A064115 -id:A064115 - cp's OEIS frontend

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).

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.

A349063 Numbers k such that k and k+1 have the same sum of powerful divisors (A183097) and this sum is larger than 1.

Original entry on oeis.org

2988, 4067, 7595, 13572, 14651, 24156, 25235, 27684, 28763, 34740, 35819, 38268, 39347, 41327, 46403, 48852, 49931, 56987, 59436, 66492, 70020, 78155, 81683, 87660, 88739, 91188, 98244, 99323, 101772, 102851, 108828, 109907, 112356, 113435, 119412, 120491, 122940

Offset: 1

Amiram Eldar, Nov 07 2021

nonn

Numbers k such that A183097(k) = A183097(k+1) > 1.

			2988 is a term since = A183097(2988) = A183097(2989) = 50 > 1.

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

Cf. A183097.

Similar sequences: A002961, A064115, A064125, A293183, A306985.

f[p_, e_] := (p^(e + 1) - 1)/(p - 1) - p; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; q[n_] := (s1 = s[n]) > 1 && s1 == s[n + 1]; Select[Range[10^5], q]

A349224 Number k such that A033634(k) = A033634(k+1).

Original entry on oeis.org

11, 14, 957, 1334, 1485, 1634, 2685, 4136, 9347, 13915, 16260, 16499, 20145, 29903, 33998, 37236, 42251, 42818, 55308, 56419, 74918, 77748, 79826, 79833, 84134, 86343, 109864, 111506, 122073, 138237, 142116, 147454, 166934, 168739, 178356, 184260, 187863, 194028

Offset: 1

Amiram Eldar, Nov 11 2021

nonn

			11 is a term since A033634(11) = A033634(12) = 12.

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

Cf. A033634.

Similar sequences: A002961, A064115, A064125, A293183, A306985.

f[e_] := If[OddQ[e], e+2, e+1]; fun[p_, e_] := 1 + (p^f[e] - p)/(p^2 - 1); s[1] = 1; s[n_] := Times @@ (fun @@@ FactorInteger[n]); Select[Range[2*10^5], s[#] == s[#+1] &]

A349283 Numbers k such that A051378(k) = A051378(k+1).

Original entry on oeis.org

14, 206, 957, 1334, 1364, 1485, 1634, 2685, 2974, 4136, 4364, 14841, 20145, 24957, 33998, 36566, 42818, 61183, 64672, 74918, 79826, 79833, 84134, 86343, 92685, 104192, 109864, 111506, 122073, 138237, 147454, 159711, 162602, 166934, 187863, 190773, 193893, 201597

Offset: 1

Amiram Eldar, Nov 13 2021

nonn

First differs from A333949 at n = 18.

			14 is a term since A051378(14) = A051378(15) = 24.

Amiram Eldar, Table of n, a(n) for n = 1..500

Cf. A051378.

Similar sequences: A002961, A064115, A064125, A293183, A306985, A333949.

s[1] = 1; s[n_] := Times @@ (1 + Sum[First[#]^d, {d, Divisors[Last[#]]}] &) /@ FactorInteger[n]; Select[Range[2*10^5], s[#] == s[#+1] &]

A355713 Numbers k such that k and k+1 have the same sum of 5-smooth divisors.

Original entry on oeis.org

175, 2224, 2575, 4975, 7024, 9424, 9775, 11824, 12175, 14224, 14575, 16975, 19024, 21424, 21775, 23824, 24175, 26224, 26575, 28975, 31024, 33424, 33775, 35824, 36175, 38224, 38575, 40975, 43024, 45424, 45775, 47824, 48175, 50224, 50575, 52975, 55024, 57424, 57775

Offset: 1

Amiram Eldar, Jul 15 2022

nonn

Numbers k such that A355584(k) = A355584(k+1).
Equivalently, numbers k such that the largest 5-smooth divisors of k and k+1, A355582(k) and A355582(k+1), have the same sum of divisors (A000203).
For all the terms k, both k and k+1 are not squarefree: each of the two largest 5-smooth divisors, of k and k+1, cannot be squarefree, since the squarefree 5-smooth numbers are the divisors of 30 = 2*3*5 (A018255) whose values of sigma (A000203), {1, 3, 4, 6, 12, 18, 24, 72}, are not shared with sigma of any other 5-smooth number.
Apparently, all the terms are of only two types: numbers k such that A355582(k) = 16 and A355582(k+1) = 25, or numbers k such that A355582(k) = 25 and A355582(k+1) = 16. Both types are infinite sequences: The first type is the sequence of numbers of the form 2224 + 2400*m, where m is not congruent to 1 (mod 5), and the second type is the sequence of numbers of the form 175 + 2400*m, where m is not congruent to 3 (mod 5). If there are no other terms, then this sequence is a linear recurrence with a signature (1,0,0,0,0,0,0,1,-1). The question of the existence of other types is equivalent to the question of the existence of two coprime 5-smooth numbers other than 16 and 25 whose sums of divisors are equal.
Are there runs of 3 consecutive numbers with the same sum of 5-smooth divisors? There are no such runs below 5*10^10.

			175 is a term since A355584(175) = A355584(176) = 31.

