A350072 -id:A350072 - cp's OEIS frontend

A349162 a(n) = sigma(n) / gcd(sigma(n), A003961(n)), where A003961 shifts the prime factorization of n one step towards larger primes, and sigma is the sum of divisors function.

1, 1, 4, 7, 6, 4, 8, 5, 13, 6, 12, 28, 14, 8, 24, 31, 18, 13, 20, 2, 32, 12, 24, 4, 31, 14, 8, 56, 30, 24, 32, 7, 48, 18, 48, 91, 38, 20, 56, 10, 42, 32, 44, 28, 78, 24, 48, 124, 57, 31, 72, 98, 54, 8, 72, 40, 16, 30, 60, 8, 62, 32, 104, 127, 12, 48, 68, 14, 96, 48, 72, 13, 74, 38, 124, 140, 96, 56, 80, 62, 121, 42

Offset: 1

Antti Karttunen, Nov 09 2021

Denominator of ratio A003961(n) / A000203(n).
Small values are rare, but are not limited to the beginning. For example in range 1 .. 2^25, a(n) = 4 at n = 3, 6, 24, 792, 2720, 122944, 31307472.
Question: Would it be possible to prove that a(n) > 1 for all n > 2?
Obviously, 1's may occur only on squares & twice squares (A028982). See also comments in A350072. - Antti Karttunen, Feb 16 2022

Cf. A000203, A003961, A028982 (positions of odd terms), A319630, A336702, A342671, A348992 (the odd part), A348993, A349161 (numerators), A349163, A349164, A349627, A349628, A350072 [= a(n^2)].

Cf. also A349745, A351551, A351554.

Array[#1/GCD[##] & @@ {DivisorSigma[1, #], If[# == 1, 1, Times @@ Map[NextPrime[#1]^#2 & @@ # &, FactorInteger[#]]]} &, 82] (* Michael De Vlieger, Nov 11 2021 *)

A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A349162(n) = { my(s=sigma(n)); (s/gcd(s,A003961(n))); };

a(n) = A000203(n) / A342671(n) = A000203(n) / gcd(A000203(n), A003961(n)).

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.

A350073 a(n) = A064989(sigma(n)), where A064989 is multiplicative with a(2^e) = 1 and a(p^e) = prevprime(p)^e for odd primes p.

Original entry on oeis.org

1, 2, 1, 5, 2, 2, 1, 6, 11, 4, 2, 5, 5, 2, 2, 29, 4, 22, 3, 10, 1, 4, 2, 6, 29, 10, 3, 5, 6, 4, 1, 20, 2, 8, 2, 55, 17, 6, 5, 12, 10, 2, 7, 10, 22, 4, 2, 29, 34, 58, 4, 25, 8, 6, 4, 6, 3, 12, 6, 10, 29, 2, 11, 113, 10, 4, 13, 20, 2, 4, 4, 66, 31, 34, 29, 15, 2, 10, 3, 58, 49, 20, 10, 5, 8, 14, 6, 12, 12, 44, 5, 10

Offset: 1

Antti Karttunen, Dec 12 2021

Cf. A000203, A064989, A161942, A337343 (a(n) >= n), A348748, A348749.

Cf. also A326042, A350072.

f[2, e_] := 1; f[p_, e_] := NextPrime[p, -1]^e; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[DivisorSigma[1, n]]; Array[a, 100] (* Amiram Eldar, Dec 12 2021 *)

A064989(n) = { my(f = factor(n)); for (i=1, #f~, f[i,1] = if(2==f[i, 1],1,precprime(f[i, 1]-1))); factorback(f); };
A350073(n) = A064989(sigma(n));

Multiplicative with a(p^e) = A064989(1 + p + p^2 + ... + p^e).

a(n) = A064989(A000203(n)) = A064989(A161942(n)).

A349756 Numbers k such that the odd part of sigma(k) is equal to gcd(sigma(k), A003961(k)), where A003961 is fully multiplicative with a(p) = nextprime(p), and sigma is the sum of divisors function.

