A145899 -id:A145899 - cp's OEIS frontend

A206027 a(n) = the number of solutions to sigma(x) = A145899(n).

1, 2, 3, 5, 6, 7, 8, 9, 10, 11, 15, 21, 29, 35, 37, 49, 58, 59, 68, 79, 98, 111, 160, 224, 256, 317, 382, 426, 431, 563, 576, 624, 774, 865, 883, 1050, 1195, 1265, 1371, 1376, 1742, 1755, 1935, 2095, 2437, 2447, 2944, 3055, 3318, 3324, 3366, 4289, 4369, 4502

Offset: 1

Jaroslav Krizek, Feb 03 2012

nonn

			a(4) = 5 because the 5 numbers x such that sigma(x) = A145899(4) = 72 are x = 30, 46, 51, 55, 71.

Donovan Johnson, Table of n, a(n) for n = 1..100

Cf. A000203 (sigma = sum of divisors of n), A145899, A206026.

t = DivisorSigma[1, Range[10^6]]; t2 = Sort[Tally[t]]; mn = 0; t3 = {}; Do[If[t2[[n]][[2]] > mn, mn = t2[[n]][[2]]; AppendTo[t3, t2[[n]][[2]]]], {n, Length[t2]}]; t3 (* T. D. Noe, Feb 03 2012 *)

a(n) = A054973(A145899(n)). - Michel Marcus, Oct 22 2013

Extended to 1376 by T. D. Noe, Feb 04 2012

Terms a(41) and beyond from Donovan Johnson, Feb 04 2012

A289132 Indices of records in A063974.

Original entry on oeis.org

1, 12, 24, 60, 72, 216, 240, 720, 1440, 2160, 2880, 4320, 8640, 10080, 12960, 17280, 20160, 25920, 30240, 40320, 43200, 51840, 60480, 90720, 103680, 120960, 181440, 241920, 302400, 362880, 483840, 604800, 725760, 1088640, 1209600, 1451520, 1814400, 2419200

Offset: 1

Amiram Eldar, Jun 25 2017

nonn

Numbers n such that usigma(x) = n has more solutions x than any smaller n, where usigma(x) is the sum of unitary divisors of x (A034448).
The unitary version of A145899.
The corresponding number of solutions for each term is: 1, 2, 3, 4, 6, 7, 11, 18, 27, 30, 32, 48, 63, 65, 71, 88, 89, 102, 121, 122, 131, 144, 188, 190, 203, 262, 313, 364, 377, 472, 483, 584, 668, 725, 810, 928, 1076, 1138.
Is this a subsequence of A025487? - David A. Corneth, Jun 25 2017

			There are 3 solutions to usigma(x) = 24: usigma(14) = usigma(15) = usigma(23) = 24. For all m < 24, there are 2 or fewer solutions to usigma(x) = m, thus 24 is in the sequence.

Amiram Eldar, Table of n, a(n) for n = 1..53

Cf. A025487, A034448, A063974, A145899.

usigma[n_] := Block[{d = Divisors[n]}, Plus @@ Select[d, GCD[#, n/#] == 1 &]]; t = Map[usigma, Range[10^7]]; t2 = Sort[Tally[t]]; mn = 0; t3 = {}; Do[If[t2[[n]][[2]] > mn, mn = t2[[n]][[2]]; AppendTo[t3, t2[[n]][[1]]]], {n, Length[t2]}]; t3 (* after T. D. Noe at A145899 *)

A305676 a(n) = A305671(A305672(n)).

Original entry on oeis.org

7, 24, 72, 144, 240, 336, 504, 720, 576, 720, 1440, 2016, 2880, 4320, 5760, 10080, 10080, 10080, 10080, 8640, 8640, 10080, 17280, 15120, 17280, 17280, 17280, 20160, 30240, 40320, 60480, 120960, 181440, 241920, 362880, 483840, 725760, 1451520, 2177280, 2419200, 2419200

Offset: 1

Felix Fröhlich, Jun 08 2018

nonn

Do any numbers x exist that never occur in this sequence? Can this be proven for any particular value of x? For example, does 96 occur in this sequence? Note that 96 occurs four times as a value of sigma among the initial 100 composites.