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

Cf. A000203, A002961, A018255, A355584.
Subsequence of A013929 and A068781.
Similar sequences: A002961, A064115, A064125, A293183, A306985, A333949.

f[p_, e_] := If[p > 5, 1, (p^(e + 1) - 1)/(p - 1)]; s[1] = 1; s[n_] := Times @@ (f @@@ FactorInteger[n]); Select[Range[10^5], s[#] == s[# + 1] &]

s(n) = (2^(valuation(n, 2) + 1) - 1) * (3^(valuation(n, 3) + 1) - 1) * (5^(valuation(n, 5) + 1) - 1) / 8;
s1 = s(1); for(k = 2, 6e4, s2 = s(k); if(s1 == s2, print1(k-1,", ")); s1 = s2);

A360527 Numbers k such that A360522(k) = A360522(k+1).

Original entry on oeis.org

4, 8, 14, 176, 895, 956, 957, 1334, 1634, 1724, 1725, 1844, 1934, 2685, 2871, 3404, 3759, 4047, 4136, 5175, 7004, 7315, 7599, 8055, 12104, 13760, 18415, 20145, 29392, 32944, 33998, 42818, 44095, 44516, 49599, 60356, 74918, 79826, 79833, 84134, 85172, 85744, 86343

Offset: 1

Amiram Eldar, Feb 10 2023

nonn

Numbers k such that A360522(k) = A360522(k+1) = A360522(k+2) exist: 956 and 1724. Are there any other terms like these? There are none below 1.8*10^10.

			4 is a term since A360522(4) = A360522(5) = 6.

Amiram Eldar, Table of n, a(n) for n = 1..2500

Cf. A360522.

Similar sequences: A002961, A064115, A064125, A293183, A306985.

f[p_, e_] := p^e + e; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; Position[Partition[Array[s, 10^5], 2, 1], _?(SameQ @@ # &)] // Flatten

s(n) = {my(f = factor(n)); prod(i = 1, #f~, f[i,1]^f[i,2] + f[i,2]);}
lista(nmax) = {my(s1 = s(1), s2); for(n=2, nmax, s2=s(n); if(s1 == s2, print1(n-1, ", ")); s1 = s2); }

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); }

A379032 Numbers k such that k and k+1 have an equal sum of modified exponential divisors: A241405(k) = A241405(k+1).

Original entry on oeis.org

14, 44, 957, 1334, 1485, 1634, 1652, 2204, 2685, 3195, 3451, 3956, 4136, 5547, 8495, 8636, 8907, 9844, 11515, 12256, 14876, 15608, 19491, 20145, 20155, 27519, 27643, 33998, 35235, 36575, 38180, 41265, 41547, 42818, 45716, 48364, 74918, 79316, 79826, 79833, 84134

Offset: 1

Amiram Eldar, Dec 14 2024

nonn

Amiram Eldar, Table of n, a(n) for n = 1..1918 (terms below 10^10)

Cf. A241405.

Similar sequences: A002961, A064115, A064125, A293183, A306985.

f[p_, e_] := DivisorSum[e + 1, p^(# - 1) &]; mesigma[1] = 1; mesigma[n_] := mesigma[n] = Times @@ f @@@ FactorInteger[n]; Select[Range[10^5], mesigma[#] == mesigma[#+1] &]

mesigma(n) = {my(f=factor(n)); prod(i=1, #f~, sumdiv(f[i, 2]+1, d, f[i, 1]^(d-1))); }
lista(kmax) = {my(m1 = 1, m2); for(k = 2, kmax, m2 = mesigma(k); if(m1 == m2, print1(k-1, ", ")); m1 = m2);}

A333954 Numbers k such that A330575(k) = A330575(k+1).

Original entry on oeis.org

14, 16101, 72926, 97101, 2872701, 7610324

Offset: 1

Amiram Eldar, Apr 11 2020

a(7) > 6*10*8.

			14 is a term since A330575(14) = A330575(15) = 26.

Thomas Fink, Recursively divisible numbers, arXiv:1912.07979 [math.NT], 2019.

Cf. A330575.

Similar sequences: A002961, A064115, A064125, A293183, A306985.

s[1] = 1; s[n_] := s[n] = n + DivisorSum[n, s[#] &, # < n &]; seq = {}; s1 = s[1]; Do[s2 = s[n]; If[s1 == s2, AppendTo[seq, n-1]]; s1 = s2, {n, 2, 10^5}]; seq

A348628 Numbers k such that k and k+1 have the same sum of nonexponential divisors (A160135).

Original entry on oeis.org

1, 2, 3, 4, 15, 44, 674, 478899

Offset: 1

Amiram Eldar, Oct 26 2021

Numbers k such that A160135(k) = A160135(k+1).

a(9) > 1.6 * 10^11, if it exists.

			2 is a term since A160135(2) = A160135(3) = 1.
15 is a term since A160135(15) = A160135(16) = 9.

Cf. A160135.

Similar sequences: A002961, A064115, A064125, A293183, A306985, A348346.

esigma[n_] := Times @@ (Sum[First[#]^d, {d, Divisors[Last[#]]}] &) /@ FactorInteger[n]; s[1] = 1; s[n_] := DivisorSigma[1, n] - esigma[n]; Select[Range[500000], s[#] == s[# + 1] &]

cp's OEIS Frontend

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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A349063 Numbers k such that k and k+1 have the same sum of powerful divisors (A183097) and this sum is larger than 1.

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A349224 Number k such that A033634(k) = A033634(k+1).

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A349283 Numbers k such that A051378(k) = A051378(k+1).

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A355713 Numbers k such that k and k+1 have the same sum of 5-smooth divisors.

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A360527 Numbers k such that A360522(k) = A360522(k+1).

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

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A379032 Numbers k such that k and k+1 have an equal sum of modified exponential divisors: A241405(k) = A241405(k+1).

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