A000203 -id:A000203 - cp's OEIS frontend

A340793 Sequence whose partial sums give A000203.

1, 2, 1, 3, -1, 6, -4, 7, -2, 5, -6, 16, -14, 10, 0, 7, -13, 21, -19, 22, -10, 4, -12, 36, -29, 11, -2, 16, -26, 42, -40, 31, -15, 6, -6, 43, -53, 22, -4, 34, -48, 54, -52, 40, -6, -6, -24, 76, -67, 36, -21, 26, -44, 66, -48, 48, -40, 10, -30, 108, -106, 34, 8

Offset: 1

Omar E. Pol, Jan 21 2021

Essentially a duplicate of A053222.
Convolved with the nonzero terms of A000217 gives A175254, the volume of the stepped pyramid described in A245092.
Convolved with the nonzero terms of A046092 gives A244050, the volume of the stepped pyramid described in A244050.
Convolved with A000027 gives A024916.
Convolved with A000041 gives A138879.
Convolved with A000070 gives the nonzero terms of A066186.
Convolved with the nonzero terms of A002088 gives A086733.
Convolved with A014153 gives A182738.
Convolved with A024916 gives A000385.
Convolved with A036469 gives the nonzero terms of A277029.
Convolved with A091360 gives A276432.
Convolved with A143128 gives the nonzero terms of A000441.
For the correspondence between divisors and partitions see A336811.

1 together with A053222.
Cf. A000203 (partial sums).
Cf. A000027, A000041, A000070, A000217, A000385, A000441, A002088, A002961, A014153, A024916, A036469, A046092, A053223, A053224, A053225, A053226, A053227, A066186, A086733, A091360, A138879, A143128, A175254, A182738, A237593, A244050, A245092, A276432, A277029, A336811.

a:= n-> (s-> s(n)-s(n-1))(numtheory[sigma]):
seq(a(n), n=1..77);  # Alois P. Heinz, Jan 21 2021

Join[{1}, Differences @ Table[DivisorSigma[1, n], {n, 1, 100}]] (* Amiram Eldar, Jan 21 2021 *)

a(n) = if (n==1, 1, sigma(n)-sigma(n-1)); \\ Michel Marcus, Jan 22 2021

a(n) = A053222(n-1) for n>1. - Michel Marcus, Jan 22 2021

A362451 Gilbreath transform of {sigma(i), i >= 1} (cf. A000203).

Original entry on oeis.org

1, 2, 1, 1, 1, 2, 1, 0, 0, 1, 0, 4, 0, 3, 0, 2, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 1, 1, 0, 1, 1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 0, 1, 1, 1, 68, 0, 14, 0, 7, 0, 2, 0, 21, 1, 8, 1, 9, 1, 0, 1, 18, 0, 7, 0, 2, 0, 1, 0, 13, 1, 1, 1, 2, 1, 1

Offset: 1

N. J. A. Sloane, May 03 2023

nonn

Given a sequence {u(i), i >= o} with offset o, its absolute difference sequence is the sequence {v(i) = |u(i+1)-u(i)|, i >= o}.
The Gilbreath transform of a sequence s = {s(i), i >= o} is constructed as follows.
Form an array A in which the initial row is s and each subsequence row is the absolute difference sequence of the previous row. The sequence of leading terms of the rows of A is the Gilbreath transform of s.
If "absolute difference sequence" is changed to the familiar "first differences", instead of the Gilbreath transform we get the usual inverse binomial transform.
It appears that the terms are mostly 0's and 1's, with occasional eruptions of "geysers". See A362456, A362457.

			We give two examples. (1) For the Gilbreath transform of the sequence of primes (cf. A000040), the array A is given in A036262. The Gilbreath transform begins {2, 1, 1, 1, 1, ...}, and the famous Gilbreath conjecture is that every term after the initial 2 is equal to 1.
(2) For the Gilbreath transform of {tau(i), i >= 1} (cf. A000005), the array A is given in A362450, and the Gilbreath transform is given in A361897. The authors of the latter sequence conjecture that its terms are just 0's and 1's.
See A362452 for a further example.

