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 1-4 of 4 results.

A350934 a(n) is the smallest number m such that tau(m - 1) = tau(m + 1) = tau(m) + n or 0 if no such m exists, where tau(k) = A000005(k).

Original entry on oeis.org

34, 9, 7, 964324, 19, 3822025, 41, 15129, 341, 427166224, 199, 700569, 1241, 11923111249, 919, 12376324, 6641, 34539129, 12221, 363016809, 3401, 56776225, 5741, 199809, 52865, 48045571249, 47081, 3764067904, 21113, 19035769, 18089, 145371249, 59291, 2219069449
Offset: 0

Views

Author

Jaroslav Krizek, Jan 25 2022

Keywords

Comments

Corresponding values of tau(a(n)): 4, 3, 2, 9, 2, 27, 2, 9, 4, 15, 2, 21, 4, 27, 2, 9, 4, 15, 6, 45, 4, 27, 2, 9, 8, 15, 6, 21, 4, 3, 2, 9, 4, 15, ...
Triples of [tau(a(n) - 1), tau(a(n)), tau(a(n) + 1)] = [tau(a(n)) + n, tau(a(n)), tau(a(n)) + n]: [4, 4, 4], [4, 3, 4], [4, 2, 4], [12, 9, 12], [6, 2, 6], [32, 27, 32], [8, 2, 8], [16, 9, 16], [12, 4, 12], ...
If n is odd then a(n) is a square. - Amiram Eldar, Jan 26 2022

Examples

			a(3) = 964324 because 964324 is the smallest number m such that tau(m-1) = tau(m+1) = tau(m)+3; tau(964323) = tau(964325) = tau(964324)+3 = 9+3 = 12.

Crossrefs

Cf. A000005, A190646, A350132, A350935, A350936.

Programs

  • Magma
    Ax:=func; [Ax(n): n in [0..8]]
  • Mathematica
    seq[m_, nmax_] := Module[{s = Table[0, {m + 1}], c = 0, d1 = 1, d2 = 2, n = 3, d, k}, While[c < m + 1 && n < nmax, d = DivisorSigma[0, n]; If[d1 == d, k = d - d2 + 1; If[k >= 1 && k <= m + 1 && s[[k]] == 0, s[[k]] = n - 1; c++]]; n++; d1 = d2; d2 = d]; TakeWhile[s, # > 0 &]]; seq[8, 10^7] (* Amiram Eldar, Jan 26 2022 *)

Extensions

More terms from Amiram Eldar, Jan 26 2022

A350935 a(n) is the smallest number m such that tau(m-1) = tau(m+1) = n*tau(m) or 0 if no such m exists, where tau(k) = A000005(k).

Original entry on oeis.org

34, 7, 19, 41, 6641, 199, 640063, 919, 17299, 22193, 350632961, 5741, 57394565119, 2345921, 3568049, 18089, 55171346530303, 41651, 193405731995647, 252881, 88099649, 1439024129, 916791443027132417, 90271, 821128751, 20969598977, 3959299, 2319679, 190190725057515297439745, 7860401
Offset: 1

Views

Author

Jaroslav Krizek, Jan 25 2022

Keywords

Comments

Corresponding values of tau(a(n)): 4, 2, 2, 2, 4, 2, 4, 2, 2, 2, 4, 2, 8, 2, 4, 2, 8, 2, 4, 2, 4, 4, 8, 2, 4, 4, 2, 2, 8, 2, ...
Triples of [tau(a(n) - 1), tau(a(n)), tau(a(n) + 1)] = [n * tau(a(n)), tau(a(n)), n * tau(a(n))]: [4, 4, 4], [4, 2, 4], [6, 2, 6], [8, 2, 8], [20, 4, 20], [12, 2, 12], [28, 4, 28], [16, 2, 16], [18, 2, 18], [20, 2, 20], ...

Examples

			a(3) = 19 because 19 is the smallest number m such that tau(m-1) = tau(m+1) = 3 * tau(m); tau(18) = tau(20) = 3 * tau(19) = 3 * 2 = 6.

