A053222 -id:A053222 - cp's OEIS frontend

A064431 Numbers n such that |A053222(n)| = tau(n).

6, 21, 38, 44, 45, 81, 164, 261, 278, 4388, 9009, 114302, 41623514, 440169957, 19033618221, 175036435525, 341846222877

Offset: 1

Jason Earls, Sep 30 2001

t(n) = abs(sigma(n+1)-sigma(n)) for(n=1,10^6, if(t(n)==numdiv(n),print(n)))

a(13)-a(17) from Giovanni Resta, Feb 08 2014

A002961 Numbers k such that k and k+1 have same sum of divisors.

Original entry on oeis.org

14, 206, 957, 1334, 1364, 1634, 2685, 2974, 4364, 14841, 18873, 19358, 20145, 24957, 33998, 36566, 42818, 56564, 64665, 74918, 79826, 79833, 84134, 92685, 109214, 111506, 116937, 122073, 138237, 147454, 161001, 162602, 166934

Offset: 1

N. J. A. Sloane, Mira Bernstein, Robert G. Wilson v

For the values of n < 2*10^10 in this sequence, sigma(n)/n is between 1.5 and 2.25. - T. D. Noe, Sep 17 2007
Whether this sequence is infinite is an unsolved problem, as noted in many of the references and links. - Franklin T. Adams-Watters, Jan 25 2010
144806446575 is the first term for which sigma(n)/n > 2.25. All n < 10^12 have sigma(n)/n > 3/2. - T. D. Noe, Feb 18 2010
A053222(a(n)) = 0. - Reinhard Zumkeller, Dec 28 2011
Numbers n such that n + 1 = antisigma(n+1) - antisigma(n), where antisigma(n) = A024816(n) = the sum of the non-divisors of n that are between 1 and n. Example for n = 14: 15 = antisigma(15) - antisigma(14) = 96 - 81. - Jaroslav Krizek, Nov 10 2013
Up to 10^13, the value of the sigma(n)/n varies between 1417728000/945151999 (attained for n = 2835455997) and 2913242112/1263730145 (for n = 5174974943775). - Giovanni Resta, Feb 26 2014
Also numbers n such that A242962(n) = A242962(n+1), with A242962(n) = T(n) mod antisigma(n), where T(n) = A000217(n) is the n-th triangular number and antisigma(n) = A024816(n) is the sum of numbers less than n which do not divide n. - Jaroslav Krizek, May 29 2014
Guy and Shanks construct 5559060136088313 as a term of this sequence. - Michel Marcus, Dec 29 2014
Note that in all cases, n and n+1 are composite. - Zak Seidov, May 03 2016

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 840.
R. K. Guy, Unsolved Problems in Theory of Numbers, Sect. B13.
W. Sierpiński, A Selection of Problems in the Theory of Numbers. Macmillan, NY, 1964, p. 110.
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..10135 (terms < 10^13; first 4804 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 [alternative scanned copy].
Jonathan Bayless and Paul Kinlaw, On repeated values of sigma and multiperfect numbers, Journal of Combinatorics and Number Theory, Vol. 7, No. 3 (2015), pp. 177-189.
Lourdes Benito, Solutions of the problem of Erdős-Sierpiński: sigma(n)=sigma(n+1), arXiv:0707.2190 [math.NT], 2007.
Richard Guy and Daniel Shanks, A Constructed Solution of sigma(n) = sigma(n+1), The Fibonacci Quarterly, Volume 12, Number 3, October 1974, 299.
A. Makowski, On Some Equations Involving Functions phi(n) and sigma(n), The American Mathematical Monthly, Vol. 67, No. 7 (Aug. - Sep., 1960), pp. 668-670.
N. J. A. Sloane & D. Singmaster, Correspondence 1972.
Andreas Weingartner, On the Solutions of sigma(n) = sigma(n+k), Journal of Integer Sequences, Vol. 14 (2011), #11.5.5.

