A049417 -id:A049417 - cp's OEIS frontend

A129656 Infinitary abundant numbers: integers for which A126168 (n)>n, or equivalently for which A049417 (n)>2n.

24, 30, 40, 42, 54, 56, 66, 70, 72, 78, 88, 96, 102, 104, 114, 120, 138, 150, 168, 174, 186, 210, 216, 222, 246, 258, 264, 270, 280, 282, 294, 312, 318, 330, 354, 360, 366, 378, 384, 390, 402, 408, 420, 426, 438, 440, 456, 462, 474, 480, 486, 498

Offset: 1

Ant King, Apr 29 2007

For large n, the distribution of a(n) is approximately linear and asymptotically satisfies a(n)~7.95n. It follows that the density of the infinitary abundant numbers is 1/7.95, which is about 0.126.

			The third integer that is exceeded by its proper infinitary divisor sum is 40. Hence a(3)=40.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Graeme L. Cohen, On an integer's infinitary divisors, Math. Comp., 54 (1990), 395-411.
Eric Weisstein's World of Mathematics, Infinitary Divisor.

Cf. A126168, A049417, A127666, A129657, A007357.

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}]] ]] ] Null;properinfinitarydivisorsum[k]:=Plus@@InfinitaryDivisors[k]-k;InfinitaryAbundantNumberQ[k_]:=If[properinfinitarydivisorsum[k]>k,True,False];Select[Range[500],InfinitaryAbundantNumberQ[ # ] &]
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[1000], isigma[#]>2# &] (* Amiram Eldar, May 12 2019 *)

A306985 Numbers k such that isigma(k) = isigma(k+1), where isigma(k) is the sum of the infinitary divisors of k (A049417).

Original entry on oeis.org

14, 27, 44, 459, 620, 957, 1334, 1634, 1652, 2204, 2685, 3195, 3451, 3956, 4064, 4544, 5547, 8495, 8636, 8907, 9844, 11515, 15296, 19491, 20145, 20155, 27643, 31724, 33998, 38180, 41265, 41547, 42818, 45716, 48364, 61964, 64665, 74875, 74918, 79316, 79826

Offset: 1

Amiram Eldar, Mar 18 2019

nonn

a(n) differs from A293183(n) starting at n = 15.

			14 is in the sequence since isigma(14) = isigma(15) = 24.

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

Cf. A002961, A049417, A064125, A293183.

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_] := isigma[n] == isigma[n+1]; Select[Range[1000], aQ]

A129657 Infinitary deficient numbers: integers for which A126168(n) < n, or equivalently for which A049417(n) < 2n.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 25, 26, 27, 28, 29, 31, 32, 33, 34, 35, 36, 37, 38, 39, 41, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 55, 57, 58, 59, 61, 62, 63, 64, 65, 67, 68, 69, 71, 73, 74, 75, 76, 77, 79, 80, 81, 82, 83, 84

Offset: 1

Ant King, Apr 29 2007

For large n, the distribution of a(n) is approximately linear and asymptotically satisfies a(n)~1.144n. It follows that the density of the infinitary deficient numbers is 1/1.144, which is about 0.874.

			The sixth integer that exceeds its proper infinitary divisor sum is 7. Hence a(6)=7.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Graeme L. Cohen, On an Integer's Infinitary Divisors, Mathematics of Computation, Vol. 54, No. 189, (1990), pp. 395-411.
Eric Weisstein's World of Mathematics, Infinitary Divisor.

Cf. A126168, A049417, A127666, A129656, A007357.

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}]] ]] ] Null;properinfinitarydivisorsum[k]:=Plus@@InfinitaryDivisors[k]-k;InfinitaryDeficientNumberQ[k_]:=If[properinfinitarydivisorsum[k] 0, 1 + p^(2^(m - j)), 1], {j, 1, m}]]; isigma[1] = 1; isigma[n_] := Times @@ fun @@@ FactorInteger[n]; Select[Range[100], isigma[#] < 2 # &] (* Amiram Eldar, Jun 09 2019 *)

A327634 Infinitary highly abundant numbers: numbers m such that isigma(m) > isigma(k) for all k < m, where isigma(k) is the sum of infinitary divisors of n (A049417).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 10, 12, 14, 18, 21, 22, 24, 30, 40, 42, 54, 66, 72, 78, 88, 96, 102, 114, 120, 168, 210, 216, 264, 312, 330, 360, 378, 384, 408, 456, 480, 510, 546, 552, 600, 672, 690, 696, 744, 840, 1080, 1320, 1512, 1560, 1848, 1920, 2040, 2184, 2280, 2688

Offset: 1

Amiram Eldar, Sep 20 2019

The infinitary version of A002093.

			The first 10 values of isigma(k) for k = 1 to 10 are: 1, 3, 4, 5, 6, 12, 8, 15, 10, 18. Record values are reached for all these values of k except for 7 and 9, therefore the sequence begins with 1, 2, 3, 4, 5, 6, 8, 10, ...

