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.

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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

Views

Author

Amiram Eldar, Feb 10 2023

Keywords

Comments

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.

Examples

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

Links

Crossrefs

Cf. A360522.
Similar sequences: A002961, A064115, A064125, A293183, A306985.

Programs

  • Mathematica
    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
  • PARI
    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

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); }

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

Views

Author

Amiram Eldar, Dec 14 2024

Keywords

Links

Crossrefs

Cf. A241405.
Similar sequences: A002961, A064115, A064125, A293183, A306985.

Programs

  • Mathematica
    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] &]
  • PARI
    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

Views

Author

Amiram Eldar, Apr 11 2020

Keywords

Comments

a(7) > 6*10*8.

Examples

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

Links

Crossrefs

Cf. A330575.
Similar sequences: A002961, A064115, A064125, A293183, A306985.

Programs

  • Mathematica
    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

Views

Author

Amiram Eldar, Oct 26 2021

Keywords

Comments

Numbers k such that A160135(k) = A160135(k+1).
a(9) > 1.6 * 10^11, if it exists.

Examples

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

Crossrefs

Cf. A160135.
Similar sequences: A002961, A064115, A064125, A293183, A306985, A348346.

Programs

  • Mathematica
    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] &]
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