Cf. A145899, A305671, A305672, A305673, A305675, A305676.

composite(n) = my(i=0); forcomposite(c=1, , i++; if(i==n, return(c)))
mcv(v) = my(w=vecsort(v, , 8), count=vector(#w), ind=0, i=0); for(x=1, #w, for(y=1, #v, if(w[x]==v[y], count[x]++))); for(k=1, #count, if(count[k]==vecmax(count), ind=k; i++)); if(i > 1, return(0), return(w[ind]))
a305671(n) = my(v=[]); for(k=1, n, v=concat(v, sigma(composite(k)))); mcv(v)
terms(n) = my(x=0, k=1, i=0); while(1, if(a305671(k) > 0, print1(a305671(k), ", "); i++); if(i==n, break); while(a305671(k) > 0, k++); while(a305671(k)==0, k++))
terms(11) \\ Print initial 11 terms of the sequence

A332037 Indices of records in A332036.

Original entry on oeis.org

1, 12, 24, 60, 120, 240, 360, 720, 1440, 2160, 2880, 4320, 5760, 7200, 8640, 12960, 14400, 17280, 21600, 25920, 28800, 30240, 34560, 40320, 43200, 51840, 60480, 86400, 120960, 172800, 181440, 241920, 259200, 302400, 362880, 483840, 518400, 604800, 725760, 907200

Offset: 1

Amiram Eldar, Feb 05 2020

nonn

Numbers k such that bsigma(x) = k has more solutions x than any smaller k, where bsigma(x) is the sum of bi-unitary divisors of x (A188999).
The bi-unitary version of A145899.
The corresponding number of solutions for each term is 1, 2, 3, 5, 7, 12, 13, 20, ... (see the link for more values).

			There are 3 solutions to bsigma(x) = 24: bsigma(14) = bsigma(15) = bsigma(23) = 24. For all m < 24 there are 2 or fewer solutions to bsigma(x) = m, thus 24 is in the sequence.

Amiram Eldar, Table of n, a(n) for n = 1..58
Amiram Eldar, Table of n, a(n), A332036(a(n)) for n = 1..58

Cf. A145899, A188999, A289132, A332036, A332037, A332039.

fun[p_, e_] := If[OddQ[e], (p^(e + 1) - 1)/(p - 1), (p^(e + 1) - 1)/(p - 1) - p^(e/2)]; bsigma[1] = 1; bsigma[n_] := Times @@ (fun @@@ FactorInteger[n]); m = 10000; v = Table[0, {m}]; Do[b = bsigma[k]; If[b <= m, v[[b]]++], {k, 1, m}]; s = {}; vm = -1; Do[If[v[[k]] > vm, vm = v[[k]]; AppendTo[s, k]], {k, 1, m}]; s

A332039 Indices of records in A332038.

Original entry on oeis.org

1, 12, 24, 60, 120, 240, 360, 720, 1440, 2880, 4320, 5760, 7200, 8640, 11520, 14400, 17280, 21600, 25920, 28800, 34560, 43200, 60480, 86400, 120960, 129600, 172800, 241920, 259200, 302400, 345600, 483840, 518400, 604800, 907200, 1036800, 1209600, 1814400, 2419200

Offset: 1

Amiram Eldar, Feb 05 2020

nonn

Numbers k such that isigma(x) = k has more solutions x than any smaller k, where isigma(x) is the sum of infinitary divisors of x (A049417).
The infinitary version of A145899.
The corresponding number of solutions for each term is 1, 2, 3, 5, 7, 12, 13, 20, ... (see the link for more values).

			There are 3 solutions to isigma(x) = 24: isigma(14) = isigma(15) = isigma(23) = 24. For all m < 24 there are 2 or fewer solutions to isigma(x) = m, thus 24 is in the sequence.

Amiram Eldar, Table of n, a(n) for n = 1..53
Amiram Eldar, Table of n, a(n), A332038(a(n)) for n = 1..53

Cf. A145899, A049417, A289132, A332037, A332038, A332041.

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]); m = 10000; v = Table[0, {m}]; Do[i = isigma[k]; If[i <= m, v[[i]]++], {k, 1, m}]; s = {}; vm = -1; Do[If[v[[k]] > vm, vm = v[[k]]; AppendTo[s, k]], {k, 1, m}]; s

A206026 a(n) = smallest number m such that sigma(k) = m has at least n positive solutions k.

