A002191 -id:A002191 - cp's OEIS frontend

A202273 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) = 3m.

1, 2, 5, 8, 14, 15, 20, 23, 24, 26, 30, 38, 40, 41, 45, 46, 51, 54, 55, 56, 58, 59, 60, 71, 74, 78, 87, 88, 89, 90, 92, 95, 106, 113, 118, 123, 136, 137, 143, 145, 146, 153, 167, 173, 178, 179, 215, 233, 263, 269, 303, 317, 335, 353

Offset: 1

Jaroslav Krizek, Dec 25 2011

Sequence is finite with 54 terms.

			m=1, 3m=3, sigma(h)=3 for h=2; number 2 is in sequence.
m=2, 3m=6, sigma(h)=6 for h=5; number 5 is in sequence.
m=5, 3m=15, sigma(h)=15 for h=8; number 8 is in sequence.
m=8, 3m=24, sigma(h)=24 for h=14,15,23; numbers 14,15,23 are in sequence.

Cf. A000203, A002191, A202274.

A258912 Numbers k such that A000203(x) = k has more than one solution and they all share the same largest prime factor.

Original entry on oeis.org

1178, 1364, 1408, 1656, 1767, 1836, 1922, 1984, 2108, 2196, 2328, 2368, 3162, 3336, 3410, 3996, 4096, 4123, 4144, 4278, 4898, 5064, 5076, 5084, 5248, 5456, 5488, 5673, 6014, 6208, 6504, 6784, 6816, 7416, 7998, 8618, 8896, 9088, 9184, 9517, 10048, 10292, 10864

Offset: 1

Michel Marcus, Jun 14 2015

nonn

By definition this is a subsequence of A159886.

Pollack shows that the density of such integers relative to A002191 is 1.

			The pre-image of 1178 is [592, 925], and both have greatest prime factor 37, so 1178 is in the sequence.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Paul Pollack, Remarks on fibers of the sum-of-divisors function, Analytic number theory (volume in honor of H. Maier), M. Rassias and C. Pomerance, eds., Springer, 2015, alternative link.

Cf. A000203 (sum of divisors), A002191 (possible values of sum of divisors), A159886 (sigma(x)=n has more than one solution).

isok(n) = {my(v = select(x->sigma(x)==n, vector(n, i, i))); if (#v < 2, return (0)); vgpf = vector(#v, k, fvk = factor(v[k]); fvk[#fvk~,1]); vecmin(vgpf) == vecmax(vgpf);}

A275671 Even values produced by the sigma function A000203, in increasing order.

Original entry on oeis.org

4, 6, 8, 12, 14, 18, 20, 24, 28, 30, 32, 36, 38, 40, 42, 44, 48, 54, 56, 60, 62, 68, 72, 74, 78, 80, 84, 90, 96, 98, 102, 104, 108, 110, 112, 114, 120, 124, 126, 128, 132, 138, 140, 144, 150, 152, 156, 158, 160, 162, 164, 168, 174, 176, 180, 182, 186, 192, 194

Offset: 1

Jaroslav Krizek, Aug 04 2016

nonn

Even terms of A002191.

Complement of A060657 with respect to A002191.

			8 is in the sequence because sigma(7) = 8 and it is an even number.

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

Cf. A000203, A002191, A060657, A152678.

Set(Sort([SumOfDivisors(n): n in[1..1000] | not IsOdd(SumOfDivisors(n)) and SumOfDivisors(n) le 1000]));

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

A286011 a(1)=1, and for n>1, a(n) is the maximum number of iterations of sigma resulting in n, starting at some integer k; or 0 if n cannot be reached from any k.

Original entry on oeis.org

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

Offset: 1

Michel Marcus, Apr 30 2017

nonn

a(n)=0 for n in A007369 and a(n)>0 for n in A002191.
Records are found at indices given by A007497.
The above would be correct for a(1) = 0 (in a weak sense) or rather a(1) = -1 (for infinity), but as the sequence is defined, 2 & 3 do not produce a record, so the indices of records are 1, (3), 4, 7, ... = {1} U A007497 \ {2, (3)}. - M. F. Hasler, Nov 20 2019

			a(4)=2 because 4=sigma(3), but also sigma(sigma(2)) with 2 iterations.
a(7)=3 because 7=sigma(4), but also sigma(sigma(3)), and sigma(sigma(sigma(2))), with 3 iterations.

Robert Israel, Table of n, a(n) for n = 1..10000
M. Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems: invsigma.gp, Oct. 2005

Cf. A000203, A002191, A007369, A007497, A257670.