Cf. A000203 (sigma function), A053215, A053249, A054004
Cf. A007373, A015861, A015863, A015865, A015866, A015867, A015876, A015877, A015880, A015881, A015882, A015883, A181647. - Reinhard Zumkeller, Nov 03 2010
Cf. A238380.

import Data.List (elemIndices)
a002961 n = a002961_list !! (n-1)
a002961_list = map (+ 1) $ elemIndices 0 a053222_list
-- Reinhard Zumkeller, Dec 28 2011

Flatten[Position[Partition[DivisorSigma[1,Range[170000]],2,1],{x_,x_}]] (* Harvey P. Dale, Aug 08 2011 *)
SequencePosition[DivisorSigma[1,Range[200000]],{x_,x_}][[All,1]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Mar 06 2018 *)

t1=sigma(1);for(n=2,1e6,t2=sigma(n);if(t2==t1,print1(n-1", "));t1=t2) \\ Charles R Greathouse IV, Jul 15 2011

Sum_{n>=1} 1/a(n) is in the interval (0.080958, 610837) (Bayless and Kinlaw, 2015). - Amiram Eldar, Oct 15 2020

More terms from Jud McCranie, Oct 15 1997

A053224 Numbers k for which sigma(k) < sigma(k+1).

Original entry on oeis.org

1, 2, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 47, 49, 51, 53, 55, 57, 59, 61, 62, 63, 65, 67, 69, 71, 73, 74, 75, 77, 79, 81, 83, 85, 87, 89, 91, 93, 95, 97, 99, 101, 103, 107, 109, 111, 113, 115, 119, 121, 123, 125, 127, 129, 131

Offset: 1

Asher Auel, Jan 06 2000

nonn

The asymptotic density of this sequence is 1/2 (Erdős, 1936). - Amiram Eldar, Mar 19 2021

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Paul Erdős, On a problem of Chowla and some related problems, Mathematical Proceedings of the Cambridge Philosophical Society, Vol. 32, No. 4 (1936), pp. 530-540; alternative link.

Cf. A000203, A053222, A053226, A053230 (first differences).

import Data.List (elemIndices)
a053224 n = a053224_list !! (n-1)
a053224_list = map (+ 1) $ elemIndices True $
   zipWith (<) a000203_list $ tail a000203_list
-- Reinhard Zumkeller, May 07 2012

with(numtheory): seq( `if`(sigma(i) < sigma(i+1),i,[][]), i=1..134);

Select[Range[150], DivisorSigma[1, #] < DivisorSigma[1, # + 1] &] (* Carl Najafi, Aug 16 2011 *)

is(n)=sigma(n)Charles R Greathouse IV, Mar 09 2014

A053226 Numbers k for which sigma(k) > sigma(k+1).

Original entry on oeis.org

4, 6, 8, 10, 12, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 45, 46, 48, 50, 52, 54, 56, 58, 60, 64, 66, 68, 70, 72, 76, 78, 80, 82, 84, 86, 88, 90, 92, 94, 96, 98, 100, 102, 104, 105, 106, 108, 110, 112, 114, 116, 117, 118, 120, 122, 124, 126, 128, 130

Offset: 1

Asher Auel, Jan 06 2000

nonn

The asymptotic density of this sequence is 1/2 (Erdős, 1936). - Amiram Eldar, Mar 19 2021

T. D. Noe, Table of n, a(n) for n = 1..1000
Paul Erdős, On a problem of Chowla and some related problems, Mathematical Proceedings of the Cambridge Philosophical Society, Vol. 32, No. 4 (1936), pp. 530-540; alternative link.

Cf. A000203, A053222, A053224, A053238.

import Data.List (findIndices)
a053226 n = a053226_list !! (n-1)
a053226_list = map (+ 1) $ findIndices (< 0) a053222_list
-- Reinhard Zumkeller, Oct 16 2011

with(numtheory): seq( `if`((sigma(i) > sigma(i+1)),i,print( )), i=1..139);

Select[Range[150], DivisorSigma[1, #] > DivisorSigma[1, # + 1] &] (* Carl Najafi, Aug 16 2011 *)

is(n)=sigma(n)>sigma(n+1) \\ Charles R Greathouse IV, Mar 09 2014

A340793 Sequence whose partial sums give A000203.