Original entry on oeis.org

1, 12, 24, 72, 72, 168, 240, 336, 360, 504, 576, 720, 720, 720, 720, 1440, 1440, 1440, 1440, 1440, 1440, 2880, 2880, 2880, 2880, 2880, 2880, 2880, 2880, 4320, 4320, 4320, 4320, 4320, 4320, 5760, 5760, 8640, 8640, 8640, 8640, 8640, 8640, 8640, 8640, 8640, 8640

Offset: 1

Jaroslav Krizek, Feb 03 2012

nonn

Sequence of numbers from A145899.

			a(6) = 168 because 168 is the smallest value of sigma(k) for n = 6 positive integers k such that sigma(k) = 168 has solution; k = 60, 78, 92, 123, 143, 167.

Amiram Eldar, Table of n, a(n) for n = 1..1000
Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems (invphi.gp).

Cf. A000203, A054973, A085790, A145899, A206027.

list(len) = {my(v = vector(len), k = 1, c = 0, i); while(c < len, i = invsigmaNum(k); for(j = 1, i, if(j <= len && v[j] == 0, v[j] = k; c++)); k++); v;} \\ Amiram Eldar, Dec 15 2024, using Max Alekseyev's invphi.gp

A332041 Indices of records in A332040.

Original entry on oeis.org

1, 6, 30, 330, 390, 2730, 5460, 12090, 60060, 92820, 223860, 1021020, 1922700, 3805620, 13458900, 41861820, 110362980, 113573460, 227146920, 251170920, 502341840, 563603040, 888287400, 1270629360, 1776574800, 3310889400, 23107724640, 27939071160, 33754921200, 36419783400

Offset: 1

Amiram Eldar, Feb 05 2020

nonn

Numbers k such that esigma(x) = k has more solutions x than any smaller k, where esigma(x) is the sum of exponential divisors of x (A051377).
The exponential version of A145899.
The corresponding number of solutions for each term is 1, 2, 5, 6, 8, 9, 10, 12, 15, 16, 19, 22, 27, 29, 35, 37, 38, 44, 45, 47, 50, 51, 52, 53, 66, 80, 83, 89, 95, 102.

			There are 2 solutions to esigma(x) = 6: esigma(4) = esigma(6) = 6. For all m < 6 there are no more than one solution to esigma(x) = m, thus 6 is in the sequence.

Cf. A051377, A145899, A289132, A332037, A332039, A332040.

f[p_, e_] := DivisorSum[e, p^# &]; esigma[1] = 1; esigma[n_] := Times @@ f @@@ FactorInteger[n]; m = 10000; v = Table[0, {m}]; Do[sig = esigma[k]; If[sig <= m, v[[sig]]++], {k, 1, m}]; s = {}; vm = -1; Do[If[v[[k]] > vm, vm = v[[k]]; AppendTo[s, k]], {k, 1, m}]; s

a(26)-a(30) from Giovanni Resta, Feb 06 2020

A362402 Positive numbers m such that a record number of numbers k have m as the sum of divisors of k that have a square factor (A162296).

Original entry on oeis.org

1, 4, 48, 72, 216, 288, 864, 1440, 1728, 2880, 3456, 4320, 5184, 5760, 8640, 12096, 17280, 25920, 34560, 48384, 51840, 69120, 103680, 120960, 155520, 181440, 207360, 241920, 311040, 362880, 483840, 622080, 725760, 967680, 1088640, 1209600, 1451520, 2177280, 2903040

Offset: 1

Amiram Eldar, Apr 18 2023

nonn

The value 0 appears in the range of A162296 for all squarefree numbers (A005117) and therefore it is excluded from this sequence.
The corresponding record values are in A362403.
Except for 1, a subsequence of A362401.

Amiram Eldar, Table of n, a(n) for n = 1..60

Cf. A162296, A362401, A362403.

Similar sequences: A097942, A100827, A145899, A238895.

s[n_] := Module[{f = FactorInteger[n], p, e}, p = f[[;; , 1]]; e = f[[;; , 2]]; Times @@ ((p^(e + 1) - 1)/(p - 1)) - Times @@ (p + 1)]; s[1] = 0; seq[max_] := Module[{v = Select[Array[s, max], 0 < # <= max &], sq = {1}, t, tmax = 0}, t = Sort[Tally[v]]; Do[If[t[[k]][[2]] > tmax, tmax = t[[k]][[2]]; AppendTo[sq, t[[k]][[1]]]], {k, 1, Length[t]}]; sq]; seq[10^5]

s(n) = {my(f = factor(n), p, e); prod(i = 1, #f~, p = f[i, 1]; e = f[i, 2]; ((p^(e + 1) - 1)/(p - 1))) -  prod(i = 1, #f~, f[i, 1] + 1);}
lista(kmax) = {my(v = vector(kmax), vmax = 0, i); for(k=1, kmax, i = s(k); if(i > 0 && i <= kmax, v[i]++)); print1(1, ", "); for(k=1, kmax, if(v[k] > vmax, vmax = v[k]; print1(k, ", "))); }

A247131 Numbers n > 0 such that a record number of composite numbers k have n as the sum of the nontrivial divisors of k.