N:= 100: # to get a(1)..a(N)
V:= Vector(N):
for n from 1 to N do
  s:= numtheory:-sigma(n);
  if s <= N then V[s]:= max(V[s],V[n]+1) fi
od:
convert(V,list); # Robert Israel, May 01 2017

a(n) = {if (n==1, return(1)); vn = vector(n-1, k, k+1); nb = 0; knb = 0; ok = 1; while(ok, nb++; vn = vector(#vn, k, sigma(vn[k])); svn = Set(vn); if (#select(x->x==n, svn), knb = nb); if (!#select(x->x<=n, svn), ok = 0);); knb;}

apply( A286011(n)=if(n<3,2-n, n=invsigma(n), vecmax(apply(self,n))+1), [1..99]) \\ See Alekseyev-link for invsigma(). - M. F. Hasler, Nov 20 2019

A289872 a(n) is the number of partial sums of the divisors of n that are the sum of divisors of some integer.

Original entry on oeis.org

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

Offset: 1

Michel Marcus, Jul 14 2017

nonn

			For n=2, the divisors are 1, 2; the partial sums are 1, 3; 1=sigma(1) and 3=sigma(2); so a(2)=2.
For n=10, the divisors are 1, 2, 5, 10; the partial sums are 1, 3, 8, 18; 1=sigma(1), 3=sigma(2), 8=sigma(7) and 18=sigma(10); so a(10)=4.

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

Cf. A000203, A002191, A027750, A240698.

M:= 1000: # get a(n) for n=1..m where m is the first number with sigma(m+1) > M
S:= Vector(M):
for n from 1 to M-1 do
  v:= numtheory:-sigma(n);
  if v > M then if not assigned(nmax) then nmax:= n-1 fi
  elif S[v] = 0 then S[v]:= 1
  fi;
od:
seq(add(S[i],i=ListTools:-PartialSums(sort(convert(numtheory:-divisors(n),list)))), n = 1..nmax); # Robert Israel, Jul 14 2017

s = Union@ DivisorSigma[1, Range[10^6]]; Array[Count[Accumulate@ Divisors@ #, k_ /; MemberQ[s, k]] &, 90] (* Michael De Vlieger, Jul 14 2017 *)

issigma(n) = {for (k=1, n, if (sigma(k) == n, return (1));); 0;}
a(n) = {d = divisors(n); v = vector(#d, k, sum(j=1, k, d[j])); sum(k=1, #v, issigma(v[k]));}

For n>=1 and p prime, a(p^n) = n+1.

A300779 Odd numbers x such that x and x + 2 are both sums of divisors, i.e., elements of A000203.

Original entry on oeis.org

1, 13, 91, 241573, 38152387, 139415801707, 55342019130181, 61166380109329, 417542026135897, 417542026135897, 13805828672331787

Offset: 1

Hugo Pfoertner, Mar 12 2018

If some x or x + 2 is in A300869, i.e., it has more than one representation as sigma(m), as for x = 417542026135897 = sigma((4*17*209459)^2) = sigma((5*17*209459)^2) = sigma((2*7723267)^2) - 2, then it is listed with multiplicity and all corresponding pairs of numbers are provided in A300780.

			a(1) = 1 because 1 = sigma(1) and 3 = sigma(2),
a(2) = 13: 13 = sigma(9) and 15 = sigma(8),
a(3) = 91: 91 =sigma(36), 93 = sigma(50),
a(4) = 241573: 241573 = sigma(241081), 241575 = sigma(117128),
a(5) = 38152387: 38152387 = sigma(15069924), 38152389 = sigma(23011209).

Seqfan Mailing List Thread, Consecutive sigmas, with contributions from _Franklin T. Adams-Watters_, _Hugo Pfoertner_ and _Emmanuel Vantieghem_.

Cf. A000203, A002191, A083531, A300780 (numbers corresponding to sigma values), A300869.

a(5) from Emmanuel Vantieghem

a(6)-a(11) from Giovanni Resta, Mar 13 2018

A300780 Pairs of numbers producing consecutive odd sums of divisors, i.e., sigma(a(2k)) = sigma(a(2k-1)) + 2, with sigma values given in A300779.

Original entry on oeis.org

1, 2, 9, 8, 36, 50, 241081, 117128, 15069924, 23011209, 95887457649, 92943436658, 31623999684484, 31716423048338, 48730521525625, 55647526914529, 202869088076944, 238595412613156, 316982950120225, 238595412613156, 12016309156631329, 9203885645879282

Offset: 1

Hugo Pfoertner, Mar 12 2018

So far, a(2n-1) is always a square. Will this always hold? - M. F. Hasler, Mar 12 2018

			a(1) = 1, a(2) = 2 because sigma(2) - sigma(1) = 3 - 1 = 2.
a(3) = 9, a(4) = 8 because sigma(8) - sigma(9) = 15 - 13 = 2.
a(5) = 36, a(6) = 50 because sigma(50) - sigma(36) = 93 - 91 = 2.
The first pairs have the following factorization: (3^2, 2*2^2), (6^2, 2*5^2), (491^2, 2*(2*11^2)^2), ((2*3*647)^2, (3^2*13*41)^2). - _M. F. Hasler_, Mar 12 2018

Cf. A000203, A002191, A300779.

a(11)-a(22) from Giovanni Resta, Mar 13 2018

A379655 Numbers k such that k and k+1 are both possible values of the sum of divisors function (A000203).

