A063880 -id:A063880 - cp's OEIS frontend

A097702 a(n) = (A063880(n) - 108)/216.

0, 2, 3, 5, 6, 8, 9, 11, 14, 15, 17, 18, 20, 21, 23, 26, 27, 29, 30, 32, 33, 35, 36, 38, 39, 41, 42, 44, 45, 47, 48, 50, 51, 53, 54, 56, 57, 59, 63, 65, 66, 68, 69, 71, 72, 74, 75, 77, 78, 80, 81, 83, 86, 89, 90, 92, 93, 95, 96, 98, 99, 101, 102, 104, 105, 107, 108, 110

Offset: 1

Ralf Stephan, Aug 26 2004

nonn

Conjecture: k is a term iff 6*k+3 is squarefree. - Vladeta Jovovic, Aug 27 2004
It is only a conjecture that all terms are integers (confirmed up to 10^6 by Robert G. Wilson v).
From Amiram Eldar, Aug 31 2024: (Start)
The first conjecture is true. If m = 216*k + 108 = 108 * (2*k + 1) is a term of A063880, then 2*k+1 is a squarefree number coprime to 6. This is because sigma(n)/usigma(n) is multiplicative, equals 1 if and only if n is squarefree and larger than 1 otherwise, sigma(108)/usigma(108) = 2 and sigma(3^k)/usigma(3^k) increases with k. 6*k+3 = 3*(2*k+1) is squarefree because 2*k+1 is a squarefree coprime to 6.
Assuming that (A063880(n) - 108)/216 is an integer for all n, we have a(n) = (A276378(n) - 1)/2. (End)

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

Cf. A000203, A005117, A034448, A063880, A097703, A276378.

usigma[n_] := Block[{d = Divisors[n]}, Plus @@ Select[d, GCD[ #, n/# ] == 1 &]]; (Select[ Range[ 24500], DivisorSigma[1, # ] == 2usigma[ # ] &] - 108)/216 (* Robert G. Wilson v, Aug 28 2004 *)

{u(n)=sumdiv(n,d,if(gcd(d,n/d)==1,d))}
n=2; while(n<50000,n++; if(sigma(n)==2*u(n),print1((n-108)/216", ")))

A097703 Numbers j such that m = 216j + 108 satisfies sigma(m) != 2usigma(m).

Original entry on oeis.org

1, 4, 7, 10, 12, 13, 16, 19, 22, 24, 25, 28, 31, 34, 37, 40, 43, 46, 49, 52, 55, 58, 60, 61, 62, 64, 67, 70, 73, 76, 79, 82, 84, 85, 87, 88, 91, 94, 97, 100, 103, 106, 109, 112, 115, 118, 121, 122, 124, 127, 130, 133, 136, 137, 139, 142, 144, 145, 148, 151, 154, 157

Offset: 1

Ralf Stephan, Aug 26 2004

nonn

Conjecture: all numbers of form 3k + 1 are here. Other terms are listed in A097704.
From Amiram Eldar, Aug 31 2024: (Start)
The conjecture is true. If j = 3*k+1, then m = 324*(2*k+1). Let e = A007949(2*k+1) >= 0, so 2*k+1 = 3^e * i and i coprime to 6. Then sigma(m)/(2 * usigma(m)) = (7/20) * (3^(e+5)-1)/(3^(e+4)+1) * sigma(i)/usigma(i) >= 847/820 > 1, because sigma(i)/usigma(i) >= 1 for all i.
If m = 216*j + 108 = 108*(2*j+1) then sigma(m) = 2*usigma(m) if and only if 2*j+1 is a squarefree number coprime to 3 (see A097702), i.e., 2*j+1 is a term of A276378. Therefore this sequence consists of numbers j such that 2*j+1 is either a multiple of 3 or nonsquarefree (or both). (End)

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

Complement of A097702.

Cf. A000203, A007949, A013929, A034448, A063880, A097704, A276378.

usigma[n_] := Block[{d = Divisors[n]}, Plus @@ Select[d, GCD[ #, n/# ] == 1 &]]; Complement[ Range[157], (Select[ Range[37000], DivisorSigma[1, # ] == 2usigma[ # ] &] - 108)/216] (* Robert G. Wilson v, Aug 28 2004 *)

is(k) = {my(f = factor(216*k + 108)); sigma(f) != 2 * prod(i = 1, #f~, 1 + f[i,1]^f[i,2]);} \\ Amiram Eldar, Aug 31 2024

A348505 a(n) = usigma(n) / gcd(sigma(n), usigma(n)), where sigma is the sum of divisors function, A000203, and usigma is the unitary sigma, A034448.

Original entry on oeis.org

1, 1, 1, 5, 1, 1, 1, 3, 10, 1, 1, 5, 1, 1, 1, 17, 1, 10, 1, 5, 1, 1, 1, 3, 26, 1, 7, 5, 1, 1, 1, 11, 1, 1, 1, 50, 1, 1, 1, 3, 1, 1, 1, 5, 10, 1, 1, 17, 50, 26, 1, 5, 1, 7, 1, 3, 1, 1, 1, 5, 1, 1, 10, 65, 1, 1, 1, 5, 1, 1, 1, 6, 1, 1, 26, 5, 1, 1, 1, 17, 82, 1, 1, 5, 1, 1, 1, 3, 1, 10, 1, 5, 1, 1, 1, 11, 1, 50, 10, 130

Offset: 1

Antti Karttunen, Oct 29 2021

nonn

This is not multiplicative. The first point where a(m*n) = a(m)*a(n) does not hold for coprime m and n is 72 = 8*9, where a(72) = 6 != 3*10 = a(8) * a(9).

