A231484 -id:A231484 - cp's OEIS frontend

A336547 Numbers k such that for 1 <= i < j <= h, all sigma(p_i^e_i), sigma(p_j^e_j) are pairwise coprime, when k = p_1^e_1 * ... * p_h^e_h, where each p_i^e_i is the maximal power of prime p_i dividing k.

1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 13, 14, 16, 17, 18, 19, 20, 23, 24, 25, 26, 27, 28, 29, 31, 32, 36, 37, 38, 41, 43, 44, 45, 47, 48, 49, 50, 53, 54, 56, 59, 61, 62, 63, 64, 67, 68, 71, 72, 73, 74, 75, 76, 79, 80, 81, 83, 86, 89, 92, 96, 97, 99, 100, 101, 103, 104, 107, 108, 109, 112, 113, 116, 117, 121, 122, 124, 125

Offset: 1

Antti Karttunen, Jul 25 2020

nonn

Numbers k such that A051027(k) = Product_{p^e||k} A051027(p^e) = A353802(n). Here each p^e is the maximal prime power divisor of k, and A051027(k) = sigma(sigma(k)). Numbers at which points A051027 appears to be multiplicative.
Proof that this interpretation is equal to the main definition:
  (1) If none of sigma(p_1^e_1), ..., sigma(p_k^e_k) share prime factors, then A051027(k) = sigma(sigma(p_1^e_1) * ... * sigma(p_k^e_k)) = A051027(p_1^e_1) * ... * A051027(p_k^e_k), by multiplicativity of sigma.
  (2) On the other hand, if say, gcd(sigma(p_i^e_i), sigma(p_j^e_j)) = c > 1 for some distinct i, j, then that c has at least one prime factor q, with product t = sigma(p_1^e_1) * ... * sigma(p_k^e_k) having a divisor of the form q^v (where v = valuation(t,q)), and the same prime factor q occurs as a divisor in more than one of the sigma(p_i^e_j), in the form q^k, with the exponents summing to v, then it is impossible to form sigma(q^v) = (1 + q + q^2 + ... + q^v) as a product of some sigma(q^k_1) * ... * sigma(q^k_z), i.e., as a product of (1 + q + ... + q^k_1) * ... * (1 + q + ... + q^k_z), with v = k_1 + ... + k_z, because such a product is always larger than (1 + q + ... + q^v). And if there are more such cases of "split primes", then each of them brings its own share to this monotonic inequivalence, thus Product_{p^e|n} A051027(p^e) = A353802(n) >= A051027(n), for all n.
From Antti Karttunen, May 07 2022: (Start)
Also numbers k such that A062401(k) = phi(sigma(n)) = Product_{p^e||k} A062401(p^e) = A353752(n).
Proof that also this interpretation is equal to the main definition:
  (1) like in (1) above, if none of sigma(p_1^e_1), ..., sigma(p_k^e_k) share prime factors, then by the multiplicativity of phi.
  (2) On the other hand, if say, gcd(sigma(p_i^e_i), sigma(p_j^e_j)) = c > 1 for some distinct i, j, then that c must have a prime factor q occurring in both sigma(p_i^e_i) and sigma(p_j^e_j), with say q^x being the highest power of q in the former, and q^y in the latter. Then phi(q^x)*phi(q^y) < phi(q^(x+y)), i.e. here the inequivalence acts to the opposite direction than with sigma(sigma(...)), so we have A353752(n) <= A062401(n) for all n.
(End)
From Antti Karttunen, May 22 2022: (Start)
All even perfect numbers (even terms of A000396) are included in this sequence. In general, for any perfect number n in this sequence, map k -> A026741(sigma(k)) induces on its unitary prime power divisors (p^e||n) a permutation that is a single cycle, mapping each one of them to the next larger one, except that the largest is mapped to the smallest one. Therefore, for a hypothetical odd perfect number n = x*a*b*c*d*e*f*g*h to be included in this sequence, where x is Euler's special factor of the form (4k+1)^(4h+1), and a .. h are even powers of odd primes (of which there are at least eight distinct ones, see P. P. Nielsen reference in A228058), further constraints are imposed on it: (1) that h < x < 2*a (here assuming that a, b, c, ..., g, h have already been sorted by their size, thus we have a < b < ... < g < h < x < 2*a, and (2), that we must also have sigma(a) = b, sigma(b) = c, ..., sigma(f) = g, sigma(h) = x, and sigma(x) = 2*a. Note that of the 400 initial terms of A008848, only its second term 81 is a prime power, so empirically this seems highly unlikely to ever happen.
(End)

			28 = 2^2 * 7 is present, as sigma(2^2) = 7 and sigma(7) = 8, and 7 and 8 are relatively prime (do not share prime factors). Likewise for all even terms of A000396. - _Antti Karttunen_, May 09 2022

Antti Karttunen, Table of n, a(n) for n = 1..25000
Index entries for sequences related to sigma(n)

Cf. A051027, A062401, A336546 (characteristic function), A336548 (complement).
Positions of zeros in A336562, in A353753, and in A353803.
Positions of ones in A353755, in A353784, and in A353806.
Union of A000961 and A336549.
Subsequence of A336358 and of A336557.
Cf. also A000396, A008848, A228058, A231484, A353752, A353802.