Original entry on oeis.org

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

A053223 Second differences of sigma(n).

Original entry on oeis.org

-1, 2, -4, 7, -10, 11, -9, 7, -11, 22, -30, 24, -10, 7, -20, 34, -40, 41, -32, 14, -16, 48, -65, 40, -13, 18, -42, 68, -82, 71, -46, 21, -12, 49, -96, 75, -26, 38, -82, 102, -106, 92, -46, 0, -18, 100, -143, 103, -57, 47, -70, 110, -114, 96, -88, 50, -40, 138, -214, 140, -26, 15, -66

Offset: 1

Asher Auel, Jan 06 2000

sign

T. D. Noe, Table of n, a(n) for n = 1..10000

Cf. A000203, A053222, A053228, A053229, A053247, A053248.

Indices of 0: A099632 - 1.

[DivisorSigma(1, n+2) - 2*DivisorSigma(1,n+1) + DivisorSigma(1, n): n in [1..100]]; // G. C. Greubel, Sep 03 2018

Table[DivisorSigma[1,n+2] - 2*DivisorSigma[1,n+1] + DivisorSigma[1,n], {n,1,100}] (* G. C. Greubel, Sep 03 2018 *)

a(n)=sigma(n+2)-2*sigma(n+1)+sigma(n) \\ Charles R Greathouse IV, Mar 09 2014

G.f.: -1 - 1/x + (1 - x)^2*Sum_{k>=1} k*x^(k-2)/(1 - x^k). - Ilya Gutkovskiy, Jan 29 2017

A053227 First differences of sigma(n) that are negative.

Original entry on oeis.org

1, 4, 2, 6, 14, 13, 19, 10, 12, 29, 2, 26, 40, 15, 6, 53, 4, 48, 52, 6, 6, 24, 67, 21, 44, 48, 40, 30, 106, 43, 76, 30, 72, 121, 44, 88, 65, 42, 116, 12, 90, 122, 40, 24, 154, 15, 115, 112, 18, 30, 54, 170, 64, 134, 96, 28, 2, 36, 227, 18, 68, 184, 79, 120, 176, 132, 148

Offset: 1

Asher Auel, Jan 06 2000

nonn

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A000203, A053222, A053225.

with(numtheory): seq( `if`((sigma(i) > sigma(i+1)),(sigma(i) - sigma(i+1)),print( )), i=1..200);

Abs[Select[Differences[DivisorSigma[1,Range[200]]],#<0&]] (* Harvey P. Dale, Apr 17 2022 *)

a(n) = A053222(A053226(n))

A053225 First differences of sigma(n) that are positive.

Original entry on oeis.org

2, 1, 3, 6, 7, 5, 16, 10, 7, 21, 22, 4, 36, 11, 16, 42, 31, 6, 43, 22, 34, 54, 40, 76, 36, 26, 66, 48, 10, 108, 34, 8, 23, 60, 58, 48, 123, 40, 10, 16, 72, 106, 5, 140, 24, 60, 144, 56, 16, 132, 73, 61, 114, 106, 172, 106, 96, 126, 66, 216, 53, 56, 156, 127, 76, 204, 44, 36

Offset: 1

Asher Auel, Jan 06 2000

nonn

Cf. A000203, A053222, A053227.

with (numtheory): seq( `if`((sigma(i) < sigma(i+1)),(sigma(i+1)-sigma(i)),print( )), i=1..139);

a(n) = A053222(A053224(n))

A333038 Numbers m such that sigma(m) <= sigma(m-1).

Original entry on oeis.org

5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 46, 47, 49, 51, 53, 55, 57, 59, 61, 65, 67, 69, 71, 73, 77, 79, 81, 83, 85, 87, 89, 91, 93, 95, 97, 99, 101, 103, 105, 106, 107, 109, 111, 113, 115, 117, 118, 119, 121, 123

Offset: 1

Bernard Schott, Mar 06 2020

nonn

This sequence is infinite because all primes p >= 5 are terms with sigma(p) < sigma(p-1).
The integer m is a term iff A053222(m-1) <= 0.
The numbers m such that sigma(m) = sigma(m-1) are in A231546.

