A293183 -id:A293183 - cp's OEIS frontend

A372693 Numbers k such that A372692(k) = A372692(k+1) > 1.

7380, 18755, 24804, 25631, 26299, 27467, 32799, 44891, 49196, 49725, 50940, 53603, 59652, 64386, 71027, 79739, 85788, 89300, 94275, 103212, 105056, 105875, 124992, 129348, 132011, 138060, 141899, 147100, 149435, 155484, 158147, 164196, 170324, 175571, 181620, 184283

Offset: 1

Amiram Eldar, May 10 2024

nonn

The numbers k such that A372692(k) = A372692(k+1) = 1 are in A372690.

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

Cf. A036537, A372690, A372692.
Subsequence of A068781.
A372694 is a subsequence.
Similar sequences: Cf. A002961, A064125, A293183, A306985, A343819, A348346.

f[p_, e_] := p^(2^(-1 + Position[Reverse@ IntegerDigits[e, 2], ?(# == 0 &)])); s[1] = 1; s[n] := s[n] = Times @@ (Flatten@ (f @@@ FactorInteger[n]) + 1);
Select[Range[10^5], (s1 = s[#]) > 1 && s1 == s[# + 1] &]

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

A294029 Values of bsigma(k) = bsigma(k+1), where bsigma is the sum of the bi-unitary divisors (A188999).

Original entry on oeis.org

24, 40, 60, 720, 960, 1440, 2160, 2640, 2400, 3000, 4320, 4320, 4320, 5280, 7400, 11520, 11880, 12960, 14400, 20160, 30240, 26640, 34560, 25200, 34560, 49920, 51840, 60480, 63360, 60480, 65280, 62400, 61560, 115200, 93600, 114912, 100800, 120960, 120960

Offset: 1

Amiram Eldar, Oct 22 2017

nonn

The sum of bi-unitary divisors of numbers n such that n and n+1 have the same sum (A293183).

The bi-unitary version of A053215.

			24 is in the sequence since 24 = bsigma(14) = bsigma(15).

Amiram Eldar, Table of n, a(n) for n = 1..2148

Cf. A053215, A290303, A188999, A293183.

f[n_] := Select[Divisors[n], Function[d, CoprimeQ[d, n/d]]]; bsigma[m_] := DivisorSum[m, # &, Last@Intersection[f@#, f[m/#]] == 1 &]; a = {}; b1 = 0; For[k = 0, k < 10^6, k++; b2 = bsigma[k]; If[b1 == b2, a = AppendTo[a, b1]]; b1 = b2]; a (* after Michael De Vlieger at A188999 *)

a(n) = A188999(A293183(n)).

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.

A332315 Numbers k such that k and k + 1 have the same norm of the sum of divisors in Gaussian integers.

Original entry on oeis.org

30514, 36777, 43978, 3474262, 5745125, 10628554, 16567494, 40831527, 58008301, 111798477, 142981839, 288834504, 392413941, 580867202, 650141557, 944224497, 967593411, 1874210882, 6306287377, 6442064745, 7377567197, 8121464245

Offset: 1

Amiram Eldar, Feb 09 2020

The first term, 30514, is also a number k such that k and k + 1 have the sum divisors in Gaussian integers: -54720 + 48960*i (where i is the imaginary unit). What is the next term with this property?

No more terms below 1.5*10^10.

			30514 is a term since A103230(30514) = A103230(30515) = 5391360000.

Cf. A002961, A064125, A103228, A103229, A103230, A293183, A306985.

csigma[n_] :=(Abs @ DivisorSigma[1, n, GaussianIntegers -> True])^2; seq = {}; n1 = csigma[1]; Do[n2 = csigma[n]; If[n1 == n2, AppendTo[seq, n - 1]]; n1 = n2, {n, 2, 5*10^5}]; seq

A332475 Numbers k such that k and k + 1 have the same norm of the sum of unitary divisors in Gaussian integers (A332474).

Original entry on oeis.org

5, 11, 37, 1738, 2772, 6600, 42251, 49913, 57816, 104754, 220324, 288350, 364452, 792156, 1711932, 1971475, 2607049, 2793473, 3211933, 3521148, 3526312, 4012736, 5805149, 5918276, 6522320, 6542147, 6635436, 7612267, 12604600, 14844791, 17078848, 19024332, 21177516

Offset: 1

Amiram Eldar, Feb 13 2020

nonn

			5 is a term since A332474(5) = A332474(6) = 80.

Amiram Eldar, Table of n, a(n) for n = 1..100

Cf. A002961, A064125, A293183, A306985, A332315, A332472, A332473, A332474.

f[p_, e_] := If[Abs[p] == 1, 1, (p^e + 1)]; normUsigma[n_] := Abs[Times @@ f @@@ FactorInteger[n, GaussianIntegers -> True]]^2; seq = {}; u1 = normUsigma[1]; Do[u2 = normUsigma[n]; If[u1 == u2, AppendTo[seq, n - 1]]; u1 = u2, {n, 2, 10^6}]; seq

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);

cp's OEIS Frontend

A372693 Numbers k such that A372692(k) = A372692(k+1) > 1.

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A294029 Values of bsigma(k) = bsigma(k+1), where bsigma is the sum of the bi-unitary divisors (A188999).

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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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A332315 Numbers k such that k and k + 1 have the same norm of the sum of divisors in Gaussian integers.

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A332475 Numbers k such that k and k + 1 have the same norm of the sum of unitary divisors in Gaussian integers (A332474).

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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).