PARI
```
isA336547(n) = A336546(n);
```

{k | A336562(k) == 0}. - Antti Karttunen, May 09 2022

The old definition moved to comments and replaced with an alternative definition from the comment section by Antti Karttunen, May 07 2022

A008848 Squares whose sum of divisors is a square.

Original entry on oeis.org

1, 81, 400, 32400, 1705636, 3648100, 138156516, 295496100, 1055340196, 1476326929, 2263475776, 2323432804, 2592846400, 2661528100, 7036525456, 10994571025, 17604513124, 39415749156, 61436066769, 85482555876, 90526367376, 97577515876, 98551417041

Offset: 1

N. J. A. Sloane

nonn

Solutions to sigma(x^2) = (2k+1)^2. - Labos Elemer, Aug 22 2002
Intersection of A006532 and A000290. The product of any two coprime terms is also in this sequence. - Charles R Greathouse IV, May 10 2011
Also intersection of A069070 and A000290. - Michel Marcus, Oct 06 2013
Conjectures: (1) a(2) = 81 is the only prime power (A246655) in this sequence. (2) 81 and 400 are only terms x for which sigma(x) is in A246655. (3) x = 1 is the only such term that sigma(x) is also a term. See also comments in A074386, A336547 and A350072. - Antti Karttunen, Jul 03 2023, (2) corrected in May 11 2024

			n=81: sigma(81) = 1+3+9+27+81 = 121 = 11^2.
n=400: sigma(400) = sigma(16)*sigma(25) = 31*31 = 961.
n=32400 (= 81*400): sigma(32400) = 116281 = 341^2 = 121*961.

A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 10.
I. Kaplansky, The challenges of Fermat, Wallis and Ozanam (and several related challenges): II. Fermat's second challenge, Preprint, 2002.

Donovan Johnson, Table of n, a(n) for n = 1..400

Terms of A008847 squared.
Cf. A028982, A001248, A000203, A074386, A231484 (odd terms), A246655, A336547, A350072.
Subsequence of A000290, of A006532, and of A069070.

Do[s=DivisorSigma[1, n^2]; If[IntegerQ[Sqrt[s]]&&Mod[s, 2]==1, Print[n^2]], {n, 1, 10000000}] (* Labos Elemer *)
Select[Range[320000]^2,IntegerQ[Sqrt[DivisorSigma[1,#]]]&] (* Harvey P. Dale, Feb 22 2015 *)

for(n=1,1e6,if(issquare(sigma(n^2)), print1(n^2", "))) \\ Charles R Greathouse IV, May 10 2011

a(n) = A008847(n)^2.

A351448 Odd numbers k for which A003958(sigma(k)) = 2*A003958(k), where A003958 is multiplicative with a(p^e) = (p-1)^e and sigma is the sum of divisors function.

Original entry on oeis.org

8181, 400869, 1507005, 3918213, 11151837, 22002273, 26669007, 47319957, 58170393, 73843245, 75825981, 83488077, 94338513, 108277641, 119656197, 126889821, 137740257, 163057941, 184758813, 191992437, 199226061, 202842873, 204768225, 220926933, 228160557, 258457473, 264328677, 277602471, 300496797

Offset: 1

Antti Karttunen, Feb 12 2022

nonn

Odd numbers k such that A351442(k) = 2*A003958(k).
Any hypothetical odd term of A005820, if such a term exists, should appear in this sequence, in A347391, and in A016754 (odd squares).
None of the first 33 terms is a square, and all of them except 75825981 and 204768225 are multiples of 81. Note that 81 is one of the terms of A008848 (and of A231484), squares whose sum of divisors is also square (with A000203(81) = 121).

Index entries for sequences related to sigma(n)

Odd terms in A351447.

Cf. A000203, A003958, A005820, A008848, A016754, A231484, A339905, A347391, A351442.

A003958(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]--); factorback(f); };
isA351448(n) = (n%2 && (A003958(sigma(n)) == 2*A003958(n)));

cp's OEIS Frontend

A336547 Numbers k such that for 1 <= i < j <= h, all sigma(p_i^e_i), sigma(p_j^e_j) are pairwise coprime, when k = p_1^e_1 * ... * p_h^e_h, where each p_i^e_i is the maximal power of prime p_i dividing k.

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A008848 Squares whose sum of divisors is a square.

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A351448 Odd numbers k for which A003958(sigma(k)) = 2*A003958(k), where A003958 is multiplicative with a(p^e) = (p-1)^e and sigma is the sum of divisors function.

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