A374777 -id:A374777 - cp's OEIS frontend

A374778 Denominator of the mean abundancy index of the divisors of n.

1, 4, 6, 12, 10, 24, 14, 32, 27, 8, 22, 72, 26, 56, 60, 80, 34, 54, 38, 120, 4, 88, 46, 192, 75, 104, 54, 56, 58, 48, 62, 64, 132, 136, 28, 162, 74, 152, 52, 320, 82, 16, 86, 264, 135, 184, 94, 160, 49, 30, 204, 104, 106, 216, 20, 64, 76, 232, 118, 720, 122, 248

Offset: 1

Amiram Eldar, Jul 19 2024

			For n = 2, n has 2 divisors, 1 and 2. Their abundancy indices are sigma(1)/1 = 1 and sigma(2)/2 = 3/2, and their mean abundancy index is (1 + 3/2)/2 = 5/4. Therefore a(2) = denominator(5/4) = 4.

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

Cf. A374777 (numerators).

f[p_, e_] := ((e+1)*p^2 - (e+2)*p + p^(-e))/((e+1)*(p-1)^2); a[1] = 1; a[n_] := Denominator[Times @@ f @@@ FactorInteger[n]]; Array[a, 100]

a(n) = {my(f = factor(n), p, e); denominator(prod(i = 1, #f~, p = f[i, 1]; e = f[i, 2]; ((e+1)*p^2 - (e+2)*p + p^(-e))/((e+1)*(p-1)^2)));}

A374783 Numerator of the mean unitary abundancy index of the unitary divisors of n.

Original entry on oeis.org

1, 5, 7, 9, 11, 35, 15, 17, 19, 11, 23, 21, 27, 75, 77, 33, 35, 95, 39, 99, 5, 115, 47, 119, 51, 135, 55, 135, 59, 77, 63, 65, 161, 175, 33, 19, 75, 195, 63, 187, 83, 25, 87, 207, 209, 235, 95, 77, 99, 51, 245, 243, 107, 275, 23, 255, 91, 295, 119, 231, 123, 315

Offset: 1

Amiram Eldar, Jul 20 2024

The unitary abundancy index of a number k is A034448(k)/k = A332882(k)/A332883(k).
The record values of a(n)/A374784(n) are attained at the primorial numbers (A002110).
The least number k such that a(k)/A374784(k) is larger than 2, 3, 4, ..., is A002110(9) = 223092870, A002110(314) = 7.488... * 10^878, A002110(65599) = 5.373... * 10^356774, ... .

			For n = 4, 4 has 2 unitary divisors, 1 and 4. Their unitary abundancy indices are usigma(1)/1 = 1 and usigma(4)/4 = 5/4, and their mean unitary abundancy index is (1 + 5/4)/2 = 9/8. Therefore a(4) = numerator(9/8) = 9.

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

Cf. A002110, A005117, A034444, A034448, A077610, A306633, A332882, A332883, A343525, A374784 (denominators).

Similar sequences: A374777/A374778, A374786/A374787.

f[p_, e_] := 1 + 1/(2*p^e); a[1] = 1; a[n_] := Numerator[Times @@ f @@@ FactorInteger[n]]; Array[a, 100]

a(n) = {my(f = factor(n)); numerator(prod(i = 1, #f~, 1 + 1/(2*f[i,1]^f[i,2])));}

Let f(n) = a(n)/A374784(n). Then:
f(n) = (Sum_{d|n, gcd(d, n/d) = 1} usigma(d)/d) / ud(n), where usigma(n) is the sum of unitary divisors of n (A034448), and ud(n) is their number (A034444).
f(n) is multiplicative with f(p^e) = 1 + 1/(2*p^e).
f(n) = (Sum_{d|n, gcd(d, n/d) = 1} d*ud(d))/(n*ud(n)) = A343525(n)/(n*A034444(n)).
Dirichlet g.f. of f(n): zeta(s) * zeta(s+1) * Product_{p prime} (1 - 1/(2*p^(s+1)) - 1/(2*p^(2*s+1))).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} f(k) = Product_{p prime} (1 + 1/(2*p*(p+1))) = 1.17443669198552182119... . For comparison, the asymptotic mean of the unitary abundancy index over all the positive integers is zeta(2)/zeta(3) = 1.368432... (A306633).
Lim sup_{n->oo} f(n) = oo (i.e., f(n) is unbounded).
f(n) <= A374777(n)/A374778(n) with equality if and only if n is squarefree (A005117).