Original entry on oeis.org

1, 2, 3, 6, 7, 14, 20, 21, 24, 27, 31, 42, 54, 57, 60, 62, 93, 114, 120, 127, 140, 160, 168, 186, 189, 216, 217, 220, 237, 254, 264, 301, 378, 381, 399, 408, 420, 434, 460, 474, 480, 513, 540, 552, 602, 620, 651, 660, 744, 762, 792, 798, 837, 840, 889, 903, 940, 1026, 1080, 1120, 1128, 1140, 1302, 1320, 1380, 1392, 1512

Offset: 1

Antti Karttunen, Dec 03 2021

nonn

Numbers k for which A161942(k) = A342671(k).
From Antti Karttunen, Jul 23 2022: (Start)
Numbers k such that k is a multiple of A350073(k).
For any square s in this sequence, A349162(s) = 1, i.e. sigma(s) divides A003961(s), and also A286385(s). Question: Is 1 the only square in this sequence? (see the conjecture in A350072).
If both x and y are terms and gcd(x, y) = 1, then x*y is also present.
After 2, the only primes present are Mersenne primes, A000668.
(End)

Positions of 1's in A348992.
Positions where the powers of 2 (A000079) occur in A349162.
Cf. A000203, A003961, A161942, A286385, A342671, A350072, A350073, A355946 (characteristic function).
Cf. A000668, A046528 (subsequences).
Cf. also A348943.

f[p_, e_] := NextPrime[p]^e; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; oddpart[n_] := n/2^IntegerExponent[n, 2]; q[n_] := oddpart[(sigma = DivisorSigma[1, n])] == GCD[sigma, s[n]]; Select[Range[1500], q] (* Amiram Eldar, Dec 04 2021 *)

A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A355946(n) = { my(s=sigma(n)); !(A003961(n)%((s>>=valuation(s,2)))); };
isA349756(n) = A355946(n);

A350071 a(n) = gcd(sigma(n^2), A003961(n^2)), where A003961 shifts the prime factorization of n one step towards larger primes, and sigma is the sum of divisors function.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 7, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 3, 1, 7, 1, 1, 13, 1, 1, 1, 1, 3, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 7, 1, 3, 1, 1, 7, 3, 1, 1, 1, 1, 1, 3, 121, 1, 1, 13, 1, 1, 1, 21, 1, 1, 1, 3, 1, 3, 1, 3, 1, 7, 1, 1, 1, 3, 1, 3, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1

Offset: 1

Antti Karttunen, Dec 12 2021

nonn

Cf. A000203, A000290, A003961, A065764, A350072.

f1[p_, e_] := (p^(2*e + 1) - 1)/(p - 1); f2[p_, e_] := NextPrime[p]^(2*e); a[1] = 1; a[n_] := GCD[Times @@ f1 @@@ (f = FactorInteger[n]), Times @@ f2 @@@ f]; Array[a, 100] (* Amiram Eldar, Dec 12 2021 *)

A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A342671(n) = gcd(sigma(n), A003961(n));
A350071(n) = A342671(n^2);

a(n) = A342671(n^2).

a(n) = A065764(n) / A350072(n).

A364131 Numbers k for which A348717(k) is a multiple of A348717(sigma(k)).

Original entry on oeis.org

1, 2, 4, 9, 16, 25, 64, 81, 289, 324, 400, 484, 729, 1681, 2401, 3481, 4096, 5041, 7921, 10201, 15625, 17161, 27889, 28561, 29929, 39204, 65536, 83521, 85849, 146689, 262144, 279841, 458329, 491401, 531441, 552049, 579121, 597529, 683929, 703921, 707281, 734449, 829921, 1190281, 1203409, 1352569, 1394761, 1423249, 1481089

Offset: 1

Antti Karttunen, Jul 11 2023

nonn

Conjecture: All terms apart from a(2) = 2 are squares.

Cf. A000203, A008848, A023194 (subsequence), A348717, A350072.

A348717(n) = { my(f=factor(n)); if(#f~>0, my(pi1=primepi(f[1, 1])); for(k=1, #f~, f[k, 1] = prime(primepi(f[k, 1])-pi1+1))); factorback(f); }; \\ From A348717
isA364131(n) = !(A348717(n)%A348717(sigma(n)));

cp's OEIS Frontend

A349162 a(n) = sigma(n) / gcd(sigma(n), A003961(n)), where A003961 shifts the prime factorization of n one step towards larger primes, and sigma is the sum of divisors function.

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A008848 Squares whose sum of divisors is a square.

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A350073 a(n) = A064989(sigma(n)), where A064989 is multiplicative with a(2^e) = 1 and a(p^e) = prevprime(p)^e for odd primes p.

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A349756 Numbers k such that the odd part of sigma(k) is equal to gcd(sigma(k), A003961(k)), where A003961 is fully multiplicative with a(p) = nextprime(p), and sigma is the sum of divisors function.

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A350071 a(n) = gcd(sigma(n^2), A003961(n^2)), where A003961 shifts the prime factorization of n one step towards larger primes, and sigma is the sum of divisors function.

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A364131 Numbers k for which A348717(k) is a multiple of A348717(sigma(k)).

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