A238380 -id:A238380 - cp's OEIS frontend

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

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

A338452 Numbers k such that k and k+1 have the same total binary weight of their divisors (A093653).

Original entry on oeis.org

3, 4, 7, 20, 31, 57, 94, 98, 118, 122, 127, 201, 213, 218, 230, 242, 243, 244, 334, 384, 393, 423, 429, 481, 565, 603, 633, 694, 704, 729, 766, 844, 921, 1138, 1141, 1221, 1262, 1401, 1533, 1654, 1726, 1761, 1837, 1838, 1862, 1882, 1942, 2162, 2245, 2361, 2362

Offset: 1

Amiram Eldar, Oct 28 2020

Numbers k such that A093653(k) = A093653(k+1).

The Mersenne primes (A000668) are terms since if 2^p - 1 is a prime then A093653(2^p-1) = A093653(2^p) = p+1.

			3 is a term since A093653(3) = A093653(4) = 3.

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

Cf. A000120, A093653.
A000668 is a subsequence.
Similar sequences: A002961, A005237, A054004, A064125, A238380, A293183, A306985.

f[n_] := DivisorSum[n, DigitCount[#, 2, 1] &]; s = {}; f1 = f[1]; Do[f2 = f[n]; If[f1 == f2, AppendTo[s, n - 1]]; f1 = f2, {n, 2, 240}]; s

A322668 Numbers k such that (k, k+2) are not twin primes yet sigma(k+2)/d(k+2) - sigma(k)/d(k) = 1.

Original entry on oeis.org

1, 350, 6497, 12317, 133787, 181427, 404471, 439097, 485237, 501182, 549378, 1410119, 2696807, 6220607, 6827369, 6954767, 9770027, 10302419, 10449347, 10887977, 11014007, 16745387, 18959111, 25883519, 27334469, 39508037, 40311149, 40551617, 42561437, 44592209

Offset: 1

Amiram Eldar, Jan 11 2019

nonn

A variation of A050507 with average of the divisors instead of their sum.

Amiram Eldar, Table of n, a(n) for n = 1..241 (terms below 10^10)

Cf. A000005 (number of divisors), A000203 (sum of divisors).

Cf. A050507, A056774, A057020, A057021, A238380.

f[n_] := DivisorSigma[1, n]/DivisorSigma[0, n]; aQ[n_] := f[n + 2] - f[n] ==  1 && !(PrimeQ[n] && PrimeQ[n + 2]); Select[Range[1000000], aQ]

isok(k) = !(isprime(k) && isprime(k+2)) && (sigma(k+2)/numdiv(k+2) - sigma(k)/numdiv(k) == 1); \\ Michel Marcus, Jan 22 2019

A348415 Numbers k such that k and k+1 have the same denominator of the harmonic means of their divisors.

Original entry on oeis.org

12, 88, 180, 266, 321, 604, 4277, 4364, 8632, 15861, 18720, 28461, 47613, 63546, 97412, 98907, 135078, 137333, 154132, 179621, 185776, 192699, 203709, 265489, 284883, 344217, 383466, 517610, 604197, 876469, 1089604, 1277518, 1713865, 1839123, 1893268, 2349390

Offset: 1

Amiram Eldar, Oct 17 2021

nonn

Numbers k such that A099378(k) = A099378(k+1).
The common denominators of k and k+1 are 7, 45, 91, 30, 36, 133, 96, 637, ...
Can 3 consecutive numbers have the same denominator of harmonic mean of divisors? There are no such numbers below 10^10.

			12 is a term since the harmonic means of the divisors of 12 and 13 are 18/7 and 13/7, respectively, and both have the denominator 7.

Amiram Eldar, Table of n, a(n) for n = 1..300

Cf. A099377, A099378.

Similar sequences: A002961, A238380.

dh[n_] := Denominator[DivisorSigma[0, n]/DivisorSigma[-1, n]]; Select[Range[10^6], dh[#] == dh[# + 1] &]

f(n) = my(d=divisors(n)); denominator(#d/sum(k=1, #d, 1/d[k])); \\ A099378
isok(k) = f(k) == f(k+1); \\ Michel Marcus, Oct 20 2021

A335291 Numbers m such that the delta(m) = abs(h(m+1) - h(m)) is smaller than delta(k) for all k < m, where h(m) is the harmonic mean of the divisors of m.

Original entry on oeis.org

1, 2, 4, 91, 272, 20118, 20712, 33998, 42818, 61695, 25274946, 27194929, 34883654, 40406622, 43176318, 47350866, 52680050, 149736013, 154957034, 162929406, 171560153, 187012577, 208015843, 267361097, 300087726, 325189758, 355153181, 443360633, 584803578, 605883413

Offset: 1

Amiram Eldar, May 30 2020

nonn

Apparently, most of the terms m have h(m+1) > h(m) and numerator(delta(m)) = 1.