Amiram Eldar, Table of n, a(n) for n = 1..930

Cf. A002093, A037992, A049417, A285614, A292983.

f[p_, e_] := p^(2^(-1 + Position[Reverse @ IntegerDigits[e, 2], ?(# == 1 &)])); isigma[1] = 1; isigma[n] := Times @@ (Flatten @ (f @@@ FactorInteger[n]) + 1); seq = {};sm = 0; Do[s = isigma[n]; If[s > sm, sm = s; AppendTo[seq, n]], {n, 1, 10^4}]; seq

A185373 The numerator of the fraction |n^2/A049417(n)-A064380(n)|.

Original entry on oeis.org

1, 1, 1, 1, 0, 1, 4, 1, 5, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 7, 5, 1, 2, 1, 2, 9, 3, 1, 1, 1, 4, 11, 11, 25, 2, 1, 1, 9, 2, 1, 11, 1, 4, 3, 7, 1, 2, 1, 2, 1, 13, 1, 3, 1, 13, 49, 17, 1, 0, 1, 1, 49, 16, 25, 1, 1, 17, 19, 35, 1, 14, 1, 2

Offset: 2

Vladimir Shevelev, Feb 17 2011

n^2/A049417(n) is a multiplicative function, whereas A064380 is not. This sequence here measures the (small) differences n^2/A049417(n)-A064380(n) = 1/3, 1/4, 1/5, 1/6, 0, 1/8, 4/15, 1/10, 5/9, 1/12, 1/5 ...

Amiram Eldar, Table of n, a(n) for n = 2..10000

Cf. A064380, A049417, A185383 (denominators)

f[p_, e_] := p^(2^(-1 + Position[Reverse @ IntegerDigits[e, 2], ?(# == 1 &)])); isigma[1] = 1; isigma[n] := Times @@ (Flatten@(f @@@ FactorInteger[n]) + 1);
infCoprimeQ[n1_, n2_] := Module[{g = GCD[n1, n2]}, If[g == 1, True, AllTrue[ FactorInteger[g][[;; , 1]], BitAnd @@ IntegerExponent[{n1, n2}, #] == 0 &]]];
a[n_] := Abs[Numerator[n^2 / isigma[n] - Sum[Boole[infCoprimeQ[j, n]], {j, 1, n-1}]]]; Array[a, 100, 2] (* Amiram Eldar, Mar 20 2025 *)

A185383 a(n) is the denominator of the fraction |n^2/A049417(n)-A064380(n)|.

Original entry on oeis.org

3, 4, 5, 6, 1, 8, 15, 10, 9, 12, 5, 14, 6, 8, 17, 18, 5, 20, 3, 32, 9, 24, 5, 26, 21, 40, 5, 30, 2, 32, 51, 16, 27, 48, 25, 38, 15, 56, 9, 42, 8, 44, 15, 4, 18, 48, 17, 50, 39, 8, 35, 54, 10, 72, 15, 80, 45, 60, 1, 62, 24, 80, 85, 84, 4, 68, 45, 32, 36, 72, 25, 74, 57

Offset: 2

Vladimir Shevelev, Feb 17 2011

If A185373(n)=0, then we accept a(n)=1.

Amiram Eldar, Table of n, a(n) for n = 2..10000

Cf. A185373, A064380, A049417, A050376.

a(n)=n+1 iff n is in A050376.

a(73) corrected by Amiram Eldar, Sep 18 2019

A334901 Infinitary practical numbers: numbers m such that every number 1 <= k <= isigma(m) is a sum of distinct infinitary divisors of m, where isigma is A049417.

Original entry on oeis.org

1, 2, 6, 8, 24, 30, 40, 42, 54, 56, 66, 72, 78, 88, 104, 120, 128, 168, 210, 216, 264, 270, 280, 312, 330, 360, 378, 384, 390, 408, 440, 456, 462, 480, 504, 510, 520, 546, 552, 570, 594, 600, 616, 640, 672, 680, 690, 696, 702, 714, 728, 744, 750, 760, 792, 798

Offset: 1

Amiram Eldar, May 16 2020

nonn

Includes the powers of 2 of the form 2^(2^k - 1) for k = 0, 1, ... (A058891). The other terms are a subset of infinitary abundant numbers (A129656) and infinitary pseudoperfect numbers (A306983).

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

The infinitary version of A005153.

Cf. A049417, A058891, A129656, A286652, A306983, A334898.

bin[n_] := 2^(-1 + Position[Reverse @ IntegerDigits[n, 2], ?(# == 1 &)] // Flatten); f[p, e_] := p^bin[e]; icomp[n_] := Flatten[f @@@ FactorInteger[n]]; 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]; infPracQ[n_] := Module[{f, p, e, prod = 1, ok = True}, If[n < 1 || (n > 1 && OddQ[n]), False, If[n == 1, True, r = Sort[icomp[n]]; Do[If[r[[i]] > 1 + isigma[prod], ok = False; Break[]]; prod = prod*r[[i]], {i, Length[r]}]; ok]]]; Select[Range[1000], infPracQ]

A334974 Infinitary admirable numbers: numbers k such that there is a proper infinitary divisor d of k such that isigma(k) - 2d = 2k, where isigma is the sum of infinitary divisors function (A049417).

