A160135 -id:A160135 - cp's OEIS frontend

A348604 Nonexponential abundant numbers: numbers k such that A160135(k) > k.

24, 30, 42, 48, 54, 60, 66, 70, 72, 78, 84, 90, 96, 102, 114, 120, 126, 132, 138, 150, 156, 160, 162, 168, 174, 180, 186, 192, 198, 210, 216, 222, 224, 240, 246, 258, 264, 270, 280, 282, 288, 294, 300, 312, 318, 320, 330, 336, 352, 354, 360, 366, 378, 384, 390

Offset: 1

Amiram Eldar, Oct 25 2021

nonn

The smallest odd term is a(1357) = 8505.

The numbers of terms not exceeding 10^k, for k = 1, 2, ..., are 0, 13, 148, 1595, 15688, 158068, 1578957, 15762209, 157745113, 1577808429, ... Apparently this sequence has an asymptotic density 0.157...

			24 is a term since A160135(24) = 30 > 24.

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

Cf. A160135, A348601.
Subsequence of A005101.
Similar sequences: A034683, A064597, A129575, A129656, A292982, A348274.

esigma[n_] := Times @@ (Sum[First[#]^d, {d, Divisors[Last[#]]}] &) /@ FactorInteger[n]; q[n_] := DivisorSigma[1, n] - esigma[n] > n; Select[Range[400], q]

A348601 Nonexponential multiply-perfect numbers: numbers k such that k | A160135(k).

Original entry on oeis.org

1, 6, 40, 234, 588, 89376, 10805558400

Offset: 1

Amiram Eldar, Oct 25 2021

The corresponding quotients A160135(k)/k are 1, 1, 1, 1, 1, 2, 3, ...

a(8) > 1.5*10^10, if it exists.

			6 is a term since its nonexponential divisors are 1, 2 and 3, so A160135(6) = 1 + 2 + 3 = 6 which is divisible by 6.
40 is a term since its nonexponential divisors are 1, 2, 4, 5, 8 and 20, so A160135(40) = 1 + 2 + 4 + 5 + 8 + 20 = 40 which is divisible by 40.

Cf. A160135.

Similar sequences: A007691, A064594, A064595, A189000, A327158.

esigma[n_] := Times @@ (Sum[First[#]^d, {d, Divisors[Last[#]]}] &) /@ FactorInteger[n]; Select[Range[1000], Divisible[DivisorSigma[1, #] - esigma[#], #] &]

A348605 Odd nonexponential abundant numbers: odd numbers k such that A160135(k) > k.

Original entry on oeis.org

8505, 10395, 12285, 15015, 16065, 17955, 19635, 21735, 21945, 23205, 25515, 25935, 26565, 28875, 31185, 31395, 33495, 33915, 34125, 35805, 36855, 39585, 41055, 42315, 42735, 45885, 47355, 48195, 49665, 50505, 51765, 53865, 54285, 55965, 56595, 58695, 61215, 64155

Offset: 1

Amiram Eldar, Oct 25 2021

nonn

The odd terms of A348604.

The numbers of terms not exceeding 10^k, for k = 4, 5, ..., are 1, 51, 360, 4117, 39803, 418663, 4099004, ... Apparently this sequence has an asymptotic density 0.0004...

			8505 is a term since A160135(8505) = 8862 > 8505.

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

Cf. A160135.
Subsequence of A005231 and A348604.
Similar sequences: A094889, A127666, A129485, A293186, A321147, A348275.

esigma[n_] := Times @@ (Sum[First[#]^d, {d, Divisors[Last[#]]}] &) /@ FactorInteger[n]; q[n_] := DivisorSigma[1, n] - esigma[n] > n; Select[Range[1, 65000, 2], q]

A348629 Nonexponential highly abundant numbers: numbers m such that nesigma(m) > nesigma(k) for all k < m, where nesigma(k) is the sum of nonexponential divisors of n (A160135).

Original entry on oeis.org

1, 6, 10, 12, 18, 24, 30, 42, 48, 54, 60, 78, 84, 90, 96, 120, 168, 192, 210, 240, 270, 312, 330, 360, 384, 420, 480, 630, 672, 840, 960, 1056, 1080, 1248, 1320, 1440, 1560, 1680, 1890, 1920, 2280, 2310, 2400, 2520, 2640, 2688, 3000, 3120, 3240, 3360, 4200, 4320

Offset: 1

Amiram Eldar, Oct 26 2021

nonn

The corresponding record values are 1, 6, 8, 10, 15, 30, 42, 54, 58, 60, 78, ... (see the link for more values).

			The first 6 values of nesigma(k), for k = 1 to 6 are 1, 1, 1, 1, 1 and 6. The record values, 1 and 6, occur at 1 and 6, the first 2 terms of this sequence.

