A049417 -id:A049417 - cp's OEIS frontend

A300664 Infinitary 3-abundant numbers: numbers n such that isigma(n) >= 3n, where isigma is the sum of infinitary divisors of n (A049417).

120, 840, 1080, 1320, 1512, 1560, 1848, 1890, 1920, 2040, 2184, 2280, 2376, 2688, 2760, 2856, 3000, 3192, 3480, 3720, 4440, 4920, 5160, 5640, 5880, 6360, 7080, 7320, 7560, 8040, 8520, 8760, 9240, 9480, 9720, 9960, 10680, 10920, 11640, 11880, 12120, 12360

Offset: 1

Amiram Eldar, Mar 10 2018

nonn

Analogous to 3-abundant numbers (A023197) with isigma (A049417) instead of sigma (A000203).

			840 is in the sequence since isigma(840) = 2880 > 3 * 840.

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

Cf. A007357, A023197, A049417, A129656, A129657.

ExponentList[n_Integer, factors_List] := {#, IntegerExponent[n, #]} & /@ factors; InfinitaryDivisors[1] := {1}; InfinitaryDivisors[n_Integer ? Positive] := Module[{factors = First /@ FactorInteger[n], d = Divisors[n]}, d[[Flatten[ Position[ Transpose[ Thread[Function[{f, g}, BitOr[f, g] == g][#, Last[#]]] & /@ Transpose[ Last /@ ExponentList[#, factors] & /@ d]], ?(And @@ # &), {1}]]]]]; properinfinitarydivisorsum[k] := Plus @@ InfinitaryDivisors[k] - k; Infinitary3AbundantNumberQ[k_] :=  If[properinfinitarydivisorsum[k] >= 2 k, True, False]; Select[Range[15000], Infinitary3AbundantNumberQ[#] &] (* after Ant King at A129656 *)

A318182 Numbers m such that A049417(A049417(m)) = k*m for some k where A049417 is the infinitary sigma function.

Original entry on oeis.org

1, 2, 8, 9, 10, 15, 18, 24, 30, 60, 720, 1020, 4080, 8925, 14688, 14976, 16728, 17850, 35700, 36720, 37440, 66912, 71400, 285600, 308448, 381888, 428400, 602208, 636480, 763776, 856800, 1321920, 1505520, 3011040, 3084480, 21679488, 22276800, 30844800

Offset: 1

Michel Marcus, Aug 20 2018

nonn

a(86) > 3*10^11. All the prime factors of the first 85 terms belong to the set {2, 3, 5, 7, 11, 13, 17, 41, 43, 257}. - Giovanni Resta, Aug 25 2018
Like in A019278, here there are many instances where if x is a term, then A049417(x) is also a term.
Additionally, there exist longer chains of 3 or 4 elements like:
- 8 (3), 15 (4), 24 (5), 60 (6);
- 9 (2), 10 (3), 18 (4), 30 (5);
- 31615920 (4), 50585472 (5), 126463680 (6), 252927360 (12);
- 963407296051200 (16), 3134896756992000 (17), 15414516736819200 (18);
- 3541951043592192 (5), 8854877608980480 (6), 17709755217960960 (12), 53129265653882880 (20);
- 4829933241262080 (11), 17709755217960960 (12), 53129265653882880 (20);
  7871002319093760 (9), 26564632826941440 (10), 70839020871843840 (13), 265646328269414400 (14).

Giovanni Resta, Table of n, a(n) for n = 1..85 (first 58 terms from Tomohiro Yamada)
Tomohiro Yamada, Infinitary superperfect numbers, Annales Mathematicae et Informaticae, 47 (2017) pp. 211-218. See Table 1.
Michel Marcus, Unexhaustive list of terms

Cf. A049417 (infinitary sigma).

Cf. A019278 (analog for sigma), A318175 (analog for bi-unitary sigma).

a049417(n) = {my(b, f=factorint(n)); prod(k=1, #f[, 2], b = binary(f[k, 2]); prod(j=1, #b, if(b[j], 1+f[k, 1]^(2^(#b-j)), 1)));}
isok(n) = frac(a049417(a049417(n))/n) == 0;

More terms from Giovanni Resta, Aug 25 2018

A327566 Partial sums of the infinitary divisors sum function: a(n) = Sum_{k=1..n} isigma(k), where isigma is A049417.