A374784 Denominator of the mean unitary abundancy index of the unitary divisors of n.

Original entry on oeis.org

1, 4, 6, 8, 10, 24, 14, 16, 18, 8, 22, 16, 26, 56, 60, 32, 34, 72, 38, 80, 4, 88, 46, 96, 50, 104, 54, 112, 58, 48, 62, 64, 132, 136, 28, 16, 74, 152, 52, 160, 82, 16, 86, 176, 180, 184, 94, 64, 98, 40, 204, 208, 106, 216, 20, 224, 76, 232, 118, 160, 122, 248

Offset: 1

Amiram Eldar, Jul 20 2024

			For n = 4, 4 has 2 unitary divisors, 1 and 4. Their unitary abundancy indices are usigma(1)/1 = 1 and usigma(4)/4 = 5/4, and their mean unitary abundancy index is (1 + 5/4)/2 = 9/8. Therefore a(4) = denominator(9/8) = 8.

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

Cf. A034444, A034448 (usigma), A077610, A374783 (numerators).

Similar sequences: A374777/A374778, A374786/A374787.

f[p_, e_] := 1 + 1/(2*p^e); a[1] = 1; a[n_] := Denominator[Times @@ f @@@ FactorInteger[n]]; Array[a, 100]

a(n) = {my(f = factor(n)); denominator(prod(i = 1, #f~, 1 + 1/(2*f[i,1]^f[i,2])));}

A374786 Numerator of the mean infinitary abundancy index of the infinitary divisors of n.

Original entry on oeis.org

1, 5, 7, 9, 11, 35, 15, 45, 19, 11, 23, 21, 27, 75, 77, 33, 35, 95, 39, 99, 5, 115, 47, 105, 51, 135, 133, 135, 59, 77, 63, 165, 161, 175, 33, 19, 75, 195, 63, 99, 83, 25, 87, 207, 209, 235, 95, 77, 99, 51, 245, 243, 107, 665, 23, 675, 91, 295, 119, 231, 123, 315

Offset: 1

Amiram Eldar, Jul 20 2024

The infinitary abundancy index of a number k is A049417(k)/k.
The record values of a(n)/A374787(n) are attained at the terms of A037992.
The least number k such that a(k)/A374787(k) is larger than 2, 3, 4, ..., is A037992(6) = 7560, A037992(33) = 1370819010042780920891599455129161859473627856000, ... .

			For n = 4, 4 has 2 infinitary divisors, 1 and 4. Their infinitary abundancy indices are isigma(1)/1 = 1 and isigma(4)/4 = 5/4, and their mean infinitary abundancy index is (1 + 5/4)/2 = 9/8. Therefore a(4) = numerator(9/8) = 9.

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

Cf. A005117, A037445, A037992, A049417, A050376, A077609, A138302, A327574, A374787 (denominators).

Similar sequences: A374777/A374778, A374783/A374784.

f[p_, e_] := p^(2^(-1 + Position[Reverse@IntegerDigits[e, 2], ?(# == 1 &)])); a[1] = 1; a[n] := Numerator[Times @@ (1 + 1/(2*Flatten@ (f @@@ FactorInteger[n])))]; Array[a, 100]

a(n) = {my(f = factor(n), b); numerator(prod(i = 1, #f~, b = binary(f[i, 2]); prod(k=1, #b, if(b[k], 1 + 1/(2*f[i, 1]^(2^(#b-k))), 1))));}

