A307112 -id:A307112 - cp's OEIS frontend

A328135 Exponential 3-abundant numbers: numbers m such that esigma(m) >= 3m, where esigma(m) is the sum of exponential divisors of m (A051377).

901800900, 1542132900, 1926332100, 2153888100, 2690496900, 2822796900, 3942584100, 4487660100, 4600908900, 5127992100, 6267888900, 6742052100, 7162236900, 7305120900, 8421732900, 8969984100, 9866448900, 10203020100, 10718460900, 11723411700, 11787444900, 12528324900

Offset: 1

Amiram Eldar, Oct 04 2019

nonn

Aiello et al. found bounds on e-multiperfect numbers, i.e., numbers m such that esigma(m) = k * m for k > 2: 2 * 10^7 for k = 3, and 10^85, 10^320, and 10^1210 for k = 4, 5, and 6. The data of this sequence raise the bound for exponential 3-perfect numbers to 3 * 10^10.
The least odd term is (59#/2)^2 = 924251841031287598942273821762233522616225. The least term which is coprime to 6 is (239#/6)^2 = 3.135... * 10^190.
The least exponential 4-abundant number (esigma(m) >= 4m) is (31#)^2 = 40224510201185827416900. In general, the least exponential k-abundant number (esigma(m) >= k*m), for k > 2, is (A002110(A072986(k)))^2.
The asymptotic density of this sequence is Sum_{n>=1} f(A383699(n)) = 1.325...*10^(-9), where f(n) = (6/(Pi^2*n))*Product_{prime p|n}(p/(p+1)). - Amiram Eldar, May 06 2025

Amiram Eldar, Table of n, a(n) for n = 1..10000
W. Aiello, G. E. Hardy, and M. V. Subbarao, On the existence of e-multiperfect numbers, Fibonacci Quarterly, Vol. 25, No. 1 (1987), pp. 65-71.

Subsequence of A129575.
A383699 is a subsequence.
Cf. A002110, A051377, A072986, A321147.
Cf. A023197, A307112, A285615 (unitary), A293187 (bi-unitary), A300664 (infinitary).

f[p_, e_] := DivisorSum[e, p^# &]; esigma[1] = 1; esigma[n_] := Times @@ f @@@ FactorInteger[n]; Select[Range[10^10], esigma[#] >= 3 # &]

A340109 Coreful 3-abundant numbers: numbers k such that csigma(k) > 3*k, where csigma(k) is the sum of the coreful divisors of k (A057723).

Original entry on oeis.org

5400, 7200, 10800, 14400, 16200, 18000, 21168, 21600, 27000, 28800, 32400, 36000, 37800, 42336, 43200, 48600, 50400, 54000, 56448, 57600, 59400, 63504, 64800, 70200, 72000, 75600, 79200, 81000, 84672, 86400, 88200, 90000, 91800, 93600, 97200, 98784, 100800, 102600

Offset: 1

Amiram Eldar, Dec 28 2020

nonn

A coreful divisor d of a number k is a divisor with the same set of distinct prime factors as k, or rad(d) = rad(k), where rad(k) is the largest squarefree divisor of k (A007947).
Analogous to A068403 as A308053 is analogous to A005101.
Apparently, the least odd term in this sequence is 3^4 * 5^3 * 7^3 * 11^2 * 13^2 * 17^2 * 19^2 * 23^2 * 29^2 = 3296233276111741840875.
The asymptotic density of this sequence is Sum_{n>=1} f(A364991(n)) = 0.0004006..., where f(n) = (6/(Pi^2*n)) * Product_{prime p|n} (p/(p+1)). - Amiram Eldar, Aug 15 2023

			5400 is a term since csigma(5400) = 16380 > 3 * 5400.

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

Subsequence of A308053.
Cf. A007947, A057723, A364991 (primitive terms).
Similar sequences: A068403, A285615, A293187, A300664, A307112, A328135.

f[p_, e_] := (p^(e + 1) - 1)/(p - 1) - 1; s[1] = 1; s[n_] := Times @@ (f @@@ FactorInteger[n]); Select[Range[10^5], s[#] > 3*# &]

s(n) = {my(f = factor(n)); prod(i = 1, #f~, sigma(f[i, 1]^f[i, 2]) - 1);}
is(n) = s(n) > 3*n; \\ Amiram Eldar, Aug 15 2023

A372298 Primitive infinitary abundant numbers (definition 1): infinitary abundant numbers (A129656) whose all proper infinitary divisors are infinitary deficient numbers.

Original entry on oeis.org

40, 56, 70, 72, 88, 104, 756, 924, 945, 1092, 1188, 1344, 1386, 1428, 1430, 1596, 1638, 1760, 1870, 2002, 2016, 2080, 2090, 2142, 2176, 2210, 2394, 2432, 2470, 2530, 2584, 2720, 2750, 2944, 2990, 3040, 3128, 3190, 3200, 3230, 3250, 3400, 3410, 3496, 3712, 3770

Offset: 1

Amiram Eldar, Apr 25 2024

			40 is a term since it is an infinitary abundant number and all its proper infinitary divisors, {1, 2, 4, 5, 8, 10, 20}, are infinitary deficient numbers.
24 and 30, which are infinitary abundant numbers, are not primitive, because they are divisible by 6 which is an infinitary perfect number.