Original entry on oeis.org

1, 4, 8, 13, 19, 31, 39, 54, 64, 82, 94, 114, 128, 152, 176, 193, 211, 241, 261, 291, 323, 359, 383, 443, 469, 511, 551, 591, 621, 693, 725, 776, 824, 878, 926, 976, 1014, 1074, 1130, 1220, 1262, 1358, 1402, 1462, 1522, 1594, 1642, 1710, 1760, 1838, 1910, 1980

Offset: 1

Amiram Eldar, Sep 17 2019

nonn

Differs from A307159 at n >= 16.

Steven R. Finch, Mathematical Constants II, Cambridge University Press, 2018, section 1.7.5, pp. 53-54.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Graeme L. Cohen and Peter Hagis, Jr., Arithmetic functions associated with infinitary divisors of an integer, International Journal of Mathematics and Mathematical Sciences, Vol. 16, No. 2 (1993), pp. 373-383.

Cf. A049417 (isigma), A327574.

Cf. A024916 (all divisors), A064609 (unitary), A307042 (exponential), A307159 (bi-unitary).

f[p_, e_] := p^(2^(-1 + Position[Reverse @ IntegerDigits[e, 2], ?(# == 1 &)])); isigma[1] = 1; isigma[n] := Times @@ (Flatten @ (f @@@ FactorInteger[n]) + 1); Accumulate[Array[isigma, 52]]

a(n) ~ c * n^2, where c = 0.730718... (A327574).

A185288 Numbers n for which the terms of the multiplicative sequence {n^2/A049417(n)} are integers.

Original entry on oeis.org

1, 6, 60, 90, 120, 180, 360, 540, 816, 840, 1080, 1740, 1980, 2280, 2520, 3060, 3960, 5712, 6120, 8280, 9540, 11880, 12240, 16920, 18360, 19260, 24480, 25296, 25560, 32760, 36720, 42840, 48960, 54672, 57240, 63700, 73440, 74256, 84360, 85680, 97920, 103320, 115560

Offset: 1

Vladimir Shevelev, Feb 20 2011

nonn

The sequence contains all infinitary perfect numbers (see A007357).

			Let n=120. Its representation over distinct terms of A050376 is 2*3*4*5. Therefore A049417(n)=(2+1)*(3+1)*(4+1)*(5+1)=360. Since 360 is a divisor of 120^2, 120 is in the sequence.

Amiram Eldar, Table of n, a(n) for n = 1..1000

Cf. A007357, A049417, A064380, A185079, A185088, A185098, A185373, A185383.

fun[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 @@ fun @@@ FactorInteger[n]; aQ[n_] := Divisible[n^2, isigma[n]]; Select[Range[58000], aQ] (* Amiram Eldar, Jul 21 2019 *)

More terms from Nathaniel Johnston, Mar 16 2011

More terms from Amiram Eldar, Jul 21 2019

A240111 Numbers for which the value of the Dedekind psi function (A001615) are less than the value of the infinitary Dedekind psi function (A049417).

Original entry on oeis.org

8, 24, 27, 32, 40, 54, 56, 72, 88, 96, 104, 120, 125, 128, 135, 136, 152, 160, 168, 184, 189, 200, 216, 224, 232, 243, 248, 250, 264, 270, 280, 296, 297, 312, 328, 343, 344, 351, 352, 360, 375, 376, 378, 384, 392, 408, 416, 424, 440, 456, 459, 472, 480, 486

Offset: 1

Vladimir Shevelev, Apr 01 2014

nonn

Numbers k for which Product_{p|k} (1 + 1/p) < Product_{q is in Q_k} (1 + 1/q), where {p} are primes, {q} are terms of A050376 and Q_k is the set of distinct q's whose product is k.

The numbers of terms that do not exceed 10^k, for k = 1, 2, ..., are 1, 10, 108, 1072, 10679, 106722, 1067287, 10672851, 106728514, 1067285714, ... . Apparently, the asymptotic density of this sequence exists and equals 0.1067285... . - Amiram Eldar, Feb 13 2025

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Peter J. C. Moses)

Cf. A001615, A049417, A050376.