Crossrefs

Cf. A000005, A190646, A350132, A350934, A350936.

Programs

  • Magma
    Ax:=func; [Ax(n): n in [1..10]]

Extensions

a(11)-a(16) from Jon E. Schoenfield, Jan 26 2022
More terms from Jon E. Schoenfield and David A. Corneth, Jan 27 2022

A350936 a(n) is the smallest number m such that tau(m) = n*tau(m-1) = n*tau(m+1) or 0 if no such m exists, where tau(k) = A000005(k).

Original entry on oeis.org

34, 6, 12, 30, 816, 60, 192, 270, 180, 240, 56320, 420, 233472, 2112, 1620, 1320, 2162688, 2340, 786432, 3120, 4800, 15360, 62914560, 3360, 172368, 724992, 6300, 29760, 24964497408, 12240, 35433480192, 7560, 599040, 15138816, 81648, 21600, 7215545057280
Offset: 1

Views

Author

Jaroslav Krizek, Jan 25 2022

Keywords

Comments

Corresponding values of tau(a(n)): 4, 4, 6, 8, 20, 12, 14, 16, 18, 20, 44, 24, 52, 28, 30, 32, 68, 36, 38, 40, 42, 44, 92, 48, 100, 52, 54, 56, 116, 60, 124, 64, 132, 136, 70, 72, 296, ...
Triples of [tau(a(n) - 1), tau(a(n)), tau(a(n) + 1)] = [tau(a(n)) / n, tau(a(n)), tau(a(n)) / n]: [4, 4, 4], [2, 4, 2], [2, 6, 2], [2, 8, 2], [4, 20, 4], [2, 12, 2], [2, 14, 2], [2, 16, 2], [2, 18, 2], [2, 20, 2], [4, 44, 4], ...

Examples

			a(3) = 12 because 12 is the smallest number m such that tau(m) = 3 * tau(m-1) = 3 * tau(m+1); tau(12) = 3 * tau(11) = 3 * tau(13) = 3 * 2 = 6.

Crossrefs

Cf. A000005, A190646, A350132, A350934, A350935.

Programs

  • Magma
    Ax:=func; [Ax(n): n in [1..16]]

Extensions

a(23)-a(37) from Jon E. Schoenfield, Jan 25 2022

A190821 Prime numbers p where d(p-1) = d(p+1) increases to a record.

Original entry on oeis.org

7, 19, 41, 199, 919, 5741, 18089, 41651, 90271, 446081, 1276001, 27033161, 43220449, 53308529, 109245401, 512669249, 663929729, 2266639649, 2560742911, 2969200961, 8505402751, 32540473601, 61573368401, 74335064959, 109494811999
Offset: 1

Views

Author

Juri-Stepan Gerasimov, May 21 2011

Keywords

Comments

a(26) <= 354208192001. - Donovan Johnson, Jun 03 2011

Examples

			a(1) = 7 because 7 is prime and d(6) = 4 = d(8).

Crossrefs

Cf. A145337, A190646 (numbers n such that d(n-1)=d(n+1) increases to a record).

Programs

  • Mathematica
    s = Select[Prime@ Range@ 1000000, DivisorSigma[0, # - 1] == DivisorSigma[0, # + 1] &]; t = DivisorSigma[0, # - 1] & /@ s; a = {0}; b = {0}; Do[If[t[[k]] > Max@ b, AppendTo[a, s[[k]]]]; AppendTo[b, t[[k]]], {k, Length@ s}]; a (* Michael De Vlieger, Oct 30 2015 *)
  • PARI
    r=0; forprime(p=2,4e9,t=numdiv(p-1);if(t>r&t==numdiv(p+1),r=t; print1(p", "))) \\ Charles R Greathouse IV, May 27 2011

Extensions

a(14)-a(21) from Charles R Greathouse IV, May 27 2011
a(22) from Charles R Greathouse IV, May 31 2011
a(23)-a(25) from Donovan Johnson, Jun 03 2011
Showing 1-4 of 4 results.

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