N. J. A. Sloane, Table of n, a(n) for n = 1..60000
N. J. A. Sloane, Maple code for Gilbreath transform and related arrays
N. J. A. Sloane, Transforms
N. J. A. Sloane, New Gilbreath Conjectures, Sum and Erase, Dissecting Polygons, and Other New Sequences, Doron Zeilberger's Exper. Math. Seminar, Rutgers, Sep 14 2023: Video, Slides, Updates. (Mentions this sequence.)
Paolo Xausa, Table of n, a(n) for n = 1..1000000
Paolo Xausa, Logarithmic scatterplot for n = 1..1100000
Index entries for sequences related to Gilbreath conjecture and transform

Cf. A000005, A000040, A000203, A036262, A361897, A362450, A362452, A362456, A362457, A362464.

# To get M terms of the Gilbreath transform of s, assuming offset is 1:
GT := proc(s,M) local G,u,i;
u := [seq(s[i],i=1..M)];
G:=[s[1]];
for i from 1 to M-1 do
u:=[seq(abs(u[i+1]-u[i]),i=1..nops(u)-1)];
G:=[op(G),u[1]]; od:
G;
end;
# For the present sequence:
GT(numtheory[sigma],150);
# See link for a more comprehensive Maple program

A362451[nmax_]:=Module[{d=DivisorSigma[1,Range[nmax]]},Join[{1},Table[First[d=Abs[Differences[d]]],nmax-1]]];A362451[200] (* Paolo Xausa, May 07 2023 *)

lista(nn) = my(v=apply(sigma, [1..nn]), list = List(), nb=nn); listput(list, v[1]); for (n=2, nn, nb--; my(w = vector(nb, k, abs(v[k+1]-v[k]))); listput(list, w[1]); v = w; ); Vec(list);
lista(200) \\ (based on PARI program in A361897)

More than the usual number of terms are displayed in order to go out beyond the long initial 0,1 subsequence.

A019294 Number (> 0) of iterations of sigma (A000203) required to reach a multiple of n when starting with n.

Original entry on oeis.org

1, 2, 4, 2, 5, 1, 5, 2, 7, 4, 15, 3, 13, 3, 2, 2, 13, 4, 12, 5, 2, 13, 16, 2, 17, 4, 9, 1, 78, 7, 10, 4, 17, 11, 6, 5, 28, 22, 4, 7, 39, 2, 16, 16, 16, 10, 32, 5, 13, 17, 9, 3, 58, 11, 19, 5, 13, 67, 97, 2, 23, 5, 16, 2, 4, 8, 101, 21, 19, 11, 50, 4, 20, 20, 23, 14, 21, 10, 36, 5, 15

Offset: 1

N. J. A. Sloane

Let sigma^m(n) be result of applying sum-of-divisors function m times to n; sequence gives m(n) = min m such that n divides sigma^m(n).
Perfect numbers require one iteration.
It is conjectured that the sequence is finite for all n.
See also the Cohen-te Riele links under A019276.
a(A111227(n)) > A111227(n). - Reinhard Zumkeller, Aug 02 2012
a(659) > 870. - Michel Marcus, Jan 04 2017

			If n = 9 the iteration sequence is s(9) = {9, 13, 14, 24, 60, 168, 480, 1512, 4800, 15748, 28672} and Mod[s(9), 9] = {0, 4, 5, 6, 6, 6, 3, 0, 3, 7, 7}. The first iterate which is a multiple of 9 is the 7th = 1512, so a(9) = 7. For n = 67, the 101st iterate is the first, so a(67) = 101. Usually several iterates are divisible by the initial value. E.g., if n = 6, then 91 of the first 100 iterates are divisible by 6.
A difficult term to compute: a(461) = 557. - _Don Reble_, Jun 23 2005

Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004. See Section B41, p. 147.

