A002191 -id:A002191 - 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

A362401 Numbers in the range of A162296, where A162296(n) is the sum of divisors of n that have a square factor larger than 1.

Original entry on oeis.org

0, 4, 9, 12, 16, 24, 25, 27, 28, 32, 36, 48, 49, 54, 56, 60, 72, 75, 79, 80, 96, 100, 108, 112, 117, 120, 121, 124, 126, 128, 144, 147, 150, 152, 162, 168, 169, 176, 180, 183, 192, 196, 199, 200, 216, 224, 240, 248, 252, 268, 270, 272, 288, 289, 294, 296, 300

Offset: 1

Amiram Eldar, Apr 18 2023

nonn

Possible values of A162296 in increasing order.

			0 is a term since A162296(k) = 0 if k is squarefree (A005117).

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

Cf. A005117, A162296.

Similar sequences: A078923, A002191, A002202, A002174, A274790.

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; m = 300; Select[Union[Array[s, m]], # <= m &]

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) = select(x -> (x < kmax), Set(vector(kmax, k, s(k))))

A038688 Squares that are the sum of the divisors of some number.

Original entry on oeis.org

1, 4, 36, 121, 144, 256, 324, 400, 576, 784, 900, 961, 1024, 1296, 1600, 1764, 1936, 2304, 2704, 2916, 3136, 3600, 3844, 4096, 4356, 4624, 4900, 5184, 5776, 6084, 6400, 7056, 7744, 8100, 9216, 9604, 10000, 10404, 10816, 11664, 12544, 12996, 14400

Offset: 1

David W. Wilson

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)
Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems: Inversion of Multiplicative Functions (invphi.gp).

Intersection of A000290 and A002191.

Cf. A000203, A006532, A228061.

nn = 14400; t = Select[Union[DivisorSigma[1, Range[nn]]], IntegerQ[Sqrt[#]] &]; t = Select[t, # <= nn &] (* T. D. Noe, Sep 04 2013 *)

lista(kmax) = for(k = 1, kmax, if(invsigmaNum(k^2) > 0, print1(k^2, ", "))); \\ Amiram Eldar, Aug 12 2024, using Max Alekseyev's invphi.gp

a(n) = A228061(n)^2. - Amiram Eldar, Aug 12 2024

A060657 Odd values of the sum-of-divisors function sigma (A000203) (without repetitions).

Original entry on oeis.org

1, 3, 7, 13, 15, 31, 39, 57, 63, 91, 93, 121, 127, 133, 171, 183, 195, 217, 255, 307, 363, 381, 399, 403, 465, 511, 549, 553, 741, 781, 819, 847, 855, 871, 921, 931, 961, 993, 1023, 1093, 1143, 1209, 1281, 1407, 1651, 1659, 1723, 1729, 1767, 1815, 1893, 1953

Offset: 1

Robert G. Wilson v, Apr 18 2001

nonn

That is, the odd values produced by the sigma function.

Odd terms of A002191. - Michel Marcus, Jun 10 2014

			a(7) = 39 because sigma(18) = 1+2+3+6+9+18 = 39, an odd number.

Giovanni Resta, Table of n, a(n) for n = 1..10000
Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems (invphi.gp).

Cf. A000203, A002191, A007369.

nn = 2000; Union[Select[DivisorSigma[1, Range[nn]], OddQ[#] && # <= nn &]]  (* Harvey P. Dale, Mar 12 2011 *)

is(k) = k % 2 && invsigmaNum(k) > 0; \\ Amiram Eldar, Dec 26 2024, using Max Alekseyev's invphi.gp

Name edited by Giovanni Resta, Jan 08 2020

A109322 a(n) is the minimum positive integer j such that [j, j+n-1] does not contain any values of sigma(k) (i.e., sum of all positive divisors of k).

Original entry on oeis.org

2, 9, 9, 49, 49, 423, 423, 423, 423, 1333, 1333, 4425, 4425, 4425, 4425, 8763, 8763, 14089, 14089, 22825, 22825, 22825, 22825, 40291, 40291, 40291, 40291, 40291, 40291, 178705, 178705, 661285, 661285, 661285, 661285, 4543141, 4543141, 4543141, 4543141, 4543141, 4543141, 4543141, 4543141, 4543141, 4543141, 4543141, 4543141

Offset: 1

Max Alekseyev and Jud McCranie, Aug 08 2005

nonn

Donovan Johnson, Table of n, a(n) for n = 1..85 (terms < 10^10)
Art of Problem Solving, Gaps in {sigma(n)}...

Cf. A002191, A083531, A109323, A109324, A110875.

A111865 Expansion of g.f. Product_{k>=1} 1/(1-x^sigma(k)).

Original entry on oeis.org

1, 1, 1, 2, 3, 3, 5, 7, 9, 11, 14, 17, 24, 29, 36, 46, 57, 66, 85, 103, 125, 151, 182, 213, 264, 310, 368, 440, 524, 604, 724, 849, 998, 1164, 1363, 1573, 1854, 2136, 2481, 2879, 3336, 3807, 4427, 5079, 5844, 6698, 7695, 8754, 10072, 11451, 13075, 14898, 16988

Offset: 0

Jon Perry, Nov 23 2005

nonn

Number of partitions of n into parts of size p = sigma(k) for some k, when there are A054973(p) kinds of part p.

			a(6) = 5 : We have sigma(1)=1, sigma(2)=3, sigma(3)=4, sigma(5)=6 so 111111, 1113, 114, 6 and 33.