Amiram Eldar, Table of n, a(n) for n = 1..577
Amiram Eldar, Table of n, a(n), A160135(a(n)) for n = 1..577

Cf. A160135, A348604.
The nonexponential version of A002093.
Similar sequences: A285614, A292983, A327634, A328134, A329883, A348272.

esigma[n_] := Times @@ (Sum[First[#]^d, {d, Divisors[Last[#]]}] &) /@ FactorInteger[n]; s[1] = 1; s[n_] := DivisorSigma[1, n] - esigma[n]; seq = {}; sm = -1; Do[s1 = s[n]; If[s1 > sm, sm = s1; AppendTo[seq, n]], {n, 1, 10^4}]; seq

A348602 Smaller member of a nonexponential amicable pair: numbers (k, m) such that nesigma(k) = m and nesigma(m) = k, where nesigma(k) is the sum of the nonexponential divisors of k (A160135).

Original entry on oeis.org

198, 18180, 142310, 1077890, 1156870, 1511930, 1669910, 2236570, 2728726, 3776580, 4246130, 4532710, 5123090, 5385310, 6993610, 7288930, 8619765, 8754130, 8826070, 9478910, 10254970, 14426230, 17041010, 17257695, 21448630, 30724694, 34256222, 35361326, 37784810

Offset: 1

Amiram Eldar, Oct 25 2021

nonn

The larger counterparts are in A348603.

			198 is a term since A160135(198) = 204 and A160135(204) = 198.

Cf. A160135, A348601, A348603.

Similar sequences: A002025, A002952, A126165, A126169, A259038, A292980, A322541, A324708, A348343.

esigma[n_] := Times @@ (Sum[First[#]^d, {d, Divisors[Last[#]]}] &) /@ FactorInteger[n]; s[n_] := DivisorSigma[1, n] - esigma[n]; seq = {}; Do[m = s[n]; If[m > n && s[m] == n, AppendTo[seq, n]], {n, 1, 1.7*10^6}]; seq

A348603 Larger member of a nonexponential amicable pair: numbers (k, m) such that nesigma(k) = m and nesigma(m) = k, where nesigma(k) is the sum of the nonexponential divisors of k (A160135).

Original entry on oeis.org

204, 19332, 168730, 1099390, 1292570, 1598470, 2062570, 2429030, 3077354, 3903012, 4488910, 6135962, 5504110, 5812130, 7158710, 8221598, 9627915, 10893230, 10043690, 11049730, 10273670, 18087818, 19150222, 17578785, 23030090, 32174506, 35997346, 40117714, 39944086

Offset: 1

Amiram Eldar, Oct 25 2021

nonn

The terms are ordered according to their smaller counterparts (A348602).

			204 is a term since A160135(204) = 198 and A160135(198) = 204.

Cf. A160135, A348601, A348602.

Similar sequences: A002046, A002953, A126166, A126170, A259039, A292981, A322542, A324709, A348344.

esigma[n_] := Times @@ (Sum[First[#]^d, {d, Divisors[Last[#]]}] &) /@ FactorInteger[n]; s[n_] := DivisorSigma[1, n] - esigma[n]; seq = {}; Do[m = s[n]; If[m > n && s[m] == n, AppendTo[seq, m]], {n, 1, 1.7*10^6}]; seq

A348628 Numbers k such that k and k+1 have the same sum of nonexponential divisors (A160135).

Original entry on oeis.org

1, 2, 3, 4, 15, 44, 674, 478899

Offset: 1

Amiram Eldar, Oct 26 2021

Numbers k such that A160135(k) = A160135(k+1).

a(9) > 1.6 * 10^11, if it exists.

			2 is a term since A160135(2) = A160135(3) = 1.
15 is a term since A160135(15) = A160135(16) = 9.

Cf. A160135.

Similar sequences: A002961, A064115, A064125, A293183, A306985, A348346.

esigma[n_] := Times @@ (Sum[First[#]^d, {d, Divisors[Last[#]]}] &) /@ FactorInteger[n]; s[1] = 1; s[n_] := DivisorSigma[1, n] - esigma[n]; Select[Range[500000], s[#] == s[# + 1] &]

A348630 Nonexponential superabundant numbers: numbers m such that nesigma(m)/m > nesigma(k)/k for all k < m, where nesigma(m) is the sum of nonexponential divisors of m (A160135).

Original entry on oeis.org

1, 24, 30, 96, 120, 480, 840, 3360, 13440, 30240, 36960, 120960, 147840, 272160, 332640, 1330560, 2993760, 4324320, 17297280, 38918880, 73513440, 220540320, 294053760, 661620960, 1396755360, 2646483840, 5587021440, 12570798240

Offset: 1

Amiram Eldar, Oct 26 2021

The least term k with nesigma(k)/k > m for m = 2, 3, 4, ... is 480, 332640, 1396755360, ...

