A007368 -id:A007368 - cp's OEIS frontend

A300869 Odd numbers m such that sigma(x) = m has more than 1 solution.

31, 399, 403, 1767, 3751, 4123, 5187, 5673, 9517, 11811, 12369, 17143, 22971, 27001, 30783, 33883, 34671, 43617, 48279, 53413, 53599, 54873, 58683, 68859, 69967, 73017, 73749, 80199, 86831, 88753, 109771, 117273, 122493, 123721, 141267, 152019, 153543, 158503, 160797

Offset: 1

M. F. Hasler, following a suggestion from Altug Alkan, Mar 16 2018

nonn

Goormaghtigh conjecture implies that 31 is the only prime in this sequence. - Jianing Song, Apr 27 2019

			a(1) = 31 = A123523(2), the smallest odd number m for which sigma(x) = m has (at least, and also exactly) two solutions, x = 16 and x = 25.
a(56) = 347529 = A123523(3) is the smallest odd m for which sigma(x) = m has (at least, and also exactly) three solutions, x = 406^2, x = 2*319^2 and x = 489^2.

Robert Israel, Table of n, a(n) for n = 1..9260
Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems (invphi.gp).
Wikipedia, Goormaghtigh conjecture.

Odd terms in A159886.
Cf. A000203 (sigma), A002191, A007368.
A123523 is a subsequence, except for the initial 1.
Cf. A331036.

N:= 200000: # for terms <= N
Res:= NULL: count:= 0:
for m from 1 to floor(sqrt(N)) by 2 do
sm:= numtheory:-sigma(m^2);
for k from 1 to floor(log[2](N/sm+1)) do
   v:= sm*(2^k-1);
   if v <= N then Res:= Res, v; count:= count+1 fi;
od
od:
B:= sort([Res]):
Dups:= select(t -> B[t+1]=B[t], [$1..nops(B)-1]):
sort(convert(convert(B[Dups],set),list)); # Robert Israel, Jan 15 2020

With[{s = PositionIndex@ Array[DivisorSigma[1, #] &, 10^6]}, Keys@ KeySort@ KeySelect[s, And[OddQ@ #, Length@ Lookup[s, #] > 1] &]] (* Michael De Vlieger, Mar 16 2018 *)

MAX=1e6; LIM=1e4; b=0; A300869=[]; for(x=1,LIM, for(i=1,2, (s=sigma(i*x^2))>MAX && next(2); bittest(b,s\2) && (setsearch(A300869,s) || S=setunion(A300869,[s])) || b+=1<<(s\2)))

is(k) = k%2 && invsigmaNum(k) > 1; \\ Amiram Eldar, Dec 16 2024, using Max Alekseyev's invphi.gp

A202275 Differences between A074753 (number of integers k such that sigma(k) <= n) and A202276 (number of integers k <= n such that sigma(x) = k has no solution); sigma = A000203.

Original entry on oeis.org

1, 0, 1, 2, 1, 2, 3, 4, 3, 2, 1, 3, 4, 5, 6, 5, 4, 6, 5, 6, 5, 4, 3, 6, 5, 4, 3, 4, 3, 4, 6, 8, 7, 6, 5, 6, 5, 6, 7, 8, 7, 10, 9, 10, 9, 8, 7, 10, 9, 8, 7, 6, 5, 7, 6, 8, 9, 8, 7, 10, 9, 10, 11, 10, 9, 8, 7, 8, 7, 6, 5, 10, 9, 10, 9, 8, 7, 8, 7, 9, 8, 7, 6, 9

Offset: 1

Jaroslav Krizek, Dec 25 2011

sign

Conjectures: Max a(n) = 15 for n = 195, 403, 434. For n >= 687, a(n) < 0.

First term < 0: a(538) = -1.

Jaroslav Krizek, Table of n, a(n) for n = 1..10000

Cf. A074753, A202277, A202276, A175523, A007369, A007368, A007609.

a(n) = A074753(n) - A202276(n).

A202276 Number of integers k <= n such that sigma(x) = k has no solution, sigma = A000203.