Let f(n) = a(n)/A374787(n). Then:
f(n) = (Sum_{d infinitary divisor of n} isigma(d)/d) / id(n), where isigma(n) is the sum of infinitary divisors of n (A049417), and id(n) is their number (A037445).
f(n) is multiplicative with f(p^e) = Product{k>=1, e_k=1} (1 + 1/(2*p^(2^(k+1)))), where e = Sum_{k} e_k * 2^k is the binary representation of e, i.e., e_k is bit k of e.
f(n) = (Sum_{d infinitary divisor of n} d*id(d)) / (n*id(n)).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} f(k) = Product_{P} (1 + 1/(2*P*(P+1))) = 1.21407233718434377029..., where P are numbers of the form p^(2^k) where p is prime and k >= 0 (A050376). For comparison, the asymptotic mean of the infinitary abundancy index over all the positive integers is 1.461436... = 2 * A327574.
Lim sup_{n->oo} f(n) = oo (i.e., f(n) is unbounded).
f(n) <= A374777(n)/A374778(n) with equality if and only if n is squarefree (A005117).
f(n) >= A374783(n)/A374784(n) with equality if and only if n is in A138302.

A374787 Denominator of the mean infinitary abundancy index of the infinitary divisors of n.

Original entry on oeis.org

1, 4, 6, 8, 10, 24, 14, 32, 18, 8, 22, 16, 26, 56, 60, 32, 34, 72, 38, 80, 4, 88, 46, 64, 50, 104, 108, 112, 58, 48, 62, 128, 132, 136, 28, 16, 74, 152, 52, 64, 82, 16, 86, 176, 180, 184, 94, 64, 98, 40, 204, 208, 106, 432, 20, 448, 76, 232, 118, 160, 122, 248

Offset: 1

Amiram Eldar, Jul 20 2024

			For n = 4, 4 has 2 infinitary divisors, 1 and 4. Their infinitary abundancy indices are isigma(1)/1 = 1 and isigma(4)/4 = 5/4, and their mean infinitary abundancy index is (1 + 5/4)/2 = 9/8. Therefore a(4) = denominator(9/8) = 8.

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

Cf. A037445, A049417 (isigma), A077609, A374786 (numerators).

Similar sequences: A374777/A374778, A374783/A374784.

f[p_, e_] := p^(2^(-1 + Position[Reverse@IntegerDigits[e, 2], ?(# == 1 &)])); a[1] = 1; a[n] := Denominator[Times @@ (1 + 1/(2*Flatten@ (f @@@ FactorInteger[n])))]; Array[a, 100]

a(n) = {my(f = factor(n), b); denominator(prod(i = 1, #f~, b = binary(f[i, 2]); prod(k=1, #b, if(b[k], 1 + 1/(2*f[i, 1]^(2^(#b-k))), 1))));}

A374779 Numbers whose divisors have a mean abundancy index that is larger than 3.

Original entry on oeis.org

10886400, 13305600, 14515200, 18144000, 19958400, 21772800, 23587200, 23950080, 24192000, 25401600, 26611200, 27216000, 29030400, 29937600, 30481920, 31449600, 31933440, 32659200, 33264000, 33868800, 35380800, 35925120, 36288000, 37739520, 38102400, 39312000, 39916800

Offset: 1

Amiram Eldar, Jul 19 2024

nonn

Numbers k such that A374777(k)/A374778(k) > 3.
The numbers whose mean abundancy index of divisors is larger than 2 are in A245214.
The least odd term in this sequence is 84712751711029943302437712454902728115050897458369518458984375.

			10886400 is a term since A374777(10886400)/A374778(10886400) = 70644571/23514624 = 3.004... > 3.

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

Subsequence of A245214.

Cf. A068403, A374777, A374778.

f[p_, e_] := ((e+1)*p^2 - (e+2)*p + p^(-e))/((e+1)*(p-1)^2); s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; Select[Range[4*10^7], s[#] > 3 &]

is(n) = {my(f = factor(n)); prod(i = 1, #f~, p=f[i,1]; e=f[i,2]; (-2*p - e*p + p^2 + e*p^2 + p^(-e))/((e + 1)*(p - 1)^2)) > 3;}

A374781 Numbers whose divisors have a record value of the mean abundancy index.