Can two consecutive numbers have the same harmonic mean of divisors? If yes, then this sequence is finite.

			The values of delta(k) for the first terms are 0.333..., 0.166..., 0.047..., 0.0357..., ...

Amiram Eldar, Table of n, a(n) for n = 1..86

Cf. A000005, A000203, A099377, A099378.

Cf. A238380, A335071.

h[n_] := n * DivisorSigma[0, n]/DivisorSigma[1, n]; dm = 1; h1 = h[1]; s = {}; Do[h2 = h[n]; d = Abs[h2 - h1]; If[d < dm, dm = d; AppendTo[s, n-1]]; h1 = h2, {n, 2, 10^5}]; s

A349222 Numbers k such that k and k+1 have the same average of unitary divisors.

Original entry on oeis.org

5, 14, 44, 55, 152, 1334, 1634, 1652, 2204, 2232, 2295, 2685, 3195, 4256, 7191, 8216, 9144, 9503, 9844, 10152, 18423, 19491, 20118, 27404, 30247, 33998, 38180, 42818, 45716, 48364, 51624, 79316, 79338, 84134, 117116, 122073, 124676, 125811, 139460, 157640, 166624

Offset: 1

Amiram Eldar, Nov 11 2021

nonn

The average of the unitary divisors of k is equal to A034448(k)/A034444(k).
Terms k such that k and k+1 are squarefree are also terms of A238380. The terms that are not in A238380 are 44, 55, 152, 1652, 2204, 2232, 2295, 3195, 4256, ...
The average is an integer for the first 1000 terms. Are there terms with a noninteger average?

			5 is a term since the average of the unitary divisors of 5 is (1 + 5)/2 = 3, and the average of the unitary divisors of 6 is (1 + 2 + 3 + 6)/4 = 3.
44 is a term since the average of the unitary divisors of 44 is (1 + 4 + 11 + 44)/4 = 15, and the average of the unitary divisors of 45 is (1 + 5 + 9 + 45)/4 = 15.

Amiram Eldar, Table of n, a(n) for n = 1..1000

Cf. A007674, A034444, A034448, A103826, A238380.

m[1] = 1; m[n_] := (Times @@ (1 + Power @@@ (f = FactorInteger[n])))/2^Length[f]; Select[Range[10^5], m[#] == m[# + 1] &]

A274774 Least k such that sigma(kn)/tau(kn) = sigma(kn+1)/tau(kn+1), or 0 if no such k exists.

Original entry on oeis.org

5, 7, 895, 1363, 1, 3353, 2, 2589, 1007, 10341, 1265, 1726, 7, 1, 179, 6634, 10052, 5745, 86, 53389, 958, 12165, 58, 863, 649, 250017, 2395, 6103, 46, 3447, 2714, 3317, 8110, 5026, 22653, 2812637, 94, 43, 16795, 58069, 61693, 479, 38, 52790, 1437, 29, 74, 2027510, 122367, 70545

Offset: 1

Altug Alkan, Jul 28 2016

nonn

Corresponding averages are 3, 6, 540, 840, 3, 2880, 6, 3240, 1170, 8640, 1596, 3240, 28, 6, 540, 9072, 15120, 8640, 330, 55440, 2880, 21924, 270, 3240, 1860, 875070, 7200, ...

			a(13) = 7 because sigma(7*13)/tau(7*13) = sigma(7*13+1)/tau(7*13+1).

Cf. A238380.

a[n_] := Block[{k=1}, While[! Equal @@ (DivisorSigma[1, n*k + {0,1}] / DivisorSigma[ 0, n*k + {0,1}]), k++]; k]; Array[a, 20] (* Giovanni Resta, Jul 28 2016 *)

a(n) = {my(k=1); while (sigma(k*n)/numdiv(k*n) != sigma(k*n+1)/numdiv(k*n+1), k++); k; }

cp's OEIS Frontend

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

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A338452 Numbers k such that k and k+1 have the same total binary weight of their divisors (A093653).

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A322668 Numbers k such that (k, k+2) are not twin primes yet sigma(k+2)/d(k+2) - sigma(k)/d(k) = 1.

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A348415 Numbers k such that k and k+1 have the same denominator of the harmonic means of their divisors.

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A335291 Numbers m such that the delta(m) = abs(h(m+1) - h(m)) is smaller than delta(k) for all k < m, where h(m) is the harmonic mean of the divisors of m.

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A349222 Numbers k such that k and k+1 have the same average of unitary divisors.

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A274774 Least k such that sigma(k*n)/tau(k*n) = sigma(k*n+1)/tau(k*n+1), or 0 if no such k exists.

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A274774 Least k such that sigma(kn)/tau(kn) = sigma(kn+1)/tau(kn+1), or 0 if no such k exists.