Original entry on oeis.org

0, 1, 1, 1, 2, 2, 2, 2, 3, 4, 5, 5, 5, 5, 5, 6, 7, 7, 8, 8, 9, 10, 11, 11, 12, 13, 14, 14, 15, 15, 15, 15, 16, 17, 18, 18, 19, 19, 19, 19, 20, 20, 21, 21, 22, 23, 24, 24, 25, 26, 27, 28, 29, 29, 30, 30, 30, 31, 32, 32, 33, 33, 33, 34, 35, 36, 37, 37, 38, 39

Offset: 1

Jaroslav Krizek, Dec 25 2011

nonn

Partial sums of A175253.

			a(9) = 3 because sigma(x) = k has no solution for k = 2, 5 and 9.

Jaroslav Krizek, Table of n, a(n) for n = 1..10000

Cf. A074753, A202277, A202275, A175523, A007369, A007368, A007609.

seq[max_] := Module[{t = Table[0, {max}]}, t[[Complement[Range[max], Table[ DivisorSigma[1, n], {n, 1, max}]]]] = 1; Accumulate@ t]; seq[100] (* Amiram Eldar, Dec 20 2024 *)

A123523 Smallest odd number k such that sigma(x) = k has exactly n solutions.

Original entry on oeis.org

1, 31, 347529, 10773399, 4104665019, 77253471477, 28732655133, 35492068813383, 108695634368139, 461396894573979, 68452476460273269, 2529134502772059, 99414440839732473

Offset: 1

T. D. Noe, Oct 02 2006

Note that sigma(x) is odd iff x is in A028982 (numbers of the form m^2 or 2m^2 for m > 0).
a(14) > 10^18. a(15) = 175792216832685999. a(16) > 10^18. - Donovan Johnson, Jun 09 2011
From David A. Corneth, Apr 27 2019: (Start)
The least common divisor of the first 13 terms is k = 63540409508528099686942221. Checking the divisors of k to see if they give an upper bound for some a(n) gives these upper bounds:
a(14) <= 2489145199534927711323, for n = 16..27, a(n) <= 30520233337797869211, 1292387730916522149, 3939513268555279291149, 1066776514086397590567, 7538497634436073695117, 1629700928685734429889, 7217246969893966760937, 136456488459785229549035859, 396763033391372299743, 2215694819757447795607659, 500318185106520469975923, 5916133590898752361467873 respectively.
All these listed upper bounds are divisors of 12302819034343122006137404371659222028537. No more divisors of this number are an upper bound for any n.
This method doesn't give a stronger lower bound except that it tells us that a new upper bound for some term is no divisor of k. (End)

			For n = 3, sigma(x) = 347529 has exactly three solutions x = 164836, 203522, 239121.

Max Alekseyev, Computing the Inverses, their Power Sums, and Extrema for Euler's Totient and Other Multiplicative Functions, Journal of Integer Sequences 19 (2016), Article 16.5.2
David A. Corneth, PARI program, partially written by Max Alekseyev

Cf. A007368, A300869.

Different from A123524.

Needs["Statistics`DataManipulation`"]; nn=10^6; t1=DivisorSigma[1,Range[nn]^2]; t2=DivisorSigma[1,2*Range[nn/Sqrt[2]]^2]; t=Join[t1,t2]; {u,v}=Transpose[Sort[Frequencies[t]]]; Table[p=Position[u,i,1,1][[1,1]]; v[[p]], {i,Length[Union[u]]}]

a(8) from Martin Fuller, Oct 07 2006
a(9)-a(10) from Donovan Johnson, Dec 09 2008
a(11)-a(13) from Donovan Johnson, Jun 09 2011

A115374 Least prime p such that sigma(x)=sigma(p) has exactly n solutions.

Original entry on oeis.org

2, 11, 23, 179, 71, 167, 239, 431, 359, 503, 3167, 1511, 4679, 2687, 719, 9719, 4799, 16319, 5471, 10559, 1439, 26399, 24623, 3359, 15359, 3023, 7559, 6719, 2879, 26783, 10799, 13103, 5039, 6047, 45863, 29759, 61559, 18719, 27647, 99839, 22679, 68543

Offset: 1

T. D. Noe, Jan 21 2006

nonn

For 11? It also appears that for each n there are an infinite number of primes p such that sigma(x)=sigma(p) has exactly n solutions.