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

Subsequence of A129656 and A372299.
A372300 is a subsequence.
Cf. A007357, A077609, A129657.
Similar sequences: A071395, A298973, A302573, A307112, A307114, A307115.

f[p_, e_] := Module[{b = IntegerDigits[e, 2]}, m = Length[b]; Product[If[b[[j]] > 0, 1 + p^(2^(m - j)), 1], {j, 1, m}]];
isigma[1] = 1; isigma[n_] := Times @@ f @@@ FactorInteger[n]; idefQ[n_] := isigma[n] < 2*n; idivs[1] = {1};
idivs[n_] := Sort@ Flatten@ Outer[Times, Sequence @@ (FactorInteger[n] /. {p_, e_Integer} :> p^Select[Range[0, e], BitOr[e, #] == e &])];
q[n_] := Module[{d = idivs[n]}, Total[d] > 2*n && AllTrue[Most[d], idefQ]]; Select[Range[4000], q]

isidiv(d, f) = {if (d==1, return (1)); for (k=1, #f~, bne = binary(f[k, 2]); bde = binary(valuation(d, f[k, 1])); if (#bde < #bne, bde = concat(vector(#bne-#bde), bde)); for (j=1, #bne, if (! bne[j] && bde[j], return (0)); ); ); return (1); }
idivs(n) = {my(f = factor(n), d = divisors(f), idiv = []); for (k=1, #d, if (isidiv(d[k], f), idiv = concat(idiv, d[k])); ); idiv; } \\ Michel Marcus at A077609
isigma(n) = {my(f = factor(n), b); prod(i=1, #f~, b = binary(f[i, 2]); prod(k=1, #b, if(b[k], 1+f[i, 1]^(2^(#b-k)), 1)))} ;
is(n) = isigma(n) > 2*n && select(x -> x < n && isigma(x) >= 2*x, idivs(n)) == [];

A383699 Primitive exponential 3-abundant numbers: the powerful terms of A328135.

Original entry on oeis.org

901800900, 1542132900, 1926332100, 2153888100, 2690496900, 2822796900, 3942584100, 4487660100, 4600908900, 5127992100, 6267888900, 6742052100, 7162236900, 7305120900, 8421732900, 8969984100, 9866448900, 10203020100, 10718460900, 11723411700, 11787444900, 12528324900

Offset: 1

Amiram Eldar, May 06 2025

nonn

Subsequence of A328135 and first differ from it at n = 25: A328135(25) = 15330615300 is not a term of this sequence.

For squarefree numbers k, esigma(k) = k, where esigma is the sum of exponential divisors function (A051377). Thus, if m is a term (esigma(m) >= 3*m) and k is a squarefree number coprime to m, then esigma(k*m) = esigma(k) * esigma(m) = k * esigma(m) >= 3*k*m, so k*m is an exponential 3-abundant number. Therefore, the sequence of exponential 3-abundant numbers (A328135) can be generated from this sequence by multiplying with coprime squarefree numbers.

			901800900 is a term since esigma(901800900) = 2905943040 > 3 * 901800900 = 2705402700, and 901800900 = 2^2 * 3^2 * 5^2 * 7^2 * 11^2 * 13^2 is a powerful number.

Amiram Eldar, Table of n, a(n) for n = 1..10136 (terms below 10^16)

Intersection of A001694 and A328135.
Subsequence of A328136.
Cf. A051377, A307112.

pows[max_] := Union[Flatten[Table[i^2*j^3, {j, 1, Surd[max, 3]}, {i, 1, Sqrt[max/j^3]}]]];
f[p_, e_] := DivisorSum[e, p^# &]; esigma[1] = 1; esigma[n_] := Times @@ f @@@ FactorInteger[n];
seq[max_] := Select[pows[max], esigma[#] >= 3 # &]; seq[10^10]

A307111 a(n) is the least primitive n-abundant number k with the largest possible abundancy index sigma(k)/k.

Original entry on oeis.org

3465, 6930, 19399380, 8172244080

Offset: 2

Amiram Eldar, Mar 25 2019

A primitive n-abundant number k is a number with sigma(k)/k > n and sigma(d)/d < n for all the proper divisors d of k.

Cohen proved that for any given eps > 0 there are only finitely many primitive n-abundant numbers k with sigma(k)/k >= n + eps. Thus for each n there is a maximal value of the abundancy index sigma(k)/k for primitive n-abundant numbers k. If this maximum occurs at more than one number the least of them is given in this sequence.

			3465 is in the sequence since it is the primitive abundant (A071395) number with the largest possible abundancy index among the primitive abundant numbers: sigma(3465)/3465 = 832/385 = 2.161003... The abundancy indices of the next terms are 1248/385 = 3.241558..., 193536/46189 = 4.190088..., 642816/124729 = 5.153701...

Graeme L. Cohen, Primitive alpha-abundant numbers, Mathematics of Computation, Vol. 43, No. 167 (1984), pp. 263-270.

Cf. A000203, A071395, A307112, A307114, A307115.

cp's OEIS Frontend

A328135 Exponential 3-abundant numbers: numbers m such that esigma(m) >= 3m, where esigma(m) is the sum of exponential divisors of m (A051377).

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A340109 Coreful 3-abundant numbers: numbers k such that csigma(k) > 3*k, where csigma(k) is the sum of the coreful divisors of k (A057723).

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A372298 Primitive infinitary abundant numbers (definition 1): infinitary abundant numbers (A129656) whose all proper infinitary divisors are infinitary deficient numbers.

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A383699 Primitive exponential 3-abundant numbers: the powerful terms of A328135.

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A307111 a(n) is the least primitive n-abundant number k with the largest possible abundancy index sigma(k)/k.

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