Original entry on oeis.org

3, 6, 7, 12, 13, 14, 30, 31, 38, 39, 56, 62, 90, 120, 126, 127, 132, 182, 194, 216, 255, 306, 307, 363, 380, 398, 399, 402, 464, 510, 511, 548, 552, 740, 780, 846, 847, 854, 920, 930, 960, 961, 992, 1022, 1023, 1092, 1093, 1280, 1407, 1650, 1658, 1722, 1723, 1728

Offset: 1

Amiram Eldar, Jan 03 2025

Numbers k such that k and k+1 are both in A002191.

			3 is a term since 3 = sigma(2) and 3 + 1 = 4 = sigma(3).
6 is a term since 6 = sigma(5) and 6 + 1 = 7 = sigma(4).

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

Cf. A000203, A342555, A342560.

Subsequence of A002191.

seq[lim_] := Module[{v = Select[Union[DivisorSigma[1, Range[lim]]], # <= lim &]}, v[[Position[Differences[v], 1] // Flatten]]]; seq[2000]

isA002191(n) = invsigmaNum(n) > 0; \\ using Max Alekseyev's invphi.gp
list(lim) = my(q1 = isA002191(1), q2); for(k = 2, lim, q2 = isA002191(k); if(q1 && q2, print1(k-1, ", ")); q1 = q2);

A379723 Possible values of the sum of squares of divisors function (A001157).

Original entry on oeis.org

1, 5, 10, 21, 26, 50, 85, 91, 122, 130, 170, 210, 250, 260, 290, 341, 362, 455, 500, 530, 546, 610, 651, 820, 842, 850, 962, 1050, 1220, 1300, 1365, 1370, 1450, 1682, 1700, 1810, 1850, 1911, 2210, 2366, 2451, 2500, 2562, 2650, 2810, 2900, 3172, 3255, 3410, 3482

Offset: 1

Amiram Eldar, Jan 03 2025

The distinct values of the sigma_2(n) function, in ascending order.
The asymptotic density of this sequence is 0 (Niven, 1951).
5460 = sigma_2(60) and 5461 = sigma_2(64) are two consecutive integers in this sequence. Are there any other such pairs? There are none below 10^10.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems: Inversion of Multiplicative Functions (invphi.gp).
Ivan Niven, The asymptotic density of sequences, Bull. Amer. Math. Soc., Vol. 57 (1951), pp. 420-434. See Theorem 3, p. 429.

Cf. A001157, A002191, A002202.
A066872 is a subsequence.
Subsequence of A211347.

seq[lim_] := Select[Union[DivisorSigma[2, Range[lim]]], # <= lim &]; seq[3500]

is(n) = invsigmaNum(n, 2) > 0; \\ Amiram Eldar, Jan 03 2025, using Max Alekseyev's invphi.gp

A072612 Minimal value of { abs(n-sigma(k)) : k>0 }.

Original entry on oeis.org

0, 1, 0, 0, 1, 0, 0, 0, 1, 2, 1, 0, 0, 0, 0, 1, 1, 0, 1, 0, 1, 2, 1, 0, 1, 2, 1, 0, 1, 0, 0, 0, 1, 2, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 2, 1, 0, 1, 2, 3, 2, 1, 0, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 2, 3, 1, 0, 1, 2, 1, 0, 1, 0, 1, 2, 1, 0, 1, 0, 1, 2, 1, 0, 1, 2, 3, 2, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 1, 2, 1, 0, 1, 0, 1

Offset: 1

Benoit Cloitre, Aug 11 2002

a(A002191(k)) = 0

a(n)=vecmin(vector(n,k,abs(n-sigma(k))))

Conjecture : S(n) = sum( k=1, n, a(k) ) = n + O(sqrt(n)) and more precisely for n large enough : n - (5/2)*sqrt(n) < S(n) < n - (3/2)*sqrt(n)

cp's OEIS Frontend

A202273 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) = 3m.

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A258912 Numbers k such that A000203(x) = k has more than one solution and they all share the same largest prime factor.

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PARI

A275671 Even values produced by the sigma function A000203, in increasing order.

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Magma

PARI

A286011 a(1)=1, and for n>1, a(n) is the maximum number of iterations of sigma resulting in n, starting at some integer k; or 0 if n cannot be reached from any k.

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Maple

PARI

PARI

A289872 a(n) is the number of partial sums of the divisors of n that are the sum of divisors of some integer.

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Maple

Mathematica

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A300779 Odd numbers x such that x and x + 2 are both sums of divisors, i.e., elements of A000203.

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Extensions

A300780 Pairs of numbers producing consecutive odd sums of divisors, i.e., sigma(a(2*k)) = sigma(a(2*k-1)) + 2, with sigma values given in A300779.

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Extensions

A379655 Numbers k such that k and k+1 are both possible values of the sum of divisors function (A000203).

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Mathematica

A300780 Pairs of numbers producing consecutive odd sums of divisors, i.e., sigma(a(2k)) = sigma(a(2k-1)) + 2, with sigma values given in A300779.