Donovan Johnson, Table of n, a(n) for n = 1..1199

Cf. A007368 (least k such that sigma(x)=k has n solutions), A066075 (number of solutions to sigma(x)=sigma(prime(n))).

s=DivisorSigma[1, Range[100000]]; t=Table[Length[Position[s, Prime[n]+1]], {n, PrimePi[Length[s]]}]; u=Union[t]; nLast=First[Complement[Range[u[[ -1]]], u]]-1; Flatten[Table[Prime[Position[t, n, 1, 1]], {n, nLast}]]

sigv(n) =  select(i->sigma(i) == n, vector(n, i, i));
a(n) = {p = 2; while (#(sigv(p+1))! = n, p = nextprime(p+1)); p;} \\ Michel Marcus, May 01 2014

A175253 a(n) = characteristic function of numbers k such that A000203(m) = k has no solution for any m, where A000203(m) = sum of divisors of m.

Original entry on oeis.org

0, 1, 0, 0, 1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 0, 0, 0, 1, 1, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1

Offset: 1

Jaroslav Krizek, Mar 14 2010

nonn

a(n) = characteristic function of numbers from A007369(n). a(n) = 1 if A000203(m) not equal to n for any m, else 0. a(n) = 1 for such n that A054973(n) = 0. a(n) = 0 for such n that A054973(n) >= 1. a(n) + A175192(n) = A000012(n).

Jaroslav Krizek, Table of n, a(n) for n = 1..10000

Cf. A074753, A202275, A202276, A175523, A007369, A007368, A007609.

seq[max_] := Module[{t = Table[0, {max}]}, t[[Complement[Range[max], Table[ DivisorSigma[1, n], {n, 1, max}]]]] = 1; t]; seq[100] (* Amiram Eldar, Mar 22 2024 *)

More terms from Jaroslav Krizek, Dec 25 2011.

A202277 Numbers m such that number of integers k such sigma(k) <= m (A074753) is equal to number of integers k <= m such that sigma(x) = k has no solution (A202276).

Original entry on oeis.org

2, 537, 639, 647, 653, 655, 657, 661, 663, 672, 674, 684, 686

Offset: 1

Jaroslav Krizek, Dec 25 2011

nonn

Numbers m such that A074753(m) - A202276(m) = 0.
Numbers m such that A202275(m) = 0 (see graph of A202275).
Conjecture: sequence is finite with 13 terms.

Cf. A074753, A202275, A202276, A175523, A007369, A007368, A007609.

A090553 Sigma(A090552(n)).

Original entry on oeis.org

1, 3, 6, 12, 12, 24, 24, 24, 72, 72, 72, 72, 144, 144, 144, 144, 144, 288, 288, 288, 576, 576, 576, 576, 576, 576, 576, 576, 1152, 1152, 1152, 1152, 1152, 1152, 1152, 2304, 2304, 2304, 2304, 2304, 2304, 2304, 2304, 2304, 2304, 2304, 2304, 2304, 4608, 4608

Offset: 1

Amarnath Murthy, Dec 09 2003

nonn

Cf. A007368, A090552.

lista(nn) = {a = 1; print1(a, ", "); for (n = 1, nn, sa = sigma(a); a++; while (sigma(a) % sa, a++); print1(sigma(a), ", "););} \\ Michel Marcus, Oct 12 2013

a(n) = A000203(A090552(n)). - Michel Marcus, Oct 12 2013

Corrected and extended by David Wasserman, Jan 04 2006

A239635 Common Sigma, Uncommon Clique Numbers: a(n) is the minimal s for which there exists a set of n pairwise relatively prime integers with a sigma value of s.