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537
Index entries for sequences related to sigma(n)

Cf. A000203, A005117, A034448, A048146, A063880, A348503, A348504, A348506 (positions of ones).

Cf. also A344697, A348049.

f1[p_, e_] := p^e + 1; f2[p_, e_] := (p^(e + 1) - 1)/(p - 1); a[1] = 1; a[n_] := (usigma = Times @@ f1 @@@ (fct = FactorInteger[n])) / GCD[usigma, Times @@ f2 @@@ fct]; Array[a, 100] (* Amiram Eldar, Oct 29 2021 *)

A034448(n) = { my(f=factorint(n)); prod(k=1, #f~, 1+(f[k, 1]^f[k, 2])); }; \\ After code in A034448
A348505(n) = { my(u=A034448(n)); (u/gcd(u, sigma(n))); };

a(n) = A034448(n) / A348503(n) = A034448(n) / gcd(A000203(n), A034448(n)).

A348506 Numbers k such that sigma(k) is a multiple of usigma(k), where sigma is the sum of divisors function, A000203, and usigma is the unitary sigma, A034448.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 26, 29, 30, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 46, 47, 51, 53, 55, 57, 58, 59, 61, 62, 65, 66, 67, 69, 70, 71, 73, 74, 77, 78, 79, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97, 101, 102, 103, 105, 106, 107, 108, 109, 110, 111, 113, 114, 115, 118, 119

Offset: 1

Antti Karttunen, Oct 29 2021

nonn

Conjectured to be the union of A005117 and A063880.

Antti Karttunen, Table of n, a(n) for n = 1..20000

Positions of ones in A348505.
Cf. A005117 and A063880.
Cf. A000203, A034448.

A034448(n) = { my(f=factorint(n)); prod(k=1, #f~, 1+(f[k, 1]^f[k, 2])); }; \\ After code in A034448
isA348506(n) = !(sigma(n)%A034448(n));

A098241 Numbers k such that 216*k+108 is a term of A097703 and A007494 and A098240.

Original entry on oeis.org

302, 2117, 2909, 3327, 3932, 5142, 5747, 6957, 8772, 9377, 11192, 12402, 13007, 14217, 14547, 16032, 17847, 18452, 20267, 20366, 21477, 22082, 23292, 23897, 25107, 25403, 26922, 27527, 29342, 30552, 31157, 32367, 32972, 34182, 35997, 36602, 37823, 38417, 39627

Offset: 1

Ralf Stephan and Robert G. Wilson v, Sep 15 2004

nonn

Numbers k such that m = 216*k+108 satisfies sigma(m) <> 2*usigma(m) (A097703), m is not of the form 3x+1 (A007494) and GCD(2*m+1, numerator(Bernoulli(4*m+2))) is squarefree (A098240).

Also, terms m of A097704 such that GCD(2*m+1, Bernoulli(4*m+2)) is squarefree. Most terms of A097704 are in A098240. These are the exceptions.

Amiram Eldar, Table of n, a(n) for n = 1..100

Cf. A000203, A034448, A007494, A016777, A027641, A027642, A063880, A067778, A097703, A097704, A098240.

usigma[n_] := Block[{d = Divisors[n]}, Plus @@ Select[d, GCD[ #, n/# ] == 1 &]]; lmt = 1296000; t = (Select[ Range[ lmt], DivisorSigma[1, # ] == 2usigma[ # ] &] - 108)/216; u = (Select[ Range[ Floor[(lmt - 108)/432]], !SquareFreeQ[ GCD[ #, Numerator[ BernoulliB[ 2# ]] ]] &] -1)/2; v = Table[ 3k - 2, {k, Floor[(lmt - 108)/216]}]; Complement[ Range[ Floor[ (lmt - 108)/216]], t, u, v]
q[n_] := Mod[n, 3] != 1 && (Divisible[2*n + 1, 3] || (! Divisible[2*n + 1, 3] && ! SquareFreeQ[2*n + 1])) && SquareFreeQ[GCD[2*n + 1, BernoulliB[4*n + 2]]]; Select[Range[10^4], q] (* Amiram Eldar, Aug 31 2024 *)

More terms from Amiram Eldar, Aug 31 2024

cp's OEIS Frontend

A097702 a(n) = (A063880(n) - 108)/216.

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A097703 Numbers j such that m = 216*j + 108 satisfies sigma(m) != 2*usigma(m).

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A348505 a(n) = usigma(n) / gcd(sigma(n), usigma(n)), where sigma is the sum of divisors function, A000203, and usigma is the unitary sigma, A034448.

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A348506 Numbers k such that sigma(k) is a multiple of usigma(k), where sigma is the sum of divisors function, A000203, and usigma is the unitary sigma, A034448.

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A098241 Numbers k such that 216*k+108 is a term of A097703 and A007494 and A098240.

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A097703 Numbers j such that m = 216j + 108 satisfies sigma(m) != 2usigma(m).