Alois P. Heinz, Table of n, a(n) for n = 0..10000 (first 1001 terms from Seiichi Manyama)

Cf. A000203, A002191, A007609, A054973, A305320.

with(numtheory):
seq(coeff(series(mul(1/(1-x^sigma(k)),k=1..n), x,n+1),x,n),n=0..60); # Muniru A Asiru, May 31 2018

CoefficientList[ Series[Product[1/(1 - x^DivisorSigma[1, k]), {k, 47}], {x, 0, 52}], x] (* Robert G. Wilson v, Nov 25 2005 *)

lista(nn) = Vec(prod(k=1, nn, 1/(1-x^sigma(k))+ O(x^nn))) \\ Michel Marcus, May 30 2018

G.f.: Product_{k>=1} 1/(1-x^sigma(k)).

More terms from Robert G. Wilson v, Nov 25 2005

a(0)=1 prepended by Seiichi Manyama, May 30 2018

A202274 Numbers k for which sigma(k) = 2^m - 1 for some m.

Original entry on oeis.org

1, 2, 4, 8, 16, 25, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 65536, 131072, 262144, 524288, 1048576, 2097152, 4194304, 8388608, 16777216, 33554432, 67108864, 134217728, 268435456, 536870912, 1073741824, 2147483648, 4294967296, 8589934592, 17179869184, 34359738368, 68719476736, 137438953472

Offset: 1

Jaroslav Krizek, Dec 25 2011

nonn

Original definition, now conjectural, is "Positive integers m in increasing order determined by these rules: a(1) = 1, for n>=1, if m is in the sequence then also are numbers h such that sigma(h) = 4m-1". This is certainly equal to the new definition if 25 is the only term that is not a power of 2, as there is no x such that sigma(x) = 99 = 4*25-1. - Antti Karttunen, Dec 13 2024
If 31 is only number h of form 2^k-1 for any k>=1 such that sigma(x) = h has solution for more than one value of x then a(n) is union number 25 with A000079 (powers of 2).
Numbers k such that A000203(k) is in A000225. If Goormaghtigh conjecture is valid, then it is certain that 25 is the only odd prime power (after 1) in this sequence. - Antti Karttunen, Dec 13 2024

			These examples relate to the original definition:
m=1, 4m-1=3, sigma(h)=3 for h=2; number 2 is in sequence.
m=2, 4m-1=7, sigma(h)=7 for h=4; number 4 is in sequence.
m=4, 4m-1=15, sigma(h)=15 for h=8; number 8 is in sequence.
m=8, 4m-1=31, sigma(h)=31 for h=16 and 25; numbers 16 and 25 are in sequence.

Wikipedia, Goormaghtigh conjecture

Cf. A000079 (subsequence), A000203, A000225, A002191, A292369 (conjectured subsequence).
Subsequence of A028982. Conjectured intersection of A028982 and A378983.
Positions of 0's in A336694, A336695.
Positions of 1's in A324294, A332459, A336692, A336693, A336696.
Cf. also A202273.

is_A202274(n) = ((x->!bitand(x,x+1))(sigma(n)));
for(n=1,2^20,if(is_A202274(n^2), print1(n^2,", ")); if(n>1 && is_A202274(2*((n-1)^2)), print1(2*((n-1)^2),", "))); \\ Remember to sort! - Antti Karttunen, Dec 13 2024

Data section corrected (terms 1024, 2048 were duplicated), more terms added, and the name replaced with a new definition, with the original definition moved to the comments - Antti Karttunen, Dec 13 2024

A274790 Numbers in the range of the sum of odd divisors function.

Original entry on oeis.org

1, 4, 6, 8, 12, 13, 14, 18, 20, 24, 30, 31, 32, 38, 40, 42, 44, 48, 54, 56, 57, 60, 62, 68, 72, 74, 78, 80, 84, 90, 96, 98, 102, 104, 108, 110, 112, 114, 120, 121, 124, 128, 132, 133, 138, 140, 144, 150, 152, 156, 158, 160, 164, 168, 174, 176, 180, 182, 183

Offset: 1

Timothy L. Tiffin, Jul 06 2016

nonn

Numbers which appear in A000593.

Possible values for the sum of odd divisors of the positive integers, in increasing order. - Omar E. Pol, Jul 06 2016

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000593, subsequence of A002191, A274793 (complement).

