A361174 -id:A361174 - cp's OEIS frontend

A383697 Exponential squarefree exponential abundant numbers: numbers k such that A361174(k) > 2*k.

900, 1764, 4356, 4500, 4900, 6084, 6300, 8820, 9900, 10404, 11700, 12348, 12996, 14700, 15300, 17100, 19044, 19404, 20700, 21780, 22932, 26100, 27900, 29988, 30276, 30420, 30492, 31500, 33300, 33516, 34596, 36900, 38700, 40572, 42300, 42588, 44100, 47700, 47916, 49284, 49500

Offset: 1

Amiram Eldar, May 06 2025

Subsequence of A383693 and first differs from it at n = 21.
All the terms are nonsquarefree numbers (A013929), since A361174(k) = k if k is a squarefree number (A005117).
The least odd term is a(198045) = 225450225, and the least term that is coprime to 6 is a(9.815...*10^17) = 1117347505588495206025.
The least term that is not an exponentially squarefree number (A209061) is a(8.85...*10^1324) = 2^4 * Product_{k=2..248} prime(k)^2 = 1.00786...*10^1328.
The asymptotic density of this sequence is Sum_{n>=1} f(A383698(n)) = 0.000878475..., where f(n) = (6/(Pi^2*n))*Product_{prime p|n}(p/(p+1)).

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

Subsequence of A013929, A129575 and A383693.
A383698 is a subsequence.
Cf. A005117, A209061, A361174.

f[p_, e_] := DivisorSum[e, p^# &, SquareFreeQ[#] &]; q[k_] := Times @@ f @@@ FactorInteger[k] > 2*k; Select[Range[1000], q]

ff(p, e) = sumdiv(e, d, if(issquarefree(d), p^d, 0));
isok(k) = {my(f = factor(k)); prod(i=1, #f~, ff(f[i, 1], f[i, 2])) > 2*k; }

A361175 The sum of the exponential infinitary divisors of n.

Original entry on oeis.org

1, 2, 3, 6, 5, 6, 7, 10, 12, 10, 11, 18, 13, 14, 15, 18, 17, 24, 19, 30, 21, 22, 23, 30, 30, 26, 30, 42, 29, 30, 31, 34, 33, 34, 35, 72, 37, 38, 39, 50, 41, 42, 43, 66, 60, 46, 47, 54, 56, 60, 51, 78, 53, 60, 55, 70, 57, 58, 59, 90, 61, 62, 84, 78, 65, 66, 67

Offset: 1

Amiram Eldar, Mar 03 2023

First differs from A322857 at n = 256.
The exponential infinitary divisors of n = Product_i p(i)^e(i) are all the numbers of the form Product_i p(i)^d(i) where d(i) is an infinitary divisor of e(i).
The number of exponential infinitary divisors of n is A307848(n).

Amiram Eldar, Table of n, a(n) for n = 1..10000
Andrew V. Lelechenko, Exponential and infinitary divisors, Ukrainian Mathematical Journal, Vol. 68, No. 8 (2017), pp. 1222-1237; arXiv preprint, arXiv:1405.7597 [math.NT] (2014).
Index entries for sequences related to divisors of numbers.

Cf. A077609, A307848.

Similar sequences: A051377, A322857, A323309, A361174.

idivs[1] = {1}; idivs[n_] := Sort @ Flatten @ Outer[Times, Sequence @@ (FactorInteger[n] /. {p_, e_Integer} :> p^Select[Range[0, e], BitOr[e, #] == e &])];
f[p_, e_] := Total[p^idivs[e]]; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

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); } \\ Michel Marcus at A077609
ff(p, e) = sumdiv(e, d, if(isidiv(d, factor(e)), p^d, 0));
a(n) = {my(f=factor(n)); prod(i=1, #f~, ff(f[i, 1], f[i, 2])); }

Multiplicative with a(p^e) = Sum_{d infinitary divisor of e} p^d.

A383698 Primitive exponential squarefree exponential abundant numbers: the powerful terms of A383697.

Original entry on oeis.org

900, 1764, 4356, 4500, 4900, 6084, 10404, 12348, 12996, 19044, 30276, 34596, 44100, 47916, 49284, 60516, 66564, 79092, 79524, 88200, 101124, 108900, 112500, 125316, 132300, 133956, 152100, 161604, 176868, 181476, 191844, 213444, 217800, 220500, 224676, 246924

Offset: 1

Amiram Eldar, May 06 2025

nonn

Subsequence of A383694 and first differs from it at n = 11.
The least odd term is a(1345) = 225450225, and the least term that is coprime to 6 is 1117347505588495206025.
For squarefree numbers k, essigma(k) = k, where essigma is the sum of exponential squarefree exponential divisors function (A361174). Thus, if m is a term (essigma(m) > 2*m) and k is a squarefree number coprime to m, then essigma(k*m) = essigma(k) * essigma(m) = k * essigma(m) > 2*k*m, so k*m is an exponential squarefree exponential abundant number. Therefore, the sequence of exponential squarefree exponential abundant numbers (A383697) can be generated from this sequence by multiplying with coprime squarefree numbers.

