A164557 -id:A164557 - cp's OEIS frontend

A324295 Numbers k such that s(k) = s(k+1) where s(k) is the sum of divisors of k that are smaller than sqrt(k) (A070039).

2, 3, 4, 186, 318, 434, 473, 582, 730, 978, 1024, 1035, 1245, 1357, 1397, 1506, 1661, 1902, 2085, 2116, 2224, 2329, 2453, 2505, 2506, 2770, 2954, 3144, 3345, 3377, 3624, 3641, 3765, 3790, 3882, 4037, 4172, 4438, 4898, 4938, 4975, 5221, 6126, 6285, 6312, 6356

Offset: 1

Amiram Eldar, Sep 03 2019

nonn

			186 is in the sequence since A070039(186) = A070039(187) = 12.

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

Cf. A002961, A064115, A064125, A070039, A164522, A164557, A293183, A306985, A325175.

s[n_] := DivisorSum[n, # &, # < Sqrt[n] &]; seq={}; s1 = 0; Do[s2 = s[n]; If[s1 == s2, AppendTo[seq, n - 1]]; s1 = s2, {n, 2, 6500}]; seq

A324367 Numbers k such that s(k) = s(k+1) where s(k) is the sum of divisors of k that are larger than sqrt(k) (A238535).

Original entry on oeis.org

45, 62, 15795, 355022, 14257705, 28856174, 2324581982, 103321586193

Offset: 1

Amiram Eldar, Sep 03 2019

a(9) > 2*10^11. - Giovanni Resta, Sep 06 2019

			45 is in the sequence since A238535(45) = A238535(46) = 69.

Cf. A002961, A064115, A064125, A164522, A164557, A238535, A293183, A306985, A324295.

s[n_] := DivisorSum[n, # &, # > Sqrt[n] &]; seq={}; s1 = 0; Do[s2 = s[n]; If[s1 == s2, AppendTo[seq, n - 1]]; s1 = s2, {n, 2, 10000}]; seq

a(8) from Giovanni Resta, Sep 06 2019

A327875 Numbers k such that s(k) = s(k+1) where s(k) is the sum of unitary, squarefree divisors of k, including 1 (A092261).

Original entry on oeis.org

8, 14, 288, 675, 735, 957, 1334, 1634, 2685, 2871, 5750, 8055, 9800, 12104, 12167, 20145, 33998, 42818, 71994, 74918, 79826, 79833, 84134, 111506, 122073, 138237, 144990, 147454, 166934, 201597, 235224, 274533, 289454, 324423, 332928, 347738, 383594, 400315

Offset: 1

Amiram Eldar, Sep 28 2019

nonn

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

Cf. A002961, A064115, A064125, A092261, A164522, A164557, A293183, A306985, A324295, A325175.

f[p_, e_] := If[e==1, p+1, 1]; s[n_] := Times @@ f @@@ FactorInteger[n]; s1=0; seq = {}; Do[s2 = s[n]; If[s1 == s2, AppendTo[seq, n-1]]; s1 = s2, {n,1,10000}]; seq

s(n)={sumdiv(n, d, d*issquarefree(d)*(gcd(d, n/d) == 1))}
{ for(k=1, 10^6, if(s(k)==s(k+1), print1(k, ", "))) } \\ Andrew Howroyd, Sep 28 2019

8 is in the sequence since A092261(8) = A092261(9) = 1.

A362400 Numbers k such that A162296(k) = A162296(k+1) > 0.

Original entry on oeis.org

135, 819, 1863, 9207, 10340, 41124, 75051, 95336, 278972, 305091, 465596, 544924, 570411, 711027, 903804, 977876, 1114695, 1327095, 1444779, 1520684, 1760571, 1987371, 2083491, 2303091, 2581928, 2842324, 2869011, 3062631, 3243140, 4043624, 4335848, 4469984, 4598091

Offset: 1

Amiram Eldar, Apr 18 2023

nonn

A162296(k) = A162296(k+1) = 0 if and only if k and k+1 are both squarefree (A005117), i.e., k is in A007674.

			135 is a term since A162296(135) = A162296(136) = 216.

Amiram Eldar, Table of n, a(n) for n = 1..527 (terms below 10^10)

Cf. A005117, A007674, A162296.
Subsequence of A013929 and A068781.
Similar sequences: A002961, A064115, A064125, A164522, A164557, A293183, A306985, A325175, A324295.

s[n_] := Module[{f = FactorInteger[n], p, e}, p = f[[;; , 1]]; e = f[[;; , 2]]; Times @@ ((p^(e + 1) - 1)/(p - 1)) - Times @@ (p + 1)]; Select[Range[2, 5*10^6], (sn = s[#]) > 0 && sn == s[# + 1] &]

s(n) = {my(f = factor(n), p, e); prod(i = 1, #f~, p = f[i, 1]; e = f[i, 2]; ((p^(e + 1) - 1)/(p - 1))) -  prod(i = 1, #f~, f[i, 1] + 1);}
lista(kmax) = {my(s1 = s(1), s2); for(k=2, kmax, s2 = s(k); if(s1 > 0 && s2 == s1, print1(k-1, ", ")); s1 = s2); }

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A324295 Numbers k such that s(k) = s(k+1) where s(k) is the sum of divisors of k that are smaller than sqrt(k) (A070039).

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A324367 Numbers k such that s(k) = s(k+1) where s(k) is the sum of divisors of k that are larger than sqrt(k) (A238535).

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A327875 Numbers k such that s(k) = s(k+1) where s(k) is the sum of unitary, squarefree divisors of k, including 1 (A092261).

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A362400 Numbers k such that A162296(k) = A162296(k+1) > 0.

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