A211347 -id:A211347 - cp's OEIS frontend

A002191 Possible values for sum of divisors of n.

1, 3, 4, 6, 7, 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

Offset: 1

N. J. A. Sloane

nonn

Distinct values attained by the sigma(n) function, in ascending order.

The asymptotic density of this sequence is 0 (Niven, 1951, Rao and Murty, 1979). - Amiram Eldar, Jul 23 2020

			a(100) = 272, a(10^3) = 3696, a(10^4) = 44496, a(10^5) = 510356, a(10^6) = 5691216. - _M. F. Hasler_, Nov 22 2019

J. W. L. Glaisher, Number-Divisor Tables. British Assoc. Math. Tables, Vol. 8, Camb. Univ. Press, 1940, p. 85.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Giovanni Resta, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972, p. 840.
Ivan Niven, The asymptotic density of sequences, Bull. Amer. Math. Soc., Vol. 57 (1951), pp. 420-434.
R. Sita Rama Chandra Rao and G. Sri Rama Chandra Murty, On a theorem of Niven, Canadian Mathematical Bulletin, Vol 22, No. 1 (1979), pp. 113-115.

Cf. A000203, A007609.
Complement of A007369. A175192(a(n)) = 1, A054973(a(n)) >= 1. - Jaroslav Krizek, Mar 01 2010
See A083531 for the gaps, i.e., first differences. - M. F. Hasler, Mar 12 2018
Subsequence of A211347.

N:= 1000: # to get all entries <= N
select(`<=`,{seq(numtheory[sigma](i),i=1..N)},N); # Robert Israel, Jun 16 2014

lim=1000; Select[Union[DivisorSigma[1,Range[lim]]], #<=lim &] (* T. D. Noe, May 06 2010 *)

list(lim)=select(n->n<=lim,Set(vector(lim\=1,n,sigma(n)))) \\ Charles R Greathouse IV, Nov 12 2013

A002191_upto(N,M=N\1+1)=Set(apply(t->min(sigma(t),M), [1..N\1-1]))[^-1] \\ Needs big stack for N >= 10^6; slower alternative: {A002191_upto(N)= my(L=List(1),s); for(n=2,N\=1,N<(s=sigma(n))||listput(L,s));Set(L)}
A2191=A002191_upto(1e4); A002191(n)={#A2191A002191_upto(n*logint(n,10)+n); A2191[n]} \\ - M. F. Hasler, Nov 22 2019

a(n)/n < log_10(n) + O(1) with O(1) <= 1 for all n. - M. F. Hasler, Nov 22 2019

A258777 Number of points of projective spaces on finite fields.

Original entry on oeis.org

1, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 17, 18, 20, 21, 24, 26, 28, 30, 31, 32, 33, 38, 40, 42, 44, 48, 50, 54, 57, 60, 62, 63, 65, 68, 72, 73, 74, 80, 82, 84, 85, 90, 91, 98, 102, 104, 108, 110, 114, 121, 122, 126, 127, 128, 129, 132, 133, 138, 140, 150, 152, 156, 158, 164, 168, 170, 174, 180, 182, 183, 192, 194, 198, 200

Offset: 1

Matthieu Pluntz, Jun 09 2015

List of integers of form (p^(k*n) - 1)/(p^k - 1) = sigma_k(p^(n-1)) = sum of d^k over all divisors d of p^(n-1), for some prime p and some positive integers k and n. The cardinality of the field is p^k and the dimension of the space is n-1.

In other words, numbers that are a repunit in at least one base that is a prime power (A246655). - Peter Munn, Oct 21 2020

			7 = (2^(1*3) - 1)/(2^1 - 1) so 7 is in the sequence. 10 = (3^(2*2) - 1)/(3^2 - 1) so 10 is in the sequence.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Pierre-Emmanuel Caprace, Pierre de la Harpe, Groups with irreducibly unfaithful subsets for unitary representations, arXiv:1807.04992 [math.GR], 2018.
Eric Weisstein's World of Mathematics, Repunit

Union of 1, A090503 and (A246655 + 1).
Subsequence of A211347.
Cf. A001231, A246655.