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

Intersection of A001694 and A383697.
Subsequence of A383694.
Cf. A005117, A361174.

fun[p_, e_] := DivisorSum[e, p^# &, SquareFreeQ[#] &]; q[n_] := Min[(f = FactorInteger[n])[[;; , 2]]] > 1 && Times @@ fun @@@ f > 2*n; Select[Range[250000], q]

fun(p, e) = sumdiv(e, d, if(issquarefree(d), p^d, 0));
isok(k) = {my(f = factor(k)); ispowerful(f) && prod(i = 1, #f~, fun(f[i, 1], f[i, 2])) > 2*k;}

A383761 Irregular triangle read by rows in which the n-th row lists the exponential squarefree exponential divisors of n.

Original entry on oeis.org

1, 2, 3, 2, 4, 5, 6, 7, 2, 8, 3, 9, 10, 11, 6, 12, 13, 14, 15, 2, 4, 17, 6, 18, 19, 10, 20, 21, 22, 23, 6, 24, 5, 25, 26, 3, 27, 14, 28, 29, 30, 31, 2, 32, 33, 34, 35, 6, 12, 18, 36, 37, 38, 39, 10, 40, 41, 42, 43, 22, 44, 15, 45, 46, 47, 6, 12, 7, 49, 10, 50

Offset: 1

Amiram Eldar, May 09 2025

Differs from A322791, A361255 and A383760 at rows 16, 48, 80, 81, 112, 144, 162, ... .

An exponential squarefree exponential divisor (or e-squarefree e-divisor) d of a number n is a divisor d of n such that for every prime divisor p of n, the p-adic valuation of d is a squarefree divisor of the p-adic valuation of n.

			The first 10 rows are:
  1
  2
  3
  2, 4
  5
  6
  7
  2, 8
  3, 9
  10

Amiram Eldar, Table of n, a(n) for n = 1..15331 (first 10000 rows, flattened)
László Tóth, On certain arithmetic functions involving exponential divisors, II, Annales Univ. Sci. Budapest., Sect. Comp., Vol. 27 (2007), pp. 155-166; arXiv preprint, arXiv:0708.3557 [math.NT], 2007-2009.

Cf. A278908 (row lengths), A361174 (row sums).

Cf. A322791, A361255, A383760.

sqfDivQ[n_, d_] := SquareFreeQ[d] && Divisible[n, d];
expSqfDivQ[n_, d_] := Module[{f = FactorInteger[n]}, And @@ MapThread[sqfDivQ, {f[[;; , 2]], IntegerExponent[d, f[[;; , 1]]]}]]; expSqfDivs[1] = {1};
expSqfDivs[n_] := Module[{d = Rest[Divisors[n]]}, Select[d, expSqfDivQ[n, #] &]];
Table[expSqfDivs[n], {n, 1, 70}] // Flatten

A383867 The sum of divisors d of n having the property that for every prime p dividing n the p-adic valuation of d is either 0 or a squarefree divisor of the p-adic valuation of n.

Original entry on oeis.org

1, 3, 4, 7, 6, 12, 8, 11, 13, 18, 12, 28, 14, 24, 24, 7, 18, 39, 20, 42, 32, 36, 24, 44, 31, 42, 31, 56, 30, 72, 32, 35, 48, 54, 48, 91, 38, 60, 56, 66, 42, 96, 44, 84, 78, 72, 48, 28, 57, 93, 72, 98, 54, 93, 72, 88, 80, 90, 60, 168, 62, 96, 104, 79, 84, 144, 68

Offset: 1

Amiram Eldar, May 13 2025

Analogous to the sum of (1+e)-divisors (A051378) as exponential squarefree exponential divisors (A383761, A361174) are analogous to exponential divisors (A322791, A051377).

The number of these divisors is A383863(n).

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

Cf. A051378, A209061, A322791, A361174, A383761, A383863.

f[p_, e_] := 1 + DivisorSum[e, p^# &, SquareFreeQ[#] &]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n)); prod(i = 1, #f~, 1 + sumdiv(f[i, 2], d, if(issquarefree(d), f[i, 1]^d)));}

Multiplicative with a(p^e) = 1 + Sum_{d squarefree divisor of e} p^d.
a(n) <= A051378(n), with equality if and only if n is an exponentially squarefree number (A209061).
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = Product_{p prime} f(1/p) = 1.47709589136345836345..., and f(x) = (1-x) * (1 + Sum_{k>=1} (1 + Sum{d|k, d squarefree} x^(2*k-d))).

cp's OEIS Frontend

A383697 Exponential squarefree exponential abundant numbers: numbers k such that A361174(k) > 2*k.

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A361175 The sum of the exponential infinitary divisors of n.

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A383698 Primitive exponential squarefree exponential abundant numbers: the powerful terms of A383697.

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A383761 Irregular triangle read by rows in which the n-th row lists the exponential squarefree exponential divisors of n.

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A383867 The sum of divisors d of n having the property that for every prime p dividing n the p-adic valuation of d is either 0 or a squarefree divisor of the p-adic valuation of n.

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