Complement of A240112 within the nonsquarefree numbers (A013929).

f1[p_, e_] := Module[{b = IntegerDigits[e, 2]}, m = Length[b]; Product[If[b[[j]] > 0, 1 + p^(2^(m - j)), 1], {j, 1, m}]]; f2[p_, e_] := (p+1)*p^(e-1); q[1] = False; q[n_] := Module[{fct = FactorInteger[n]}, Times @@ f2 @@@ fct < Times @@ f1 @@@ fct]; Select[Range[500], q] (* Amiram Eldar, Feb 13 2025 *)

isok(k) = {my(f = factor(k), b); prod(i=1, #f~, (f[i, 1]+1)*f[i, 1]^(f[i, 2]-1)) < prod(i=1, #f~, b = binary(f[i, 2]); prod(k=1, #b, if(b[k], 1+f[i, 1]^(2^(#b-k)), 1)));} \\ Amiram Eldar, Feb 13 2025

More terms from Peter J. C. Moses, Apr 02 2014

A327574 Decimal expansion of the constant that appears in the asymptotic formula for average order of the infinitary divisors sum function (A049417).

Original entry on oeis.org

7, 3, 0, 7, 1, 8, 2, 4, 2, 1, 2, 7, 3, 8, 4, 2, 2, 5, 8, 3, 8, 9, 7, 5, 4, 6, 8, 1, 7, 3, 5, 3, 0, 1, 6, 1, 9, 5, 7, 2, 5, 6, 4, 3, 3, 8, 6, 1, 7, 2, 7, 8, 6, 9, 7, 0, 7, 3, 3, 6, 7, 6, 2, 3, 0, 1, 0, 7, 9, 8, 8, 3, 3, 2, 8, 0, 0, 5, 3, 4, 6, 3, 7, 0, 2, 9, 9

Offset: 0

Amiram Eldar, Sep 17 2019

The asymptotic mean of the infinitary abundancy index lim_{n->oo} (1/n) * Sum_{k=1..n} A049417(k)/k = 1.461436... is twice this constant. - Amiram Eldar, Jun 13 2020

			0.730718242127384225838975468173530161957256433861727...

Steven R. Finch, Mathematical Constants II, Cambridge University Press, 2018, section 1.7.5, pp. 53-54.

Graeme L. Cohen and Peter Hagis, Jr., Arithmetic functions associated with infinitary divisors of an integer, International Journal of Mathematics and Mathematical Sciences, Vol. 16, No. 2 (1993), pp. 373-383.

Cf. A049417, A050376, A327566.

Cf. A013661 (corresponding constant for all divisors), A275480 (exponential), A306633 (unitary), A307160 (bi-unitary).

$MaxExtraPrecision = 1000; m = 1000; em = 10; f[x_] := Sum[Log[1 + x^(2^e)/(1 + 1/x^(2^e))], {e, 0, em}]; c = Rest[CoefficientList[Series[f[x], {x, 0, m}], x]*Range[0, m]]; RealDigits[(1/2) * Exp[NSum[Indexed[c, k]*PrimeZetaP[k]/k, {k, 2, m}, NSumTerms -> m, WorkingPrecision -> m]], 10, 100][[1]]

Equals Limit_{k->oo} A327566(k)/k^2.

Equals (1/2) * Product_{P} (1 + 1/(P*(P+1))), where P are numbers of the form p^(2^k) where p is prime and k >= 0 (A050376).

A332975 Solutions k of the equation isigma(k) = isigma(k-1) + isigma(k-2) where isigma(k) is the sum of the infinitary divisors of k (A049417).

Original entry on oeis.org

3, 24, 360, 5016, 28440, 42066, 50568, 60456, 187176, 998670, 1454706, 12055512, 14365608, 25726728, 27896424, 51670374, 91702962, 141084774, 236280786, 249854952, 386668344, 439362504, 792554574, 1115866152, 1931976696, 2467823442, 2496238590, 2655297558, 2715505440

Offset: 1

Amiram Eldar, Mar 04 2020

nonn

			24 is a term since isigma(24) = 60 and isigma(22) + isigma(23) = 36 + 24 = 60.

Amiram Eldar, Table of n, a(n) for n = 1..38

The infinitary version of A065900.

Cf. A049417, A065557, A075565, A076136, A076251, A145469, A291126, A291176, A292033, A294995, A332976.

fun[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 @@ fun @@@ FactorInteger[n]; Select[Range[3, 10^5], isigma[#] == isigma[# - 1] + isigma[# - 2] &]

A318272 a(n) is the least k such that A049417(A049417(k)) = n*k, where A049417 is the infinitary sigma function, or 0 if no such k exists.

