A361175 -id:A361175 - cp's OEIS frontend

A383695 Exponential infinitary abundant numbers that are not exponential unitary abundant: numbers k such that A361175(k) > 2*k >= A322857(k).

476985600, 815673600, 1018886400, 1177862400, 1493049600, 2014214400, 2373638400, 2712326400, 3756614400, 3863865600, 4744454400, 5218617600, 5246841600, 6234681600, 7928121600, 8108755200, 8245036800, 8972409600, 9062726400, 9824774400, 10502150400, 10603756800

Offset: 1

Amiram Eldar, May 05 2025

nonn

Exponential infinitary abundant numbers are numbers k such that A361175(k) > 2*k.
All the exponential unitary abundant numbers (A383693) are also exponential infinitary abundant numbers. There are numbers that are exponential infinitary abundant and not exponential unitary abundant. The least is: a(1) = 476985600, which is the 427970th exponential infinitary abundant number.
All the terms are nonsquarefree numbers (A013929), since A361175(k) = k if k is a squarefree number (A005117).
The asymptotic density of this sequence is Sum_{n>=1} f(A383696(n)) = 1.9875...*10^(-9), where f(n) = (6/(Pi^2*n))*Product_{prime p|n}(p/(p+1)). The relative density of this sequence within the exponential infinitary abundant numbers is 2.215... * 10^(-6).

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

Cf. A005117, A322857, A361175.
Subsequence of A013929 and A129575.
A383696 is a subsequence.

seq[max_] := Module[{prim = seqA383696[max], s = {}, sq}, Do[sq = Select[Range[Floor[max/p]], CoprimeQ[p, #] && SquareFreeQ[#] &]; s = Join[s, p*sq], {p, prim}]; Union[s]]; seq[10^10] (* using the function seqA383696 from A383696 *)

list(lim) = {my(p = listA383696(lim), s = []); for(i = 1, #p, s = concat(s, apply(x -> p[i]*x, select(x -> gcd(x, p[i]) == 1 && issquarefree(x), vector(lim\p[i], j, j))))); Set(s);} \\ using the function listA383696 from A383696

A361174 The sum of the exponential squarefree exponential divisors (or e-squarefree e-divisors) of n.

Original entry on oeis.org

1, 2, 3, 6, 5, 6, 7, 10, 12, 10, 11, 18, 13, 14, 15, 6, 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, 18, 56, 60, 51, 78, 53, 60, 55, 70, 57, 58, 59, 90, 61, 62, 84, 78, 65, 66, 67, 102

Offset: 1

Amiram Eldar, Mar 03 2023

The exponential squarefree exponential divisors (or e-squarefree e-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 a squarefree divisor of e(i).

The number of exponential squarefree exponential divisors of n is A278908(n).

Amiram Eldar, Table of n, a(n) for n = 1..10000
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.
Xiangzhen Zhao, Min Liu, and Yu Huang, Mean value for the function t^(e)(n) over square-full numbers, Scientia Magna, Vol. 8, No. 3 (2012), pp. 110-114.
Index entries for sequences related to divisors of numbers.

Cf. A278908.

Similar sequences: A051377, A322857, A323309, A361175.

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

ff(p, e) = sumdiv(e, d, if(issquarefree(d), 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|e, d squarefree} p^d.

Sum_{k=1..n} a(k) ~ c * n^2 / 2, c = Product_{p prime} (1 + Sum_{k>=2} (a(p^k) - p*a(p^(k-1)))/p^(2*k)) = 1.08989220899432387559... . - Amiram Eldar, Feb 13 2024

A383760 Irregular triangle read by rows in which the n-th row lists the exponential infinitary 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, 16, 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, 48, 7, 49, 10, 50

Offset: 1

Amiram Eldar, May 09 2025

First differs from A322791 and A383761 at rows 16, 48, 80, 81, 112, 144, 162, ... and from A361255 at rows 256, 768, 1280, 1792, ... .

An exponential infinitary 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 an infinitary 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..15379 (first 10000 rows, flattened)
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.

Cf. A307848 (row lengths), A361175 (row sums).

Cf. A077609, A322791, A361255, A383761.

infDivQ[n_, 1] = True; infDivQ[n_, d_] := n > 0 && d > 0 && BitAnd[IntegerExponent[n, First /@ (f = FactorInteger[d])], (e = Last /@ f)] == e;
expInfDivQ[n_, d_] := Module[{f = FactorInteger[n]}, And @@ MapThread[infDivQ, {f[[;; , 2]], IntegerExponent[d, f[[;; , 1]]]}]]; expInfDivs[1] = {1};
expInfDivs[n_] := Module[{d = Rest[Divisors[n]]}, Select[d, expInfDivQ[n, #] &]];
Table[expInfDivs[n], {n, 1, 70}] // Flatten

A383866 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 an infinitary 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, 19, 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, 76, 57, 93, 72, 98, 54, 93, 72, 88, 80, 90, 60, 168, 62, 96, 104, 79, 84, 144

Offset: 1

Amiram Eldar, May 13 2025

First differs from A383864 at n = 256.
The sum of divisors d of n such that each is a unitary divisor of an exponential infinitary divisor of n (see A383760).	
Analogous to the sum of (1+e)-divisors (A051378) as exponential infinitary divisors (A383760, A361175) are analogous to exponential divisors (A322791, A051377).
The number of these divisors is A383865(n).

