A048138 -id:A048138 - cp's OEIS frontend

A324938 a(n) = the number of values of m such that the sum of proper unitary divisors of m (A034460) is n.

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

Offset: 2

Amiram Eldar, Sep 05 2019

nonn

The unitary version of A048138.

The offset is 2 as in A048138 since there are infinitely many numbers (all the powers of primes) for which A034460 = 1.

			a(8) = 2 since 8 is the sum of the proper unitary divisors of 10 (1 + 2 + 5) and 12 (1 + 3 + 4).

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

Cf. A034448, A034460, A048138, A063948.

us[1] = 0; us[n_] := Times @@ (1 + Power @@@ FactorInteger[n]) - n;  m = 100; v = Table[0, {m}]; Do[u = us[k]; If[2 <= u <= m, v[[u]]++], {k, 1, m^2}]; Rest @ v

a(A063948(n)) = 0.

A125601 a(n) is the smallest k > 0 such that there are exactly n numbers whose sum of proper divisors is k.

Original entry on oeis.org

2, 3, 6, 21, 37, 31, 49, 79, 73, 91, 115, 127, 151, 121, 181, 169, 217, 265, 253, 271, 211, 301, 433, 379, 331, 361, 457, 391, 451, 655, 463, 541, 421, 775, 511, 769, 673, 715, 865, 691, 1015, 631, 1069, 1075, 721, 931, 781, 1123, 871, 925, 901, 1177, 991, 1297

Offset: 0

Klaus Brockhaus, Nov 27 2006

nonn

Minimal values for nodes of exact degree in aliquot sequences. Find each node's degree (number of predecessors) in aliquot sequences and choose the smallest value as the sequence member. - Ophir Spector, ospectoro (AT) yahoo.com Nov 25 2007

			a(4) = 37 since there are exactly four numbers (155, 203, 299, 323) whose sum of proper divisors is 37. For k < 37 there are either fewer or more numbers (32, 125, 161, 209, 221 for k = 31) whose sum of proper divisors is k.

Daniel Mondot, Table of n, a(n) for n = 0..5646 (First 157 terms from Ophir Spector. First 1000 terms from Ophir Spector and Donovan Johnson.)
W. Creyaufmueller, Aliquot sequences
Daniel Mondot, A-file, extended version of b-file but with some gaps towards the end.
J. O. M. Pedersen, Tables of Aliquot Cycles [Broken link]
J. O. M. Pedersen, Tables of Aliquot Cycles [Via Internet Archive Wayback-Machine]
J. O. M. Pedersen, Tables of Aliquot Cycles [Cached copy, pdf file only]
Eric Weisstein's World of Mathematics, Aliquot sequence

Cf. A001065, A048138, A070015, A123930, A080907, A115350, A121507, A037020, A126016, A057709, A057710, A063990.

{m=54;z=1500;y=600000;v=vector(z);for(n=2,y,s=sigma(n)-n; if(s

A331973 a(n) is the number of values of m such that the sum of proper infinitary divisors of m (A126168) is n.

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 2, 1, 2, 1, 2, 2, 2, 2, 2, 3, 1, 3, 2, 4, 3, 3, 2, 3, 2, 3, 3, 4, 2, 4, 1, 4, 3, 4, 3, 5, 0, 3, 2, 4, 3, 5, 1, 4, 3, 4, 2, 6, 2, 5, 2, 5, 3, 7, 1, 6, 2, 4, 2, 7, 1, 5, 4, 5, 3, 8, 0, 5, 2, 6, 1, 8, 2, 5, 4, 6, 4, 9, 0, 6, 1, 5, 3, 10, 2, 8, 2

Offset: 2

Amiram Eldar, Feb 03 2020

nonn

The infinitary version of A048138.

The offset is 2 as in A048138 since there are infinitely many numbers k (the primes and squares of primes) for which A126168(k) = 1.

			a(8) = 2 since 8 is the sum of the proper infinitary divisors of 2 numbers: 10 (1 + 2 + 5) and 12 (1 + 3 + 4).

Amiram Eldar, Table of n, a(n) for n = 2..30000

Cf. A048138, A049417, A077609, A126168, A324938, A331971.

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]); is[n_] := isigma[n] - n; m = 300; v = Table[0, {m}]; Do[i = is[k]; If[2 <= i <= m, v[[i]]++], {k, 1, m^2}]; Rest@v

A372739 a(n) is the number of possible values of k such that the sum of aliquot coreful divisors of k (A336563) is n.