Original entry on oeis.org

1, 2, 4, 6, 8, 12, 24, 48, 72, 96, 120, 144, 240, 288, 360, 480, 576, 720, 1440, 2160, 2880, 4320, 5760, 8640, 12960, 15120, 17280, 20160, 25920, 30240, 40320, 51840, 60480, 90720, 120960, 181440, 241920, 302400, 362880, 483840, 604800, 725760, 907200, 1209600

Offset: 1

Amiram Eldar, Jul 19 2024

nonn

Positions of records in {f(k) | k = 1, 2, ...}, where f(k) = (Sum_{d|k} sigma(d)/d) / tau(k) = A374777(k)/A374778(k), i.e., numbers k such that f(k) > f(m) for all m < k.
Similarly to the sequence of abundancy indices {sigma(k)/k} = {A017665(k)/A017666(k)}, the record values occur at indices that are least numbers of each prime signature (A025487). For the abundancy indices the records occur at the superabundant numbers (A004394).
The least number k such that A374777(k)/A374778(k) > m for m = 2, 3, ..., is 144, 10886400, 532197314288640000, 2205754988720678629683817883074560000000, ... .

			The values of A374777(k)/A374778(k) for k = 1..6 are {1, 5/4, 7/6, 17/12, 11/10, 35/24} = {1, 1.25, 1.166..., 1.416..., 1.1, 1.458...}. The records values, {1, 1.25, 1.416..., 1.458...} occur at k = 1, 2, 4, and 6, the first 4 terms of this sequence.

Amiram Eldar, Table of n, a(n) for n = 1..778

Subsequence of A025487.

Cf. A004394, A017665, A017666, A374777, A374778.

f[p_, e_] := ((e+1)*p^2 - (e+2)*p + p^(-e))/((e+1)*(p-1)^2); s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; seq[kmax_] := Module[{v = {}, smax = 0, s1}, Do[s1 = s[k]; If[s1 > smax, AppendTo[v, k]; smax = s1], {k, 1, kmax}]; v]; seq[10^5]

s(n) = {my(f = factor(n)); prod(i = 1, #f~, p=f[i,1]; e=f[i,2]; (-2*p - e*p + p^2 + e*p^2 + p^(-e))/((e + 1)*(p - 1)^2));}
lista(kmax) = {my(smax = 0, s1); for(k = 1, kmax, s1 = s(k); if(s1 > smax, print1(k, ", "); smax = s1));}

A374780 Odd terms in A245214.

Original entry on oeis.org

10854718875, 12131744625, 13408770375, 18091198125, 19538493975, 20219574375, 21070924875, 22347950625, 22915517625, 23880381525, 24902002125, 25327677375, 28307404125, 28733079375, 29462808375, 32564156625, 35118208125, 36395233875, 39800635875, 40226311125

Offset: 1

Amiram Eldar, Jul 19 2024

nonn

Odd numbers whose divisors have a mean abundancy index that is larger than 2.
The odd terms in A245214 are relatively rare: a(1) = A245214(276918705).
The least term that is not divisible by 3 is 26115176669245401228259189019322202117310546875.

			10854718875 is a term since it is odd and A374777(10854718875) / A374778(10854718875) = 11975203 / 5955950 = 2.0106... > 2.

Subsequence of A245214.

Cf. A005231, A374777, A374778.

f[p_, e_] := ((e+1)*p^2 - (e+2)*p + p^(-e))/((e+1)*(p-1)^2); s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; seq[kmax_] := Module[{v = {}}, Do[If[s[k] > 2, AppendTo[v, k]], {k, 1, kmax, 2}]; v]; seq[2*10^10]

is(k) = if(!(k % 2), 0, my(f = factor(k)); prod(i = 1, #f~, p=f[i,1]; e=f[i,2]; (-2*p - e*p + p^2 + e*p^2 + p^(-e))/((e + 1)*(p - 1)^2)) > 2);

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A374778 Denominator of the mean abundancy index of the divisors of n.

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A374783 Numerator of the mean unitary abundancy index of the unitary divisors of n.

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A374784 Denominator of the mean unitary abundancy index of the unitary divisors of n.

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A374786 Numerator of the mean infinitary abundancy index of the infinitary divisors of n.

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A374787 Denominator of the mean infinitary abundancy index of the infinitary divisors of n.

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A374779 Numbers whose divisors have a mean abundancy index that is larger than 3.

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A374781 Numbers whose divisors have a record value of the mean abundancy index.

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A374780 Odd terms in A245214.

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