Don Reble, Table of n, a(n) for n = 1..1578 (Terms a(1..400) from T. D. Noe, Nov 2007; a(401..659) from Michel Marcus, Jan 02 2017), Feb 20 2022.
G. L. Cohen and H. J. J. te Riele, Iterating the sum-of-divisors function, Experimental Mathematics, 5 (1996), pp. 93-100. See Table 2, p. 95.
Paul Erdős, Andrew Granville, Carl Pomerance and Claudia Spiro, On the normal behavior of the iterates of some arithmetic functions, Analytic number theory, Birkhäuser Boston, 1990, pp. 165-204.
Paul Erdős, Andrew Granville, Carl Pomerance and Claudia Spiro, On the normal behavior of the iterates of some arithmetic functions, Analytic number theory, Birkhäuser Boston, 1990, pp. 165-204. [Annotated copy with A-numbers]
Carl Pomerance, On the composition of the functions sigma and phi, Colloq. Math., 59 (1989), 11-15.
Wikipedia, Iterated function, as of Jan 02 2020.
Zeraoulia Rafik, On congruence of the iterated form sigma^k(m) = 0 mod m, arXiv:2102.09941 [math.NT], 2021.

Cf. A019295 (ratio sigma^m(n)/n), A019276 (indices of records), A019277 (records), A000396.

a019294 n = snd $ until ((== 0) . (`mod` n) . fst)
                        (\(x, i) -> (a000203 x, i + 1)) (a000203 n, 1)
-- Reinhard Zumkeller, Aug 02 2012

a:=[]; f:=func; for n in [1..81] do k:=n; s:=1; while f(k) mod n ne 0 do k:=f(k); s:=s+1; end while; Append(~a,s); end for; a; // Marius A. Burtea, Jan 11 2020

A019294 := proc(n)
    local a,nitr ;
    a := 1 ;
    nitr := numtheory[sigma](n);
    while modp(nitr,n) <> 0 do
        nitr := numtheory[sigma](nitr) ;
        a := a+1 ;
    end do:
    return a;
end proc: # R. J. Mathar, Aug 22 2016

f[n_, m_] := Block[{d = DivisorSigma[1, n]}, If[ Mod[d, m] == 0, 0, d]]; Table[ Length[ NestWhileList[ f[ #, n] &, n, # != 0 &]] - 1, {n, 84}] (* Robert G. Wilson v, Jun 24 2005 *)
Table[Length[NestWhileList[DivisorSigma[1,#]&,DivisorSigma[1,n], !Divisible[ #,n]&]],{n,90}] (* Harvey P. Dale, Mar 04 2015 *)

a(n)=if(n<0,0,c=1; s=n; while(sigma(s)%n>0,s=sigma(s); c++); c)

apply( A019294(n,s=n)=for(k=1,oo,(s=sigma(s))%n||return(k)), [1..99]) \\ M. F. Hasler, Jan 07 2020

Conjecture: lim_{n -> oo} log(Sum_{k=1..n} a(k))/log(n) = C = 1.6... - Benoit Cloitre, Aug 24 2002
From Michel Marcus, Jan 02 2017: (Start)
a(n) = 1 for n in A007691.
a(n) = 2 for n in A019278 unless it belongs to A007691.
a(n) = 3 for n in A019292 unless it belongs to A007691 or A019278. (End)

Additional comments from Labos Elemer, Jun 20 2001

Edited by M. F. Hasler, Jan 07 2020

A051281 Sum of divisors of n, sigma(n) (A000203), is a power of number of divisors of n, d(n) (A000005).

Original entry on oeis.org

1, 3, 7, 31, 127, 217, 889, 2667, 3937, 8191, 27559, 57337, 131071, 172011, 253921, 524287, 917497, 1040257, 1777447, 3670009, 4063201, 11010027, 12189603, 16252897, 16646017, 66584449, 113770279, 116522119, 225735769, 677207307, 1073602561, 2147483647, 3612185689, 4294434817, 7515217927

Offset: 1

Michel ten Voorde

All Mersenne primes (A000668) are terms.