Original entry on oeis.org

0, 1, 1, 0, 1, 3, 1, 0, 0, 2, 1, 1, 1, 3, 2, 0, 1, 1, 1, 0, 2, 2, 1, 1, 0, 2, 0, 0, 1, 6, 1, 0, 2, 2, 2, 1, 1, 2, 3, 0, 1, 5, 1, 0, 0, 2, 1, 0, 0, 0, 2, 0, 1, 0, 2, 1, 2, 2, 1, 2, 1, 3, 0, 0, 2, 4, 1, 0, 2, 4, 1, 0, 1, 2, 0, 0, 2, 5, 1, 1, 0, 2, 1, 1, 2, 2, 2

Offset: 1

Amiram Eldar, May 12 2024

nonn

A coreful divisor d of n is a divisor that is divisible by every prime that divides n (see also A307958).

			a(2) = 1 since there is 1 possible value of k, k = 4, such that A336563(k) = 2.
a(6) = 3 since there are 3 possible values of k, k = 8, 12 and 18, such that A336563(k) = 6.

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

Cf. A307958, A336563, A372740, A372742.

Similar sequences: A048138, A324938, A331971, A331973.

f[p_, e_] := (p^(e + 1) - 1)/(p - 1) - 1; s[1] = 0; s[n_] := Times @@ f @@@ FactorInteger[n] - n; seq[max_] := Module[{v = Table[0, {max}], i}, Do[i = s[k]; If[0 < i <= max, v[[i]]++], {k, 1, max^2}]; v]; seq[100]

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

a(n) = 0 if and only if n is in A372740.

a(n) = 1 if and only if n is in A372742.

A253303 Smallest integer m such that gcd{x | sum of proper divisors of x is m} is equal to 2*n, when there are at least two such x's.

Original entry on oeis.org

16, 136, 186, 1352, 1340, 1356, 1414, 1276, 1026, 15640, 6742, 1968, 35786, 924, 11076, 11812, 61714, 14556, 76862, 6440, 12774, 70444, 62446, 16080, 24904, 16152, 27900, 65464, 36026, 41100, 85810, 56008, 50142, 23152

Offset: 1

Michel Marcus, Dec 30 2014

Only integers m that satisfy A048138(m) > 1 are considered here.

			The integers whose sum of proper divisors is 16 are 12 and 26, and gcd(12, 26) is 2, so a(1) = 16 (see Example section of A152454).

Cf. A001065 (sum of proper divisors), A048138, A152454, A253302.

build(nb) = {vs = vector(nb); nc = nb^2; forcomposite(n=2, nc, val = sigma(n)-n; if (val <= nb, if (vs[val] == 0, vs[val] = -n, vs[val] = gcd(vs[val], n)););); vs[1] = 1; vs;}
lista() = {vd = build(100000); vmax = 80; vr = vector(vmax); forstep (ig=2, vmax, 2, for (i=1, #vd, w = vd[i]; if (w ==ig, vr[ig] = i; break;););); forstep (i=2, #vr, 2, if (vr[i] == 0, break, print1(vr[i], ", ")));}

a(n) = A253302(2*n).

A331971 a(n) is the number of values of m such that the sum of proper bi-unitary divisors of m (A331970) is n.

Original entry on oeis.org

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

Offset: 2

Amiram Eldar, Feb 03 2020

nonn

The bi-unitary version of A048138.

The offset is 2 as in A048138 since there are infinitely many numbers k (the primes and squares of primes) for which A331970(k) = 1.

			a(8) = 2 since 8 is the sum of the proper bi-unitary divisors of 2 numbers: 10 (1 + 2 + 5) and 12 (1 + 3 + 4).

Amiram Eldar, Table of n, a(n) for n = 2..30000

Cf. A048138, A188999, A222266, A324938, A331970, A331973.

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]); bs[n_] := bsigma[n] - n; m = 300; v = Table[0, {m}]; Do[b = bs[k]; If[2 <= b <= m, v[[b]]++], {k, 1, m^2}]; Rest @ v

A123930 a(n) = smallest k > 1 that is the sum of the proper divisors (cf. A001065) of at least n different numbers.

Original entry on oeis.org

2, 3, 6, 21, 31, 31, 49, 73, 73, 91, 115, 121, 121, 121, 169, 169, 211, 211, 211, 211, 211, 301, 331, 331, 331, 361, 391, 391, 421, 421, 421, 421, 421, 511, 511, 631, 631, 631, 631, 631, 631, 631, 721, 721, 721, 781, 781, 841, 841, 841, 841, 841, 841, 841, 841

Offset: 0

Sergio Pimentel, Nov 22 2006

nonn

			a(0)=2 because 2 is the smallest number that is not the sum of proper divisors of any number (untouchable). a(6)=49 since 49 is the smallest number that is the sum of proper divisors of 6 different numbers (75, 215, 287, 407, 527, 551).

Donovan Johnson, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Sum of Proper Divisors Function.

Cf. A001065, A070015, A048138, A125601.

{z=850;m=200000;v=vector(z);for(n=2,m,s=sigma(n)-n;if(sKlaus Brockhaus, Nov 27 2006

Edited and extended by Klaus Brockhaus, Nov 27 2006

A252997 Numbers n such that sigma(x) - x = n has at least two solutions, with each x having the same squarefree kernel, where sigma(x) is the sum of divisor function (A000203).