Cf. A160135, A348604.
Subsequence of A348629.
The nonexponential version of A004394.
Similar sequences: A002110 (unitary), A037992 (infinitary), A061742, A292984, A329882, A348273.

esigma[n_] := Times @@ (Sum[First[#]^d, {d, Divisors[Last[#]]}] &) /@ FactorInteger[n]; s[1] = 1 ;s[n_] := DivisorSigma[1, n] - esigma[n]; seq = {}; rm = -1; Do[r1 = s[n]/n; If[r1 > rm, rm = r1; AppendTo[seq, n]],{n, 1, 10^6}]; seq

A348631 Nonexponential weird numbers: nonexponential abundant numbers (A348604) that are not equal to the sum of any subset of their nonexponential divisors.

Original entry on oeis.org

70, 4030, 5830, 10430, 10570, 10990, 11410, 11690, 12110, 12530, 12670, 13370, 13510, 13790, 13930, 14770, 15610, 15890, 16030, 16310, 16730, 16870, 17570, 17990, 18410, 18830, 18970, 19390, 19670, 19810, 20510, 21490, 21770, 21910, 22190, 23170, 23590, 24290

Offset: 1

Amiram Eldar, Oct 26 2021

nonn

			70 is a term since the sum of its nonexponential divisors, {1, 2, 5, 7, 10, 14, 35}, is 74 > 70, and no subset of these divisors sums to 70.

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

Cf. A160135, A348604.

Similar sequences: A006037, A064114, A292986, A306984, A321146, A327948, A348525.

dQ[n_, m_] := (n > 0 && m > 0 && Divisible[n, m]); expDivQ[n_, d_] := Module[{ft = FactorInteger[n]}, And @@ MapThread[dQ, {ft[[;; , 2]], IntegerExponent[d, ft[[;; , 1]]]}]]; neDivs[1] = {}; neDivs[n_] := Module[{d = Divisors[n]}, Select[d, ! expDivQ[n, #] &]]; nesigma[n_] := Total@neDivs[n]; neAbundantQ[n_] := nesigma[n] > n; neWeirdQ[n_] := neAbundantQ[n] && Module[{d = neDivs[n]}, SeriesCoefficient[Series[Product[1 + x^d[[i]], {i, Length[d]}], {x, 0, n}], n] == 0]; Select[Range[6000], neWeirdQ]

A348606 Numbers k such that k and k+1 are both nonexponential abundant numbers.

Original entry on oeis.org

21735, 76544, 170624, 301664, 345344, 348704, 382304, 739935, 862784, 1218944, 1262624, 1272704, 1314495, 1370655, 1376864, 1539615, 1558304, 1707615, 1718144, 1761375, 1845375, 1890944, 1926015, 2100735, 2132864, 2223584, 2415104, 2463615, 2581215, 2675295, 2747744

Offset: 1

Amiram Eldar, Oct 25 2021

nonn

			21735 is a term since A160135(21735) = 21930 > 21735 and A160135(21736) = 23230 > 21736.

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

Cf. A160135.
Subsequence of A096399 and A348604.
Similar sequences: A318167, A327635, A327942, A331412, A348276.

esigma[n_] := Times @@ (Sum[First[#]^d, {d, Divisors[Last[#]]}] &) /@ FactorInteger[n]; q[n_] := DivisorSigma[1, n] - esigma[n] > n; Select[Range[1, 3*10^6], q[#] && q[#+1] &]

cp's OEIS Frontend

A348604 Nonexponential abundant numbers: numbers k such that A160135(k) > k.

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A348601 Nonexponential multiply-perfect numbers: numbers k such that k | A160135(k).

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A348605 Odd nonexponential abundant numbers: odd numbers k such that A160135(k) > k.

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A348629 Nonexponential highly abundant numbers: numbers m such that nesigma(m) > nesigma(k) for all k < m, where nesigma(k) is the sum of nonexponential divisors of n (A160135).

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A348602 Smaller member of a nonexponential amicable pair: numbers (k, m) such that nesigma(k) = m and nesigma(m) = k, where nesigma(k) is the sum of the nonexponential divisors of k (A160135).

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A348603 Larger member of a nonexponential amicable pair: numbers (k, m) such that nesigma(k) = m and nesigma(m) = k, where nesigma(k) is the sum of the nonexponential divisors of k (A160135).

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A348628 Numbers k such that k and k+1 have the same sum of nonexponential divisors (A160135).

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A348630 Nonexponential superabundant numbers: numbers m such that nesigma(m)/m > nesigma(k)/k for all k < m, where nesigma(m) is the sum of nonexponential divisors of m (A160135).

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A348631 Nonexponential weird numbers: nonexponential abundant numbers (A348604) that are not equal to the sum of any subset of their nonexponential divisors.

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