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

Cf. A036537, A051377, A051378, A077609, A322791, A361175, A383760, A383864, A383865.

infdivs[n_] := If[n == 1, {1}, Sort@ Flatten@ Outer[Times, Sequence @@ (FactorInteger[n] /. {p_, m_Integer} :> p^Select[Range[0, m], BitOr[m, #] == m &])]];  (* Michael De Vlieger at A077609 *)
f[p_, e_] := 1 + Total[p^infdivs[e]]; a[1] = 1; 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); }
infdivs(n) = {d = divisors(n); f = factor(n); idiv = []; for (k=1, #d, if (isidiv(d[k], f), idiv = concat(idiv, d[k])); ); idiv; } \\ Michel Marcus at A077609
a(n) = {my(f = factor(n), d); prod(i = 1, #f~, d = infdivs(f[i, 2]); 1 + sum(j = 1, #d, f[i, 1]^d[j]));}

Multiplicative with a(p^e) = 1 + Sum_{d infinitary divisor of e} p^d.
a(n) <= A051378(n), with equality if and only if all the exponents in the prime factorization of n are in A036537.
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = Product_{p prime} f(1/p) = 1.52187097260174705015..., and f(x) = (1-x) * (1 + Sum_{k>=1} (1 + Sum{d infinitary divisor of k} x^(2*k-d))).

A383696 Primitive exponential infinitary abundant numbers that are not primitive exponential unitary abundant: the powerful terms of A383695.

Original entry on oeis.org

476985600, 815673600, 1018886400, 1177862400, 1493049600, 2014214400, 2373638400, 2712326400, 3756614400, 3863865600, 4744454400, 5218617600, 6234681600, 7928121600, 9824774400, 10502150400, 12669753600, 14227718400, 15040569600, 17614598400, 19443513600, 22356230400

Offset: 1

Amiram Eldar, May 06 2025

nonn

For squarefree numbers k, eusigma(k) = eisigma(k) = k, where eusigma is the sum of exponential unitary divisors function (A322857), and eisigma is the sum of exponential infinitary divisors function (A361175). Thus, if m is a term (eisigma(m) > 2*m >= eusigma(m)) and k is a squarefree number coprime to m, then eusigma(k*m) = eusigma(k) * eusigma(m) = k * eusigma(m) <= 2*k*m, and eisigma(k*m) = eisigma(k) * eisigma(m) = k * eisigma(m) > 2*k*m, so k*m is an exponential infinitary abundant number that is not exponential unitary abundant (A383695). Therefore, the sequence A383695 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 A383695.

Cf. A005117, A322857, A361175.

idivs[1] = {1}; idivs[n_] := Sort @ Flatten @ Outer[Times, Sequence @@ (FactorInteger[n] /. {p_, e_Integer} :> p^Select[Range[0, e], BitOr[e, #] == e &])];
fi[p_, e_] := Total[p^idivs[e]]; fu[p_, e_] := DivisorSum[e, p^# &, CoprimeQ[#, e/#] &];
q[n_] := Module[{fct = FactorInteger[n]}, Times @@ fu @@@ fct <= 2*n < Times @@ fi @@@ fct];
pows[max_] := Union[Flatten[Table[i^2*j^3, {j, 1, Surd[max, 3]}, {i, 1, Sqrt[max/j^3]}]]];
seqA383696[max_] := Select[pows[max], q]; seqA383696[10^10]

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
fi(p, e) = sumdiv(e, d, if(isidiv(d, factor(e)), p^d, 0));
fu(p, e) = sumdiv(e, d, if(gcd(d, e/d)==1, p^d));
isprim(k) = {my(f = factor(k)); prod(i = 1, #f~, fu(f[i, 1], f[i, 2])) <= 2*k && prod(i = 1, #f~, fi(f[i, 1], f[i, 2])) > 2*k;}
listpows(lim) = my(v = List(), t); for(m = 1, sqrtnint(lim\1, 3), t=m^3; for(n = 1, sqrtint(lim\t), listput(v, t*n^2))); Set(v) \\ Charles R Greathouse IV at A001694
listA383696(lim) = select(x -> isprim(x), listpows(lim));

cp's OEIS Frontend

A383695 Exponential infinitary abundant numbers that are not exponential unitary abundant: numbers k such that A361175(k) > 2*k >= A322857(k).

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A361174 The sum of the exponential squarefree exponential divisors (or e-squarefree e-divisors) of n.

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A383760 Irregular triangle read by rows in which the n-th row lists the exponential infinitary divisors of n.

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A383866 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 an infinitary divisor of the p-adic valuation of n.

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A383696 Primitive exponential infinitary abundant numbers that are not primitive exponential unitary abundant: the powerful terms of A383695.

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