Original entry on oeis.org

24, 30, 40, 42, 54, 56, 66, 70, 78, 88, 96, 102, 104, 114, 120, 138, 150, 174, 186, 222, 246, 258, 270, 282, 294, 318, 354, 360, 366, 402, 420, 426, 438, 474, 486, 498, 534, 540, 582, 606, 618, 630, 642, 654, 660, 678, 726, 762, 780, 786, 822, 834, 894, 906, 942

Offset: 1

Amiram Eldar, May 18 2020

nonn

Equivalently, numbers that are equal to the sum of their proper infinitary divisors, with one of them taken with a minus sign.

Admirable numbers (A111592) whose number of divisors is a power of 2 (A036537) are also infinitary admirable numbers, since all of their divisors are infinitary. Terms with number of divisors that is not a power of 2 are 96, 150, 294, 360, 420, 486, 540, 630, 660, 726, 780, 960, 990, ...

			150 is in the sequence since 150 = 1 + 2 + 3 - 6 + 25 + 50 + 75 is the sum of its proper infinitary divisors with one of them, 6, taken with a minus sign.

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

The infinitary version of A111592.
Subsequence of A129656.
Cf. A036537, A049417, A077609, A328328, A334972.

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]; infDivQ[n_, 1] = True; infDivQ[n_, d_] := BitAnd[IntegerExponent[n, First /@ (f = FactorInteger[d])], (e = Last /@ f)] == e; infAdmQ[n_] := (ab = isigma[n] - 2 n) > 0 && EvenQ[ab] && ab/2 < n && Divisible[n, ab/2] && infDivQ[n, ab/2]; Select[Range[1000], infAdmQ]

A185079 a(n) = A064380(n) * A049417(n).

Original entry on oeis.org

3, 8, 15, 24, 36, 48, 60, 80, 90, 120, 140, 168, 192, 216, 255, 288, 330, 360, 390, 448, 504, 528, 600, 624, 672, 720, 760, 840, 936, 960, 1020, 1056, 1134, 1200, 1300, 1368, 1440, 1512, 1620, 1680, 1632, 1848, 1920, 1980, 2088, 2208, 2312, 2400, 2496, 2592, 2730, 2808, 2880, 3024, 3240, 3200, 3330

Offset: 2

Vladimir Shevelev, Feb 18 2011

nonn

Amiram Eldar, Table of n, a(n) for n = 2..10000

Cf. A064380, A049417, A007357, A185078, A185373, A185383.

a(n) = n^2 + o(n^(1+eps)).

Corrected and extended by T. D. Noe, Feb 18 2011

A240112 Numbers for which the values of the Dedekind psi function (A001615) are greater than the values of the infinitary Dedekind psi function (A049417).

Original entry on oeis.org

4, 9, 12, 16, 18, 20, 25, 28, 36, 44, 45, 48, 49, 50, 52, 60, 63, 64, 68, 75, 76, 80, 81, 84, 90, 92, 98, 99, 100, 108, 112, 116, 117, 121, 124, 126, 132, 140, 144, 147, 148, 150, 153, 156, 162, 164, 169, 171, 172, 175, 176, 180, 188, 192, 196, 198, 204, 207

Offset: 1

Vladimir Shevelev, Apr 01 2014

nonn

The first term of A072587 that is not in this sequence is 72.
On the set of the nonsquarefree numbers (A013929) it is complement to A240111.
The numbers of terms that do not exceed 10^k, for k = 1, 2, ..., are 2, 29, 284, 2845, 28527, 285352, 2853422, 28534455, 285344362, 2853443344, ... . Apparently, the asymptotic density of this sequence exists and equals 0.2853443... . - 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, A013929, A049417, A072587, A240111.

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[250], 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

cp's OEIS Frontend

A129656 Infinitary abundant numbers: integers for which A126168 (n)>n, or equivalently for which A049417 (n)>2n.

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A306985 Numbers k such that isigma(k) = isigma(k+1), where isigma(k) is the sum of the infinitary divisors of k (A049417).

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A129657 Infinitary deficient numbers: integers for which A126168(n) < n, or equivalently for which A049417(n) < 2n.

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A327634 Infinitary highly abundant numbers: numbers m such that isigma(m) > isigma(k) for all k < m, where isigma(k) is the sum of infinitary divisors of n (A049417).

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A185373 The numerator of the fraction |n^2/A049417(n)-A064380(n)|.

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A185383 a(n) is the denominator of the fraction |n^2/A049417(n)-A064380(n)|.

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A334901 Infinitary practical numbers: numbers m such that every number 1 <= k <= isigma(m) is a sum of distinct infinitary divisors of m, where isigma is A049417.

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A334974 Infinitary admirable numbers: numbers k such that there is a proper infinitary divisor d of k such that isigma(k) - 2*d = 2*k, where isigma is the sum of infinitary divisors function (A049417).

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A185079 a(n) = A064380(n) * A049417(n).

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A334974 Infinitary admirable numbers: numbers k such that there is a proper infinitary divisor d of k such that isigma(k) - 2d = 2k, where isigma is the sum of infinitary divisors function (A049417).