Original entry on oeis.org

218, 189648, 720240, 119967120, 129705984, 517941905, 707902440, 1321744320, 98890370304, 99080219520, 119922568640, 139834382688, 347612467648, 580542318720, 952717920000, 1064902900320, 1153644808680, 2255573174400, 3903820736256, 6859688278905, 10944640212480, 14424196864000

Offset: 1

Naohiro Nomoto, Dec 25 2014

nonn

Numbers n such that n = A001065(j) = A001065(k) and A007947(j) = A007947(k), where j != k.

			218 is the sum of proper divisors of 250 and 160, and rad(250) = rad(160) = 10, hence 218 is in the sequence with j=250 and k=160.
Other examples of n and j, k:
For n = 189648, j = 95832, k = 85536.
For n = 720240, j = 288120, k = 246960.
For n = 119967120, j = 38755080, k = 34398000.
For n = 129705984, j = 71614464, k = 60424704.

Carlos Rivera, Prime Puzzle 774. S(i) and Rad(i), The Prime Puzzles and Problems Connection.

Cf. A001065 (sum of proper divisors of n), A007947 (squarefree kernel of n).

Cf. A048138, A152454, A254035.

a(6) onward from Fred Schneider, Feb 07 2015

A356445 a(n) is the number of times that A064440(n) occurs as the sum of proper divisors function (A001065).

Original entry on oeis.org

2, 3, 5, 7, 13, 17, 19, 23, 31, 41, 59, 61, 67, 79, 83, 97, 101, 109, 113, 127, 131, 139, 149, 151, 193, 199, 223, 227, 229, 277, 283, 317, 397, 433, 521, 541, 577, 607, 677, 743, 757, 811, 863, 881, 911, 971, 1031, 1049, 1063, 1093, 1249, 1319, 1373, 1433, 1489

Offset: 1

Amiram Eldar, Sep 23 2022

nonn

The record values correspond to the terms of A064440.

All the terms are primes by definition.

Amiram Eldar, Table of n, a(n) for n = 1..188

Cf. A001065, A048138, A064440, A238895, A238896.

seq[max_] := Module[{s = Table[0, {n, 1, max}], i, v = {}, m = 0}, Do[If[(i = DivisorSigma[1, n] - n) <= max, s[[i]]++], {n, 2, (max - 1)^2}]; Do[If[s[[i]] > m && PrimeQ[s[[i]]], m = s[[i]]; AppendTo[v, s[[i]]]], {i, 1, max}]; v]; seq[2000]

a(n) = A048138(A064440(n)).

A160095 Terms n > 1 of A001065, sorted.

Original entry on oeis.org

3, 4, 6, 6, 7, 8, 8, 9, 10, 11, 12, 13, 13, 14, 14, 15, 15, 16, 16, 17, 17, 18, 19, 19, 20, 20, 21, 21, 21, 22, 22, 23, 23, 24, 25, 25, 25, 26, 27, 27, 28, 29, 29, 30, 31, 31, 31, 31, 31, 32, 32, 33, 33, 33, 34, 35, 35, 35, 36, 37, 37, 37, 37, 38, 39, 40, 40, 40, 41, 41, 41, 41

Offset: 1

Jaroslav Krizek, May 01 2009

nonn

Term 1 occurs infinitely often in A001065, therefore terms <= 1 are excluded.

n occurs in the present sequence with multiplicity A048138(n).

			6 occurs twice in the present sequence because there are two numbers m, viz. 6 and 25, for which A001065(m) = 6.

Cf. A001065 (sum of proper divisors of n), A048138 (a(n) = number of m such that sum of proper divisors of m is n).

k:=1700; S:=[ a: n in [1..k] | a gt 1 and a le Sqrt(k) where a is SumOfDivisors(n)-n ]; Sort(~S); S; // Klaus Brockhaus, May 20 2009

Edited and extended by Klaus Brockhaus, May 20 2009

cp's OEIS Frontend

A324938 a(n) = the number of values of m such that the sum of proper unitary divisors of m (A034460) is n.

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A125601 a(n) is the smallest k > 0 such that there are exactly n numbers whose sum of proper divisors is k.

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A331973 a(n) is the number of values of m such that the sum of proper infinitary divisors of m (A126168) is n.

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A372739 a(n) is the number of possible values of k such that the sum of aliquot coreful divisors of k (A336563) is n.

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A253303 Smallest integer m such that gcd{x | sum of proper divisors of x is m} is equal to 2*n, when there are at least two such x's.

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A331971 a(n) is the number of values of m such that the sum of proper bi-unitary divisors of m (A331970) is n.

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A123930 a(n) = smallest k > 1 that is the sum of the proper divisors (cf. A001065) of at least n different numbers.

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A252997 Numbers n such that sigma(x) - x = n has at least two solutions, with each x having the same squarefree kernel, where sigma(x) is the sum of divisor function (A000203).

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