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A002961 Numbers n such that n and n+1 have same sum of divisors.
(Formerly M4950)
48
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

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. [From 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. [From 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) = T(n) mod antisigma(n) = A242962(n+1) = T(n+1) mod antisigma(n+1), where T(n) = A000217(n) = triangular numbers and antisigma(n) = A024816(n) = sum of numbers less than n which do not divide n. - Jaroslav Krizek, May 29 2014

REFERENCES

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. Sierpi\'{n}ski, 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).

LINKS

T. D. Noe and 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].

Lourdes Benito, Solutions of the problem of Erdos-Sierpinski: sigma(n)=sigma(n+1), arXiv:0707.2190 [math.NT]

A. Weingartner, On the Solutions of sigma(n) = sigma(n+k), Journal of Integer Sequences, Vol. 14 (2011), #11.5.5.

MAPLE

with(numtheory); P:=proc(n) local a, i; for i from 1 by 1 to n do a:=sigma(i)/sigma(i+1); if a=1 then print(i); fi; od; end: P(100000); - Paolo P. Lava, Aug 23 2007

MATHEMATICA

f[n_]:=DivisorSigma[1, n]; lst={}; Do[If[f[n]==f[n+1], AppendTo[lst, n]], {n, 9!}]; lst [From Vladimir Orlovsky, Jul 22 2009]

Flatten[Position[Partition[DivisorSigma[1, Range[170000]], 2, 1], {x_, x_}]] (* Harvey P. Dale, Aug 08 2011 *)

PROG

(PARI) 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

(Haskell)

import Data.List (elemIndices)

a002961 n = a002961_list !! (n-1)

a002961_list = map (+ 1) $ elemIndices 0 a053222_list

-- Reinhard Zumkeller, Dec 28 2011

(MAGMA) [n: n in [3..100000] | ((n*(n+1)div 2) mod (n*(n+1)div 2-SumOfDivisors(n))) eq (((n+1)*(n+2)div 2) mod ((n+1)*(n+2)div 2-SumOfDivisors(n+1)))]  - Jaroslav Krizek, May 29 2014

CROSSREFS

Cf. A000203 (sigma function), A053215, A053249, A054004

Cf. A007373, A015861, A015863, A015865, A015866, A015867, A015876, A015877, A015880, A015881, A015882, A015883, A181647. [From Reinhard Zumkeller, Nov 03 2010]

Cf. A238380.

Sequence in context: A068769 A113349 A109764 * A063071 A192007 A160682

Adjacent sequences:  A002958 A002959 A002960 * A002962 A002963 A002964

KEYWORD

nonn,nice

AUTHOR

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

EXTENSIONS

More terms from Jud McCranie Oct 15 1997

STATUS

approved

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Last modified October 21 23:13 EDT 2014. Contains 248381 sequences.