cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-6 of 6 results.

A002191 Possible values for sum of divisors of n.

Original entry on oeis.org

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

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Author

N. J. A. Sloane

Keywords

Comments

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

Examples

			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

References

  • 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).

Links

Crossrefs

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.

Programs

  • Maple
    N:= 1000: # to get all entries <= N
select(`<=`,{seq(numtheory[sigma](i),i=1..N)},N); # Robert Israel, Jun 16 2014
  • Mathematica
    lim=1000; Select[Union[DivisorSigma[1,Range[lim]]], #<=lim &] (* T. D. Noe, May 06 2010 *)
  • PARI
    list(lim)=select(n->n<=lim,Set(vector(lim\=1,n,sigma(n)))) \\ Charles R Greathouse IV, Nov 12 2013
  • PARI
    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

Formula

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

A083533 First difference sequence of A002202. Difference between consecutive possible values of phi(n), the Euler totient function A000010.

Original entry on oeis.org

1, 2, 2, 2, 2, 2, 4, 2, 2, 2, 2, 4, 2, 2, 4, 4, 2, 2, 2, 2, 4, 2, 2, 2, 2, 4, 2, 4, 2, 6, 2, 2, 2, 4, 4, 4, 4, 2, 2, 2, 2, 2, 2, 4, 4, 6, 2, 2, 2, 4, 2, 2, 4, 4, 2, 6, 4, 2, 2, 2, 2, 4, 4, 2, 2, 4, 6, 2, 4, 2, 2, 4, 4, 2, 2, 4, 4, 2, 2, 2, 2, 4, 6, 2, 10, 2, 4, 4, 2, 2, 4, 2, 2, 4, 4, 2, 6, 4, 2, 2, 4, 6, 4, 2, 4
Offset: 1

Views

Author

Labos Elemer, May 20 2003

Keywords

Links

Crossrefs

Cf. A000010, A002202, A005277, A083531, A083532, A083534, A083535, A083536.

Programs

  • Haskell
    a083533 n = a083533_list !! (n-1)
a083533_list = zipWith (-) (tail a002202_list) a002202_list
-- Reinhard Zumkeller, Nov 26 2015
  • Mathematica
    t=Table[EulerPhi[w], {w, 1, 25000}]; u=Union[%]; Delete[u-RotateRight[u], 1]
  • PARI
    lista(lim) = {my(k1 = 1, k2 = 1); while(k1 < lim, until(istotient(k2), k2++); print1(k2 - k1, ", "); k1 = k2);} \\ Amiram Eldar, Nov 16 2024

Formula

a(n) = A002202(n+1) - A002202(n).

A083534 First difference sequence of A007617. Difference between consecutive values not being in the range of phi (A000010).

Original entry on oeis.org

2, 2, 2, 2, 2, 1, 1, 2, 2, 2, 2, 2, 1, 1, 2, 2, 2, 1, 1, 2, 1, 1, 2, 2, 2, 2, 2, 1, 1, 2, 2, 2, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 1, 1, 2, 2, 2, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 2, 2, 2, 2, 2, 2, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 2, 2, 2, 2, 1, 1, 2, 2, 2, 1, 1, 2, 1, 1, 2, 2, 1, 1, 1, 1, 2, 1, 1, 2, 2, 2, 2
Offset: 1

Views

Author

Labos Elemer, May 20 2003

Keywords

Comments

a(n) is either 2 or 1 since odd numbers are in A007619.
If a(n) = 1 then A007619(n+1) is an even number not in the range of phi.

Examples

			{11,13,14,15,17} are not in the range of phi and the corresponding differences are {2,1,1,2}.

Links

Crossrefs

Cf. A000010, A002202, A005277, A007617, A007619, A083531, A083532, A083533, A083535, A083536.

Programs

  • Haskell
    a083534 n = a083534_list !! (n-1)
a083534_list = zipWith (-) (tail a007617_list) a007617_list
-- Reinhard Zumkeller, Nov 26 2015
  • Mathematica
    t0[x_] := Table[j, {j, 1, x}]; t=Table[EulerPhi[w], {w, 1, 10000}]; u=Union[%]; c=Complement[t0[10000], u]; Delete[c-RotateRight[c], 1]
  • PARI
    list(lim) = {my(k1 = 3, k2 = 3); while(k1 < lim, until(!istotient(k2), k2++); print1(k2 - k1, ", "); k1 = k2); } \\ Amiram Eldar, Feb 22 2025

Formula

a(n) = A007617(n+1) - A007617(n).

A083532 First difference sequence of A007369. Differences between impossible values for sum of divisors of n.

Original entry on oeis.org

3, 4, 1, 1, 5, 1, 2, 2, 1, 1, 2, 1, 1, 2, 4, 1, 1, 2, 4, 2, 2, 1, 1, 2, 1, 1, 1, 1, 2, 3, 1, 2, 3, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 1, 3, 2, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 2, 2, 1, 1, 1, 1, 3, 1, 2, 4, 1, 1, 3, 1, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 1, 2, 2, 1, 1, 2, 2, 2, 2, 2, 1, 1, 2, 1, 2, 1
Offset: 1

Views

Author

Labos Elemer, May 20 2003

Keywords

Examples

			29 and 33 are the 15th and 16th nonsense values for sigma(x), since there exist no numbers n of which they are sums of divisors, while {30,31,32} equal sigma(x); e.g., for x = 29, 16, 31, respectively, thus 33 - 29 = 4 = a(15) = A007369(16) - A007369(15).

Links

Crossrefs

Cf. A002191, A007369, A083235, A083531, A083533, A083534, A083536.

Programs

  • Mathematica
    t0[x_] := Table[j, {j, 1, x}]; t=Table[DivisorSigma[1, w], {w, 1, 25000}]; u=Union[%]; c=Complement[t0[25000], u]; Delete[c-RotateRight[c], 1]

Formula

a(n) = A007369(n+1) - A007369(n).

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

Views

Author

Max Alekseyev and Jud McCranie, Aug 08 2005

Keywords

Links

Crossrefs

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

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

Views

Author

Hugo Pfoertner, Mar 12 2018

Keywords

Comments

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.

Examples

			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).

Links

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

Crossrefs

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

Extensions

a(5) from Emmanuel Vantieghem
a(6)-a(11) from Giovanni Resta, Mar 13 2018
Showing 1-6 of 6 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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