Subsequence of A046528 (product of distinct Mersenne primes). - Michel Marcus, Feb 15 2020

			d(217) = 4; sigma(217) = 256 = 4^4.

David A. Corneth, Table of n, a(n) for n = 1..2000

Cf. A000005, A000203, A046528, A286628, A289276.

spdQ[n_]:=Module[{sd=DivisorSigma[1,n],nd=DivisorSigma[0,n]},sd == nd^IntegerExponent[sd,nd]]; Join[{1},Select[Range[2,226000000],spdQ]] (* Harvey P. Dale, May 02 2012 *)

is(n)=my(t,e=ispower(sigma(n),,&t)); if(!e,return(n==1),nd); nd=numdiv(n); fordiv(e,d,if(t^d==nd,return(1)));0 \\ Charles R Greathouse IV, Feb 19 2013

isA051281(n) = { if(n==1, return(1)); my(sig = sigma(n), ndiv = numdiv(n), v = valuation(sig, ndiv)); (ndiv^v == sig); } \\ Antti Karttunen, Jun 30 2017

More terms from Jud McCranie
a(30)-a(32) from Donovan Johnson, Oct 03 2012
a(33)-a(35) from Michel Marcus, Feb 14 2020

A324653 a(n) = A000203(A276086(n)).

Original entry on oeis.org

1, 3, 4, 12, 13, 39, 6, 18, 24, 72, 78, 234, 31, 93, 124, 372, 403, 1209, 156, 468, 624, 1872, 2028, 6084, 781, 2343, 3124, 9372, 10153, 30459, 8, 24, 32, 96, 104, 312, 48, 144, 192, 576, 624, 1872, 248, 744, 992, 2976, 3224, 9672, 1248, 3744, 4992, 14976, 16224, 48672, 6248, 18744, 24992, 74976, 81224, 243672, 57, 171, 228

Offset: 0

Antti Karttunen, Mar 10 2019

nonn

Cf. A000203, A002110, A276086.
Cf. A267263, A276150, A324650, A324655 for omega, bigomega, phi and tau analogs, and also A324654.
Cf. also A324054.

A276086(n) = { my(i=0,m=1,pr=1,nextpr); while((n>0),i=i+1; nextpr = prime(i)*pr; if((n%nextpr),m*=(prime(i)^((n%nextpr)/pr));n-=(n%nextpr));pr=nextpr); m; };
A324653(n) = sigma(A276086(n));

a(n) = A000203(A276086(n)).

For n >= 1, a(A002110(n-1)) = 1+A000040(n).

A229276 Composite squarefree numbers n such that p-tau(n) divides n+sigma(n), where p are the prime factors of n, tau(n) = A000005(n) and sigma(n) = A000203(n).

Original entry on oeis.org

6, 10, 15, 66, 145, 231, 435, 1221, 11571, 99093, 105502, 292434, 449854, 585429, 643858, 968014, 1372494, 1787091, 1939434, 4659114, 5524014, 5654334, 6250371, 6974007, 19495374, 19821714, 28488039, 34701369, 46183893, 81133734, 213352233, 230140869

Offset: 1

Paolo P. Lava, Sep 19 2013

nonn

Subsequence of A120944.

			Prime factors of 435 are 3, 5, 29 and sigma(435) = 720, tau(435) = 8.
435 + 720 = 1155 and 1155 / (3 - 8) = -231, 1155 / (5 - 8) = -385, 1155 / (29 - 8) = 55.

