing Song, Notes on A107925

Cf. A002476, A003601, A107924.

Select[Range[1, 500, 2], Mod[DivisorSigma[1, #^2], DivisorSigma[0, #^2]]==0&]

A277793 Numbers k such that the arithmetic and geometric means of the divisors of k are both integers.

Original entry on oeis.org

1, 49, 169, 361, 961, 1369, 1849, 3721, 4489, 5329, 6241, 8281, 9409, 10609, 11881, 14641, 16129, 17689, 19321, 22801, 24649, 26569, 32761, 37249, 39601, 44521, 47089, 49729, 52441, 58081, 61009, 67081, 73441, 76729, 80089, 87616, 90601, 94249, 97969, 109561, 113569, 121801, 134689

Offset: 1

Ilya Gutkovskiy, Oct 31 2016

Intersection of A000290 and A003601.
Union of squares of A107924 and squares of A107925.
The squares of the primes == 1 (mod 6), squares of A002476, are a subsequence: 49, 169, 361,... - R. J. Mathar, May 19 2020

			a(2) = 49 because 49 has 3 divisors {1,7,49} therefore (1 + 7 + 49)/3 = 19 and (1*7*49)^(1/3) = 7 are both integers.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Divisor
Wikipedia, Arithmetic number
Index entries for sequences related to sums of divisors

Cf. A000290, A003601, A107924, A107925.

Select[Range[140000], Divisible[DivisorSigma[1, #1], DivisorSigma[0, #1]] && Mod[DivisorSigma[0, #1], 2] == 1 & ]
Select[Range[150000],AllTrue[{Mean[Divisors[#]],GeometricMean[ Divisors[ #]]}, IntegerQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Nov 21 2018 *)

A245644 Numbers n such that n^3 is an arithmetic number.

Original entry on oeis.org

1, 3, 5, 7, 11, 13, 14, 15, 17, 19, 21, 23, 24, 29, 31, 33, 35, 37, 39, 41, 42, 43, 46, 47, 51, 52, 53, 55, 56, 57, 59, 61, 62, 65, 66, 67, 69, 70, 71, 73, 77, 79, 80, 83, 85, 87, 89, 91, 93, 94, 95, 97, 101, 103, 105, 107, 109, 111, 113, 114, 115, 117, 119, 120, 123, 127, 129, 131, 133, 137, 138, 139

Offset: 1

R. J. Mathar, Jul 28 2014

nonn

A065091 is a subsequence.

Reinhard Zumkeller and Jens Kruse Andersen, Table of n, a(n) for n = 1..10000 (first 147 terms from Reinhard Zumkeller)
Wikipedia, Arithmetic number

Cf. A003601, A107924, A107925.

Cf. A245656.

a245644 n = a245644_list !! (n-1)
a245644_list = filter ((== 1) . a245656 . (^ 3)) [1..]
-- Reinhard Zumkeller, Jul 28 2014

isArithPow := proc(n,e)
    local dvs,d ;
    dvs := numtheory[divisors](n^e) ;
    add(d,d=dvs)/nops(dvs) ;
    if type(%,'integer') then
        true;
    else
        false;
    end if;
end proc:
for n to 300 do
    if isArithPow(n,3) then
        printf("%d,",n) ;
    end if;
end do:

Select[Range[120], IntegerQ[DivisorSigma[1, #^3 ]/DivisorSigma[0, #^3 ]] &] (* Michael De Vlieger, Aug 05 2014 after Stefan Steinerberger at A003601 *)

from sympy import divisors, divisor_count
[n for n in range(1,10**3) if not sum(divisors(n**3)) % divisor_count(n**3)] # Chai Wah Wu, Aug 04 2014

A245656(a(n)^3) = 1. - Reinhard Zumkeller, Jul 28 2014

A353062 Non-prime-powers k such that sigma(k^2) is divisible by d(k^2), where d = A000005, sigma = A000203; non-prime-powers k such that k^2 is in A003601.

Original entry on oeis.org

1, 91, 133, 217, 247, 259, 296, 301, 403, 427, 469, 481, 511, 536, 553, 559, 589, 632, 679, 703, 721, 763, 793, 817, 847, 871, 872, 889, 949, 973, 999, 1027, 1057, 1099, 1141, 1147, 1159, 1208, 1261, 1267, 1273, 1304, 1333, 1339, 1351, 1387, 1393, 1417, 1477

Offset: 1

Jianing Song, Apr 20 2022

nonn

Here prime powers means the numbers in A246655.
For p prime, p^(k-1) is a term in A003601 if and only if (p^k-1)/(p-1) is divisible by k. So this sequence is (A107924 U A107925) \ {p^((k-1)/2): p prime, k odd, k | (p^k-1)/(p-1)}.
It is standard that k does not divide 2^k-1 for k > 1, so no term > 1 in A003601 can be a power of 2, hence A107924 is a subsequence.
Since a,b in A003601 (resp. A107924 U A107925) and gcd(a,b) = 1 implies that a*b is in A003601 (resp. A107924 U A107925), this sequence is infinite. For example, all numbers of the form (p_1)*(p_2)*...*(p_k) are here, where p_i's are distinct primes congruent to 1 modulo 3, k >= 2.

			91 is a term since sigma(91^2) = 10431 is divisible by d(91^2) = 9.
296 is a term since sigma(296^2) = 178689 is divisible by d(296^2) = 21. 296 is the smallest term that is not a product of coprime numbers > 1 in A107924 U A107925.
999 is a term since sigma(999^2) = 1537851 is divisible by d(999^2) = 21. 999 is the smallest odd term that is not a product of coprime numbers > 1 in A107924 U A107925.

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

Equals (A107924 U A107925) \ A246655. The even terms are listed in A107924.

Cf. A003601, A000005, A000203.

Select[Range[1500], !PrimePowerQ[#] && Divisible @@ DivisorSigma[{1, 0}, #^2] &] (* Amiram Eldar, Jul 19 2024 *)

isA353062(n) = sigma(n^2)%numdiv(n^2)==0 && !isprimepower(n)

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A107925 Odd numbers k such that k^2 is an arithmetic number.

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A277793 Numbers k such that the arithmetic and geometric means of the divisors of k are both integers.

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A245644 Numbers n such that n^3 is an arithmetic number.

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A353062 Non-prime-powers k such that sigma(k^2) is divisible by d(k^2), where d = A000005, sigma = A000203; non-prime-powers k such that k^2 is in A003601.

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