Original entry on oeis.org

1, 12, 24, 72, 240, 360, 1440, 1440, 1440, 8640, 10080, 15120, 34560, 45360, 55440, 60480, 60480, 166320, 181440, 211680, 332640, 332640, 332640, 665280, 665280, 665280, 831600, 907200, 1663200, 2494800, 2661120, 2661120

Offset: 1

Fred Schneider, Mar 22 2014

The meaning of the name just involves a little wordplay.
Common Sigma: Refers to an underlying set with a common sigma.
Uncommon Clique: This set or "clique" (see link below) of numbers is "uncommon" because they are pairwise relatively prime and thus have no common factors.
Graph Theory Description:
Suppose we built an incidence graph for each positive integer s whereby each number which had sigma = s represented a vertex in the graph and the only edges were drawn between relatively prime numbers. For each s, we want to find the maximal clique (aka complete subgraph) size.
a(n) would be the minimum s where the maximal clique = n.
(Note: If a clique of size n exists for s, there exists a clique of any size < n, so any gaps in the series could be filled by S (e.g., 1440: terms 7-9).)
The first 5 terms and their solution sets (plus member factors showing they are all relatively prime to each other):
1 : 1
12 : 6 = 2 * 3, 11 = 11
24 : 14 = 2 * 7, 15 = 3 * 5, 23 = 23
72 : 46 = 2 * 23, 51 = 3 * 17, 55 = 5 * 11, 71 = 71
240 : 135 = 3^3 * 5, 158 = 2 * 79, 203 = 7 * 29, 209 = 11 * 19, 239 = 239
An exhaustive search was performed to show that these are in fact the minimal terms.
Comment with b-file submission:
I found a few useful optimizations:
1) For a sigma's possible solution set, only consider the numbers which have 3 or fewer distinct prime factors.
2) Check if there's a number which is a prime (power). Set that aside. (Call the count c1.)
3) Set aside any numbers with two prime factors (If there are duplicates, say two numbers which are multiples of two, pick the smallest exponent. The larger prime factor has less of chance of being found elsewhere in the solution set). Call this count c2.
4) Then, discard any 3-prime factor numbers which are not relatively prime with the above numbers.
5) Of those remaining numbers, determine the set of distinct primes found amongst their factors. Call this set size pc.
6) If (pc/3) + c1 + c2 > maxCliqueSize found, we have a possibility of adding to this sequence and should try to find a "sub-clique" (among the remaining 3-prime numbers) which is maxCliqueSize-c1-c2.

Fred Schneider, Table of n, a(n) for n = 1..65
Fred Schneider, Solution sets for terms 1..65
Wikipedia, Sigma
Wikipedia, Clique

Cf. sigma-related sequences: A000203, A007368.

A271442 a(n) is the smallest k such that sigma(2,x) = k has exactly n solutions, where sigma(2,x) is the sum of the squares of the divisors of x.

Original entry on oeis.org

2, 1, 50, 22100, 6409000, 32045000, 1185665000, 11856650000, 628402450000, 1169065690000, 16338463700000, 81692318500000, 875993015300000, 1388769414500000, 8054862604100000, 88701519427300000, 443507597136500000, 80548626041000000

Offset: 0

Giovanni Resta, Apr 20 2016

nonn

Does a(n) exist for every n?

			a(3) = 22100 because there are exactly 3 values x (120, 130, and 141) such that sigma(2,x) = 22100, and this property does not hold for any number smaller than 22100.

Giovanni Resta, Table of solutions corresponding to a(n), for n=0..17

Cf. A001157, A007368.

t=Gather@ Sort@ DivisorSigma[2, Range@ 800000]; Join[{2}, Table[ Select[t, k == Length@ # &, 1][[1, 1]], {k, 8}]]

cp's OEIS Frontend

A300869 Odd numbers m such that sigma(x) = m has more than 1 solution.

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A202275 Differences between A074753 (number of integers k such that sigma(k) <= n) and A202276 (number of integers k <= n such that sigma(x) = k has no solution); sigma = A000203.

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A202276 Number of integers k <= n such that sigma(x) = k has no solution, sigma = A000203.

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Mathematica

A123523 Smallest odd number k such that sigma(x) = k has exactly n solutions.

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A115374 Least prime p such that sigma(x)=sigma(p) has exactly n solutions.

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A175253 a(n) = characteristic function of numbers k such that A000203(m) = k has no solution for any m, where A000203(m) = sum of divisors of m.

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A202277 Numbers m such that number of integers k such sigma(k) <= m (A074753) is equal to number of integers k <= m such that sigma(x) = k has no solution (A202276).

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A090553 Sigma(A090552(n)).

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A239635 Common Sigma, Uncommon Clique Numbers: a(n) is the minimal s for which there exists a set of n pairwise relatively prime integers with a sigma value of s.

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