Original entry on oeis.org

1, 2, 8, 15, 24, 60, 1321920, 17850, 285600, 763776

Offset: 1

Michel Marcus, Aug 23 2018

It is also known that a(12) = 71400.
Then for higher indices n, we have:
  a(11) <= 414230544000;
  a(13) <= 2667897127526400;
  a(14) <= 446464417259520;
  a(15) <= 23613006957281280;
  a(16) <= 22227004800;
  a(17) <= 3134896756992000;
  a(18) <= 15414516736819200;
  a(20) <= 53129265653882880.
a(16) = 8420630400. - Giovanni Resta, Aug 25 2018

Tomohiro Yamada, Infinitary superperfect numbers, Annales Mathematicae et Informaticae, 47 (2017) pp. 211-218. See Table 1.

Cf. A049417, A318182.

Cf. A272930 (analog for sigma), A318242 (analog for bi-unitary sigma).

A326488 Numbers m such that A327566(m) = Sum_{k=1..m} isigma(k) is divisible by m, where isigma(k) is the sum of infinitary divisors of k (A049417).

Original entry on oeis.org

1, 2, 160, 285, 2340, 2614, 8903, 81231, 171710, 182712, 434887, 2651907, 56517068, 143714354, 922484770, 5162883263, 39421525873

Offset: 1

Amiram Eldar, Sep 20 2019

The infinitary version of A056550.

The corresponding quotients, A327566(a(n))/a(n), are 1, 2, 118, 209, 1711, 1910, 6506, 59357, 125473, 133513, 317781, 1937798, 41298052, 105014703, 674076450, 3772612983, 28806028088, ...

			2 is in the sequence since isigma(1) + isigma(2) = 1 + 3 = 4 is divisible by 2.

Cf. A049417 (isigma), A327566 (sums of isigma).

Cf. A056550 (corresponding with sigma), A064611 (unitary), A307043 (exponential), A307161 (bi-unitary).

f[p_, e_] := p^(2^(-1 + Position[Reverse @ IntegerDigits[e, 2], ?(# == 1 &)])); isigma[1] = 1; isigma[n] := Times @@ (Flatten @ (f @@@ FactorInteger[n]) + 1); seq = {}; s = 0; Do[s = s + isigma [n]; If[Divisible[s, n], AppendTo[seq, n]], {n, 1, 10^6}]; seq

A335400 Numbers m such that sigma(m)/isigma(m) > sigma(k)/isigma(k) for all k < m, where sigma(m) is the sum of divisors of m (A000203) and isigma(m) is the sum of infinitary divisors of m (A049417).

Original entry on oeis.org

1, 4, 16, 144, 1296, 3600, 20736, 32400, 176400, 518400, 1587600, 12960000, 25401600, 635040000, 3073593600

Offset: 1

Amiram Eldar, Jun 05 2020

			The ratio sigma(m)/isigma(m) for m = 1, 2, 3 and 4 is 1, 1, 1 and 7/5. The record values occur at m = 1 and 4.

Cf. A000203, A049417, A285906, A335396.

fun[p_, e_] := Module[{b = IntegerDigits[e, 2], m}, m = Length[b]; Product[ If[b[[j]] > 0, 1 + p^(2^(m - j)), 1], {j, 1, m}]]; isigma[1] = 1; isigma[n_] := Times @@ fun @@@ FactorInteger[n]; f[n_] := DivisorSigma[1,n] / isigma[n]; s={}; fm = 0; Do[f1 = f[n]; If[f1 > fm, fm = f1; AppendTo[s, n]], {n, 1, 2 * 10^5}]; s

cp's OEIS Frontend

A300664 Infinitary 3-abundant numbers: numbers n such that isigma(n) >= 3n, where isigma is the sum of infinitary divisors of n (A049417).

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A318182 Numbers m such that A049417(A049417(m)) = k*m for some k where A049417 is the infinitary sigma function.

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A327566 Partial sums of the infinitary divisors sum function: a(n) = Sum_{k=1..n} isigma(k), where isigma is A049417.

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A185288 Numbers n for which the terms of the multiplicative sequence {n^2/A049417(n)} are integers.

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A240111 Numbers for which the value of the Dedekind psi function (A001615) are less than the value of the infinitary Dedekind psi function (A049417).

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A327574 Decimal expansion of the constant that appears in the asymptotic formula for average order of the infinitary divisors sum function (A049417).

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A332975 Solutions k of the equation isigma(k) = isigma(k-1) + isigma(k-2) where isigma(k) is the sum of the infinitary divisors of k (A049417).

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A318272 a(n) is the least k such that A049417(A049417(k)) = n*k, where A049417 is the infinitary sigma function, or 0 if no such k exists.

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