N:= 10000:# to get all terms <= N
p:= 3:
S:= {1}:
while p < N do
  S:= map(s -> seq(s*(p^(e+1)-1)/(p-1), e=0..ilog[p](1+N*(p-1)/s)-1), S);
  p:= nextprime(p);
od:
sort(convert(S,list)); # Robert Israel, Jul 06 2016

Union@ Table[Total@ Select[Divisors@ n, OddQ], {n, 200}] (* Michael De Vlieger, Jul 07 2016 *)

list(lim)=lim\=1; my(v=[1]); forprime(p=3,lim\4, my(t,u=v,lm); for(e=1,logint(lim,p), t=(p^(e+1)-1)/(p-1); lm=lim\t; u=concat(u,select(k->k<=lm,v)*t)); v=Set(u)); Set(concat(v, apply(p->p+1, primes([lim\4,lim-1])))) \\ Charles R Greathouse IV, Jul 06 2016

A108348 Numbers of the form p^k + p^(k-1) + ... + p + 1 (where p is a prime and k>=0) in ascending order.

Original entry on oeis.org

1, 3, 4, 6, 7, 8, 12, 13, 14, 15, 18, 20, 24, 30, 31, 32, 38, 40, 42, 44, 48, 54, 57, 60, 62, 63, 68, 72, 74, 80, 84, 90, 98, 102, 104, 108, 110, 114, 121, 127, 128, 132, 133, 138, 140, 150, 152, 156, 158, 164, 168, 174, 180, 182, 183, 192, 194, 198, 200

Offset: 1

Franz Vrabec, Jul 01 2005

nonn

A proper subset of A002191 (e.g., 28 is in A002191, but not in this sequence). a(15)=31 admits two representations: 31=2^4+2^3+2^2+2+1=5^2+5+1. Are there other numbers with two or more representation?
I have checked all the sums of primes up to prime number 56873 to a sum total >= 10^100 and have not come across another number that has multiple representations. - Patrick Schutte (patrick(AT)onyxsa.co.za), Mar 28 2007
Goormaghtigh conjecture implies that 31 is the only term with 2 representations; see the Wikipedia link below. - Jianing Song, Nov 22 2022

			a(2)=3=2+1 since a(1)=1 and 2 is not expressible in the required form.

M. F. Hasler, Table of n, a(n) for n = 1..1000
Wikipedia, Goormaghtigh conjecture

Cf. A002191.

Cf. A000040, A090503.

SumNum := function ( FNum) local a,ap,b,bp,at,bt; a := 2; repeat at := 1; ap := 1; repeat at := at + a^ap; b := 2; repeat bt := 1; bp := 1; repeat bt := bt + b^bp; if at = bt and bp > 1 and a <> b then Print("a ",a," ap ",ap," at ", at," "); Print("b ",b," bp ",bp," bt ", bt," "); Print("---------------- "); fi; bp := bp + 1; until bt > at; b := NextPrime(b); until b >=a; ap := ap + 1; until at > 10^100; a := NextPrime(a); until a >FNum; end; # Patrick Schutte (patrick(AT)onyxsa.co.za), Mar 28 2007

a108348 n = a108348_list !! (n-1)
a108348_list = 1 : f [2..] where
   f (x:xs) = g a000040_list where
     g (p:ps) = h 0 $ map ((`div` (p - 1)) . subtract 1) $
                          iterate (* p) (p ^ 2) where
       h i (pp:pps) | pp > x    = if i == 0 then f xs else g ps
                    | pp < x    = h 1 pps
                    | otherwise = x : f xs
-- Reinhard Zumkeller, Nov 26 2013

A108348(n)={ local(m=1, a=[m]); while( #aA108348(1000) \\ M. F. Hasler

A243117 Possible values of A242962 in increasing order.

Original entry on oeis.org

0, 1, 3, 6, 8, 12, 13, 14, 15, 18, 20, 24, 28, 30, 31, 32, 36, 38, 39, 40, 42, 44, 48, 54, 56, 57, 60, 62, 63, 68, 72, 74, 78, 80, 84, 90, 91, 93, 96, 98, 102, 104, 108, 110, 112, 114, 120, 121, 124, 126, 127, 128, 132, 133, 138, 140, 144, 150, 152, 156, 158

Offset: 1

Jaroslav Krizek, May 29 2014

nonn

A242962(n) = (n*(n+1)/2) mod antisigma(n) = A000217(n) mod A024816(n); where A000217(n) = triangular numbers, A024816(n) = sum of numbers less than n which do not divide n.
Complement of A243118.
a(n) = A002191(n+1) for n >= 5; where A002191 = possible values of sigma(n).

			0 is in the sequence because there is a number m such that A242962(m) = 0; m = 3.

Cf. A000203, A002191, A242962, A242963, A243118.

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

A362401 Numbers in the range of A162296, where A162296(n) is the sum of divisors of n that have a square factor larger than 1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A038688 Squares that are the sum of the divisors of some number.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A060657 Odd values of the sum-of-divisors function sigma (A000203) (without repetitions).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A109322 a(n) is the minimum positive integer j such that [j, j+n-1] does not contain any values of sigma(k) (i.e., sum of all positive divisors of k).

Views

Author

Keywords

Links

Crossrefs

A111865 Expansion of g.f. Product_{k>=1} 1/(1-x^sigma(k)).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

Extensions

A202274 Numbers k for which sigma(k) = 2^m - 1 for some m.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Extensions

A274790 Numbers in the range of the sum of odd divisors function.

Views