Kevin P. Thompson, Table of n, a(n) for n = 1..62

Cf. A000005, A000203, A228299-A228302, A229273-A229275.

with (numtheory); P:=proc(q) local a, b, c, i, ok, p, n;
for n from 2 to q do  if not isprime(n) then a:=ifactors(n)[2]; ok:=1;
for i from 1 to nops(a) do if a[i][2]>1 or a[i][1]=tau(n) then ok:=0; break;
else if not type((n+sigma(n))/(a[i][1]-tau(n)), integer) then ok:=0; break; fi; fi; od; if ok=1 then print(n); fi; fi; od; end: P(2*10^6);

a(21)-a(33) from Giovanni Resta, Sep 20 2013

First term deleted by Paolo P. Lava, Sep 23 2013

A234516 Composite numbers n sorted by decreasing values of alpha(n) = log_n(sigma(n)) - log_n(n+1), where sigma(n) = A000203(n) = the sum of divisors of n.

Original entry on oeis.org

12, 6, 24, 36, 18, 30, 60, 8, 4, 48, 20, 72, 120, 84, 16, 42, 10, 40, 180, 90, 96, 144, 240, 168, 108, 360, 28, 54, 420, 252, 132, 80, 216, 210, 32, 126, 300, 336, 480, 56, 192, 288, 720, 840, 66, 504, 156, 540, 150, 264, 14, 600, 140, 270, 1260, 432, 78, 1080

Offset: 1

Jaroslav Krizek, Jan 03 2014

nonn

The number alpha(n)  = log_n(sigma(n)) - log_n(n+1) = log_n[sigma(n) / (n+1)] is called the alpha-deviation from primality of number n; alpha(p) = 0 for p = prime. See A234520 for definition of beta(n).
Lim_n->infinity alpha(n) = 0.
Conjecture: Every composite number n has a unique value of alpha(n).
Conjecture: sequence A234517 is not the sequence of numbers from a(n) such that a(n) > a(k) for all k < n.

			For the number 12; alpha(12) = log_12(sigma(12)) - log_12(12+1) = log_12(28) - log_12(13) = 0.308766187… = A234518 (maximal value of function alpha(n)).

Jaroslav Krizek, Table of n, a(n) for n = 1..200

Cf. A234515, A234517, A234518, A234519, A234520, A234521, A234522, A234523, A234524.

lista(nn) = {v = vector(nn, n, if ((n==1) || isprime(n), 0, log(sigma(n)/(n+1))/log(n))); v = vecsort(v,,5); for (i=1, 80, print1(v[i], ", "));} \\ Michel Marcus, Dec 10 2014

A234520 Composite numbers n sorted by decreasing values of beta(n) = sigma(n)^(1/n) - (n+1)^(1/n), where sigma(n) = A000203(n) = the sum of divisors of n.

Original entry on oeis.org

4, 6, 8, 12, 10, 18, 16, 24, 14, 20, 9, 15, 30, 36, 28, 22, 32, 40, 48, 42, 21, 26, 60, 54, 44, 27, 72, 56, 34, 50, 45, 52, 38, 66, 84, 33, 64, 90, 80, 70, 96, 78, 46, 39, 120, 68, 108, 35, 88, 76, 63, 25, 100, 58, 102, 126, 144, 112, 132, 62, 104, 75, 51, 92

Offset: 1

Jaroslav Krizek, Jan 14 2014

nonn

The number beta(n)  = sigma(n)^(1/n) - (n+1)^(1/n) is called the beta-deviation from primality of the number n; beta(p) = 0 for p = prime. See A234516 for definition of alpha(n).
For number 4; beta(4) = sigma(4)^(1/4) - (4+1)^(1/4), = 7^(1/4) - 5^(1/4) = 0,131227780… = A234522 (maximal value of function beta(n)).
Lim_n->infinity beta(n) = 0.
Conjecture: Every composite number n has a unique value of number beta(n).
See A234523 - sequence of numbers a(n) such that a(n) > a(k) for all k < n.

Jaroslav Krizek, Table of n, a(n) for n = 1..1000

Cf. A234515, A234516, A234517, A234518, A234519, A234521, A234522, A234523, A234524.