max = 200; Join[{1}, Select[{#, DivisorSigma[Range[Max[1, Log[#, max] // Floor]], #]}& /@ Range[2, max], PrimePowerQ[#[[1]]]&][[All, 2]] // Flatten // Union] // Select[#, # <= max&]& (* Jean-François Alcover, Jun 24 2015 after Giovanni Resta *)

list(lim)=my(v=List([1]),t); lim\=1; if(lim<2,lim=2); for(k=1,logint(lim - 1, 2), for(n=2,logint(lim*(2^k - 1) + 1, 2)\k, forprime(p=2,, t=(p^(k*n) - 1)/(p^k - 1); if(t>lim,break); listput(v,t)))); Set(v) \\ Charles R Greathouse IV, Jun 24 2015

A272883 Numbers m such that sigma_k(x) = m has no solution for any k > 0.

Original entry on oeis.org

2, 11, 16, 19, 22, 23, 25, 27, 29, 34, 35, 37, 41, 43, 45, 46, 47, 49, 51, 52, 53, 55, 58, 59, 61, 64, 66, 67, 69, 70, 71, 75, 76, 77, 79, 81, 83, 86, 87, 88, 89, 92, 94, 95, 97, 99, 100, 101, 103, 105, 106, 107, 109, 111, 113, 115, 116, 117, 118, 119, 123, 125

Offset: 1

Jaroslav Krizek, May 08 2016

nonn

Recall that sigma_k(n) = Sum_{d|n} d^k.
Sigma_0(n), the number of divisors of n, can be any positive integer and so is ignored in this sequence.
Complement of A211347.
Numbers n such that A271606(n) = 0.

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

Cf. Sequences of sigma_k(n) for k=0-24: A000005 (k=0), A000203 (k=1), A001157-A001160 (k=2-5), A013954-A013972 (k=6-24).

Cf. A211347, A271606.

[n: n in [1..2^7] | [n] notsubset Set(Sort([DivisorSigma(k,n): n in [1..2^7+1], k in [1..2^7+1] | DivisorSigma(k,n) lt 2^7+1]))];

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

A336488 Values taken by all the Jordan totient functions J_k(m) for k >= 1 and m >= 1.

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 8, 10, 12, 15, 16, 18, 20, 22, 24, 26, 28, 30, 31, 32, 36, 40, 42, 44, 46, 48, 52, 54, 56, 58, 60, 63, 64, 66, 70, 72, 78, 80, 82, 84, 88, 92, 96, 100, 102, 104, 106, 108, 110, 112, 116, 120, 124, 126, 127, 128, 130, 132, 136, 138, 140, 144, 148

Offset: 1

Amiram Eldar, Jul 23 2020

nonn

The asymptotic density of this sequence is 0 (Rao and Murty, 1979).

First differs from A221178 at n = 75, since a(75) = J_3(6) = 182 is not a term of A221178.

R. Sita Rama Chandra Rao and G. Sri Rama Chandra Murty, On a theorem of Niven, Canadian Mathematical Bulletin, Vol 22, No. 1 (1979), pp. 113-115.

A002202 is a subsequence.
Cf. A000010, A007434, A059376, A059377, A059378, A059379, A059380, A069091, A069092, A069093, A069094, A069095, A221178.
Similar sequence: A211347.

phiQ[m_] := Select[Range[m + 1, 2 m*Product[(1 - 1/(k*Log[k]))^(-1), {k, 2, DivisorSigma[0, m]}]], EulerPhi[#] == m &, 1] != {}; jor[k_, n_] := DivisorSum[n, #^k*MoebiusMu[n/#] &]; jorval[k_, mx_] := jor[k, #] & /@ Range[Floor@Surd[mx*Zeta[k], k]]; mx = 300; Select[Union @ Flatten[{Select[Range[mx], phiQ], jorval[#, mx] & /@ Range[2, Floor[Log2[mx]]]}], # <= mx &] (* using code by Jean-François Alcover at A002202 *)

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A002191 Possible values for sum of divisors of n.

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A258777 Number of points of projective spaces on finite fields.

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A272883 Numbers m such that sigma_k(x) = m has no solution for any k > 0.

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

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A336488 Values taken by all the Jordan totient functions J_k(m) for k >= 1 and m >= 1.

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