cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-6 of 6 results.

A325175 Numbers k such that s(k) = s(k+1) = s(k+2) = s(k+3) where s(k) is the sum of divisors of k that are smaller than sqrt(k) (A070039).

Original entry on oeis.org

2, 1190499411, 7896840789, 9190223334, 10546199943, 13239147654, 16488284454, 19818736503, 19927041063, 20401914903, 20551875063, 24219563814, 25442242503, 27391724583, 34539825543, 35714513463, 37014168183, 44120613543, 44130940614, 44514172134, 48647900639
Offset: 1

Views

Author

Giovanni Resta, Sep 05 2019

Keywords

Comments

Five consecutive coincident values of A070039 are also possible. One such occurrence starts a 211301122224870.

Examples

			2 is in the sequence since A070039(2) = A070039(3) = A070039(4) = A070039(5) = 1.

Crossrefs

Cf. A070039, A324295, A324310.

A334020 Numbers k such that s(k) = s(k+1), where s(k) is the sum of unitary divisors of k that are smaller than sqrt(k) (A334019).

Original entry on oeis.org

2, 3, 4, 7, 8, 16, 31, 127, 186, 256, 318, 434, 473, 574, 582, 588, 730, 735, 819, 978, 1245, 1357, 1374, 1397, 1420, 1421, 1500, 1506, 1661, 1694, 1902, 1956, 1988, 2059, 2085, 2147, 2166, 2329, 2453, 2505, 2506, 2534, 2754, 2770, 2868, 2954, 2988, 3345, 3377
Offset: 1

Views

Author

Amiram Eldar, Apr 12 2020

Keywords

Examples

			2 is a term since A334019(2) = A334019(3) = 1.

Links

Crossrefs

Cf. A064125, A324295, A334019,

Programs

  • Mathematica
    s[n_] := DivisorSum[n, # &, #^2 < n && CoprimeQ[#, n/#] &]; Select[Range[3000], s[#] == s[# + 1] &]

A324310 Numbers k such that s(k) = s(k+1) = s(k+2) where s(k) is the sum of divisors of k that are smaller than sqrt(k) (A070039).

Original entry on oeis.org

2, 3, 2505, 64486, 113413, 205365, 423414, 496156, 635053, 664033, 881565, 1011793, 1190685, 1306605, 1442136, 1923655, 1947766, 2286913, 2324422, 2465805, 3030733, 3291553, 3335205, 4100086, 4547353, 4648965, 4987065, 5025705, 5034904, 5069113, 5827485, 5909413
Offset: 1

Views

Author

Amiram Eldar, Sep 03 2019

Keywords

Comments

Are there 4 consecutive numbers with the same value of A070039, apart from 2, 3, 4, 5?
The next 4 consecutive numbers with the same value of A070039 start at 1190499411. - Giovanni Resta, Sep 05 2019

Examples

			2 is in the sequence since A070039(2) = A070039(3) = A070039(4) = 1.

Crossrefs

Cf. A070039, A324295, A325175.

Programs

  • Mathematica
    s[n_] := DivisorSum[n, # &, # < Sqrt[n] &]; seq={}; s1 = 0; s2=0; Do[s3 = s[n]; If[s1 == s2 && s2 == s3, AppendTo[seq, n - 2]]; s1 = s2; s2 = s3, {n, 2, 10^5}]; 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

Views

Author

Amiram Eldar, Sep 03 2019

Keywords

Comments

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

Examples

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

Crossrefs

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

Programs

  • Mathematica
    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

Extensions

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

Views

Author

Amiram Eldar, Sep 28 2019

Keywords

Links

Crossrefs

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

Programs

  • Mathematica
    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
  • PARI
    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

Formula

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

Views

Author

Amiram Eldar, Apr 18 2023

Keywords

Comments

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

Examples

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

Links

Crossrefs

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

Programs

  • Mathematica
    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] &]
  • PARI
    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); }
Showing 1-6 of 6 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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