Original entry on oeis.org

1, 2, 5, 20, 30, 48, 72, 90, 114, 120, 168, 210, 300, 330, 360, 390, 420, 510, 630, 720, 780, 840, 1050, 1260, 1470, 1560, 1680, 1890, 2100, 2310, 2520, 2730, 3150, 3360, 3570, 3990, 4200, 4410, 4620, 5250, 5460, 6090, 6510, 6720, 6930, 7770, 7980, 8190, 9030, 9240, 10710, 10920, 11550, 13020, 13650, 13860, 15540

Offset: 1

Daniel Lignon, Nov 22 2014

nonn

A prime number has no nontrivial divisors so their sum is = 0. That's why we take only composite numbers.

			For 1, there are no numbers.
For 2, there is 1 number: 4.
For 5, there are 2 numbers: 6 and 25.
For 20, there are 3 numbers: 18, 51, 91.

Amiram Eldar, Table of n, a(n) for n = 1..139

Cf. A145899 (similar but with all divisors), A238895 (similar but with proper divisors), A048050 (Chowla's function: sum of nontrivial divisors).

ch[1] = 0; ch[n_] := DivisorSigma[1, n] - n - 1; m = 300; v = Table[0, {m}]; Do[c = ch[k]; If[1 <= c <= m, v[[c]]++], {k, 1, m^2}]; s = {}; vm = -1; Do[If[v[[k]] > vm, vm = v[[k]]; AppendTo[s, k]], {k, 1, m}]; s (* Amiram Eldar, Nov 05 2019 *)

Obviously a(n) = A238895(n)-1.

A332043 Indices of records in A332042.

Original entry on oeis.org

1, 6, 12, 24, 48, 72, 144, 288, 576, 864, 1152, 1440, 1728, 2160, 2304, 2880, 4320, 5760, 8640, 17280, 25920, 34560, 51840, 69120, 103680, 120960, 138240, 155520, 181440, 207360, 241920, 311040, 362880, 414720, 483840, 622080, 725760, 967680, 1244160, 1451520

Offset: 1

Amiram Eldar, Feb 05 2020

nonn

Numbers k such that psi(x) = k has more solutions x than any smaller k, where psi(x) is the Dedekind psi function (A001615).

The corresponding number of solutions for each term is 1, 2, 4, 5, 6, 9, 11, 17, ... (see the link for more values).

			There are 2 solutions to psi(x) = 6: psi(4) = psi(5) = 6. For all m < 6 there are no more than one solution to psi(x) = m, thus 6 is in the sequence.

Amiram Eldar, Table of n, a(n) for n = 1..59
Amiram Eldar, Table of n, a(n), A332042(a(n)) for n = 1..59

Cf. A001615, A145899, A289132, A332037, A332039, A332041, A332042.

psi[1] = 1; psi[n_] := n * Times @@ (1 + 1/Transpose[FactorInteger[n]][[1]]); m = 10000; v = Table[0, {m}]; Do[i = psi[k]; If[i <= m, v[[i]]++], {k, 1, m}]; s = {}; vm = -1; Do[If[v[[k]] > vm, vm = v[[k]]; AppendTo[s, k]], {k, 1, m}]; s

cp's OEIS Frontend

A206027 a(n) = the number of solutions to sigma(x) = A145899(n).

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A289132 Indices of records in A063974.

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A305676 a(n) = A305671(A305672(n)).

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A332037 Indices of records in A332036.

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A332039 Indices of records in A332038.

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A206026 a(n) = smallest number m such that sigma(k) = m has at least n positive solutions k.

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A332041 Indices of records in A332040.

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A362402 Positive numbers m such that a record number of numbers k have m as the sum of divisors of k that have a square factor (A162296).

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