			Sigma(9) = 1+3+9 = 13 < sigma(8) = 1+2+4+8 = 15 so 9 is a term.
Sigma(15) = 1+3+5+15 = 24 = sigma(14) = 1+2+7+14 = 24 so 15 is a term.
Sigma(63) = 1+3+7+9+21+63 = 104 > sigma(62) = 1+2+31+62 = 96 and 63 is not a term.

J.-M. De Koninck & A. Mercier, 1001 Problèmes en Théorie Classique des Nombres, Problème 620 pp. 82, 280, Ellipses Paris 2004

Cf. A000203, A053222, A231546 (subsequence: sigma(m) = sigma(m-1)).

Cf. A053224 (sigma(m) < sigma(m+1)), A053226 (sigma(m) > sigma(m+1)).

filter:= m -> sigma(m) <= sigma(m-1): select(filter, [$1..500]);

Select[Range[2, 123], DivisorSigma[1, #] <= DivisorSigma[1, # - 1] &] (* Amiram Eldar, Mar 06 2020 *)
Flatten[Position[Partition[DivisorSigma[1,Range[200]],2,1],?(#[[2]]<= #[[1]]&),1,Heads->False]]+1 (* _Harvey P. Dale, Mar 28 2020 *)

isok(m) = (m>1) && (sigma(m) <= sigma(m-1)); \\ Michel Marcus, Mar 09 2020

A333039 Composites m such that sigma(m) < sigma(m-1).

Original entry on oeis.org

9, 21, 25, 27, 33, 35, 39, 45, 46, 49, 51, 55, 57, 65, 69, 77, 81, 85, 87, 91, 93, 95, 99, 105, 106, 111, 115, 117, 118, 119, 121, 123, 125, 129, 133, 141, 143, 145, 153, 155, 159, 161, 165, 166, 169, 171, 175, 177, 183, 185, 187, 189, 201

Offset: 1

Bernard Schott, Mar 12 2020

nonn

As all primes p >= 5 satisfy sigma(p) < sigma(p-1) [see A333038], this sequence is reserved for composite numbers.
This sequence is infinite because all squares of primes p, p >= 3 are terms.
Composites such that sigma(m) = sigma(m-1) are in A231546.

			sigma(77) = 1+7+11+77 = 96 < sigma(76) = 1+2+4+19+38+76 = 140, hence composite 77 is a term.
sigma(135) = 1+3+5+9+15+27+45+135 = 240 > sigma(134) = 1+2+67+134 = 204, hence composite 135 is not a term.

J.-M. De Koninck & A. Mercier, 1001 Problèmes en Théorie Classique des Nombres, Problème 620 pp. 82, 280, Ellipses Paris 2004

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

Cf. A000203, A053222, A231546, A333038.

filter:= m -> not isprime(m) and numtheory:-sigma(m) < numtheory:-sigma(m-1) : select(filter, [$1..500]);

Select[Range[200], CompositeQ[#] && DivisorSigma[1, #] < DivisorSigma[1, # - 1] &] (* Amiram Eldar, Mar 12 2020 *)

isok(m) = (m>1) && !isprime(m) && (sigma(m) < sigma(m-1)); \\ Michel Marcus, Mar 15 2020

cp's OEIS Frontend

A064431 Numbers n such that |A053222(n)| = tau(n).

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A002961 Numbers k such that k and k+1 have same sum of divisors.

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A053224 Numbers k for which sigma(k) < sigma(k+1).

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A053226 Numbers k for which sigma(k) > sigma(k+1).

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A340793 Sequence whose partial sums give A000203.

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A053223 Second differences of sigma(n).

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Magma

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A053227 First differences of sigma(n) that are negative.

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A053225 First differences of sigma(n) that are positive.

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