A248150 Numbers whose sum of divisors (A000203) is divisible by 4.

Original entry on oeis.org

3, 6, 7, 11, 12, 14, 15, 19, 21, 22, 23, 24, 27, 28, 30, 31, 33, 35, 38, 39, 42, 43, 44, 46, 47, 48, 51, 54, 55, 56, 57, 59, 60, 62, 63, 65, 66, 67, 69, 70, 71, 75, 76, 77, 78, 79, 83, 84, 85, 86, 87, 88, 91, 92, 93, 94, 95, 96, 99, 102, 103, 105, 107, 108, 110, 111, 112, 114, 115, 118

Offset: 1

M. F. Hasler, Oct 02 2014

nonn

A subsequence of A028983 (even sum of divisors) which contains all numbers but the squares and twice the squares, so no term of this sequence is of that form, either.

Any number having at least two odd prime factors to an odd power is in this sequence, therefore it has asymptotic density 1. - M. F. Hasler, Apr 26 2017

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

Cf. A000203, A028982, A028983, A248151.

First differs from A022544 by including 65.

Select[Range[200],Divisible[DivisorSigma[1,#],4]&] (* Harvey P. Dale, Feb 20 2015 *)

PARI
```
for(n=1,999,sigma(n)%4||print1(n","))
```

a(n) ~ n. - Charles R Greathouse IV, Sep 01 2015

A365398 Length of the longest subsequence of 1, ..., n on which sigma, the sum of the divisors of n (A000203), is nondecreasing.

Original entry on oeis.org

1, 2, 3, 4, 4, 5, 5, 6, 6, 7, 7, 8, 8, 8, 9, 10, 10, 11, 11, 12, 12, 12, 12, 13, 13, 13, 13, 14, 14, 15, 15, 15, 15, 15, 15, 16, 16, 16, 16, 17, 17, 18, 18, 18, 18, 18, 18, 19, 19, 19, 19, 19, 19, 20, 20, 21, 21, 21, 21, 22, 22, 22, 23, 24, 24, 25, 25, 25

Offset: 1

Peter Luschny, Sep 08 2023

nonn

The sequence was inspired by A365339. In particular, note remark (4.4) by Terence Tao in the linked paper.

Chai Wah Wu, Table of n, a(n) for n = 1..10000
Plot2, A365398 vs A365339.
Terence Tao, Monotone non-decreasing sequences of the Euler totient function, arXiv:2309.02325 [math.NT], 2023.

Cf. A000203, A000720, A365339, A365399, A365397.

Cf. A061069.

from bisect import bisect
from sympy import divisor_sigma
def A365398(n):
    plist, qlist, c = tuple(divisor_sigma(i) for i in range(1,n+1)), [0]*(n+1), 0
    for i in range(n):
        qlist[a:=bisect(qlist,plist[i],lo=1,hi=c+1,key=lambda x:plist[x])]=i
        c = max(c,a)
    return c # Chai Wah Wu, Sep 08 2023

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

a(n) >= A000720(n)+1 since A000203(p) = p+1 for p prime. - Chai Wah Wu, Sep 08 2023

cp's OEIS Frontend

A340793 Sequence whose partial sums give A000203.

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A362451 Gilbreath transform of {sigma(i), i >= 1} (cf. A000203).

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A019294 Number (> 0) of iterations of sigma (A000203) required to reach a multiple of n when starting with n.

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A051281 Sum of divisors of n, sigma(n) (A000203), is a power of number of divisors of n, d(n) (A000005).

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A324653 a(n) = A000203(A276086(n)).

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A229276 Composite squarefree numbers n such that p-tau(n) divides n+sigma(n), where p are the prime factors of n, tau(n) = A000005(n) and sigma(n) = A000203(n).

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A234516 Composite numbers n sorted by decreasing values of alpha(n) = log_n(sigma(n)) - log_n(n+1), where sigma(n) = A000203(n) = the sum of divisors of n.

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