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 1-9 of 9 results.

A101373 a(n) = A063740(A100827(n)).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 15, 19, 20, 22, 25, 28, 31, 34, 41, 42, 46, 52, 58, 59, 69, 74, 77, 83, 93, 99, 116, 130, 138, 140, 156, 165, 166, 167, 173, 192, 200, 218, 219, 223, 241, 242, 271, 276, 292, 304, 331
Offset: 1

Views

Author

Alonso del Arte, Jan 06 2005

Keywords

Comments

Record values attained by the highly cototient numbers. - Amiram Eldar, Apr 08 2023

Links

Crossrefs

Cf. A063740, A100827, A131934.

Extensions

More terms from Robert G. Wilson v, Jan 08 2005
Name corrected by Amiram Eldar, Apr 08 2023

A105440 Highly cototient numbers that are prime, or intersection of A000040 and A100827.

Original entry on oeis.org

2, 23, 47, 59, 83, 89, 113, 167, 269, 389, 419, 509, 659, 839, 1049, 1259, 1889, 2099, 2309, 2729, 3359, 3989, 4289, 4409, 5879, 6089, 6719, 9029, 9239, 10289, 10709, 11549, 13649, 13859, 15329, 15959, 20789, 21839, 23099, 25409, 27299, 30029, 34649, 43889, 51869
Offset: 1

Views

Author

Paul Muljadi, Sep 22 2005

Keywords

Comments

2 is definitely a "highly cototient prime". Oddly enough, though, PrimeFan does not list it as safe prime that is also a highly cototient number in RNTFS00013 even though he considers 1 to be a prime. - Robert Happelberg (roberthappelberg(AT)yahoo.com), Oct 04 2005
115 of the first 229 terms (just over 50%) of A100827 are prime. - Jud McCranie, May 18 2017

Examples

			269 (after 59 & 89) is the first prime in a sequence of primes which are congruent to 9 mod 10.

Links

Crossrefs

Cf. A000040, A100827.

Extensions

More terms from Robert Happelberg (roberthappelberg(AT)yahoo.com), Oct 04 2005
More terms from T. D. Noe, Mar 16 2010
More terms from Amiram Eldar, Dec 24 2020

A287056 a(n) is such that A100827(n) = A082917(n - a(n)) - 1, or -1 if there is no corresponding term.

Original entry on oeis.org

-1, -1, -1, -1, -1, 1, 1, -1, 1, 1, 1, 1, 1, 0, -1, 1, 1, 1, 1, 1, 1, -1, 2, 2, 1, -1, 1, 1, 1, 1, 1, 1, 1, 1, 1, -1, 2, 2, 2, -1, 3, 3, 3, 3, -1, 4, 4, -1, 5, 5, 5, 5, 5, 5, 5, 5, 5, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 2, 2, 2, 2, 2, 2
Offset: 1

Views

Author

Jud McCranie, May 18 2017

Keywords

Comments

Most of the known terms of A100827 are 1 less than a term in A082917, and conversely. This sequences looks at the location in the sequence of the corresponding terms. Negative terms do not occur among the known terms of this sequence. When a(n+1) is different from a(n) (and both are nonnegative), there are |a(n+1)-a(n)| terms in one of the sequences that aren't in the other. With some irregularities, this sequence generally gradually increases at first, reaching a(49)=5. Then there are 9 a(n)=5, followed by 20 a(n)=4, followed by 30 a(n)=3, and then a(n)=2 for n=108 to 229. What is the behavior of the rest of the sequence? Does it stay at a(n)=2?

Examples

			Examples: A100827(6)=47, A082917(5)=47+1, so a(6) = 6-5 = 1. A100827(23)=779, A082917(21)=779+1, so a(23) = 23-21 = 2.

Links

Crossrefs

Cf. A082917, A100827.

A362183 Unitary highly cototient numbers: numbers k that have more solutions x to the equation A323410(x) = k than any smaller k.

Original entry on oeis.org

0, 6, 10, 20, 31, 47, 53, 65, 77, 89, 113, 119, 149, 167, 179, 209, 293, 299, 329, 359, 389, 419, 479, 509, 599, 629, 779, 839, 989, 1049, 1139, 1259, 1469, 1559, 1649, 1679, 1889, 2099, 2309, 2729, 3149, 3359, 3569, 3989, 4289, 4409, 4619, 5249, 5459, 6089, 6509
Offset: 1

Views

Author

Amiram Eldar, Apr 10 2023

Keywords

Comments

Indices of records of A362181.
The corresponding numbers of solutions are 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 16, 17, 21, ... (A362184).

Links

Crossrefs

Cf. A323410, A362181, A362184.
The unitary version of A100827.
Similar sequences: A097942, A361968.

Programs

  • Mathematica
    ucototient[n_] := n - Times @@ (Power @@@ FactorInteger[n] - 1); ucototient[1] = 0; With[{max = 300}, solnum = Table[0, {n, 1, max}]; Do[If[(i = ucototient[k]) <= max, solnum[[i]]++], {k, 2, max^2}]; s = {0}; solmax=1; Do[sol = solnum[[k]]; If[sol > solmax, solmax = sol; AppendTo[s, k]], {k, 2, max}]; s]

A362213 Irregular table read by rows in which the n-th row consists of all the numbers m such that cototient(m) = n, where cototient is A051953.

Original entry on oeis.org

4, 9, 6, 8, 25, 10, 15, 49, 12, 14, 16, 21, 27, 35, 121, 18, 20, 22, 33, 169, 26, 39, 55, 24, 28, 32, 65, 77, 289, 34, 51, 91, 361, 38, 45, 57, 85, 30, 95, 119, 143, 529, 36, 40, 44, 46, 69, 125, 133, 63, 81, 115, 187, 52, 161, 209, 221, 841, 42, 50, 58, 87, 247, 961
Offset: 2

Views

Author

Amiram Eldar, Apr 11 2023

Keywords

Comments

The offset is 2 since cototient(p) = 1 for all primes p.
The 0th row consists of one term, 1, since 1 is the only solution to cototient(x) = 0.

Examples

			The table begins:
  n   n-th row
  --  -----------
   2  4;
   3  9;
   4  6, 8;
   5  25;
   6  10;
   7  15, 49;
   8  12, 14, 16;
   9  21, 27;
  10
  11  35, 121;
  12  18, 20, 22;

Links

Crossrefs

Cf. A005278, A051953, A063507, A063740 (row lengths), A063741, A063742, A063748, A100827, A101373, A131825, A131826.
Similar sequences: A032447, A361966, A362180.

Programs

  • Mathematica
    With[{max = 50}, cot = Table[n - EulerPhi[n], {n, 1, max^2}]; row[n_] := Position[cot, n] // Flatten; Table[row[n], {n, 2, max}] // Flatten]

A362487 Infinitary highly totient numbers: numbers k that have more solutions x to the equation iphi(x) = k than any smaller k, where iphi is the infinitary totient function A091732.

Original entry on oeis.org

1, 6, 12, 24, 48, 96, 144, 240, 288, 480, 576, 720, 1152, 1440, 2880, 4320, 5760, 8640, 11520, 17280, 34560, 51840, 69120, 103680, 120960, 172800, 207360, 241920, 345600, 362880, 414720, 483840, 725760, 967680, 1209600, 1451520, 1935360, 2419200, 2903040, 3628800
Offset: 1

Views

Author

Amiram Eldar, Apr 22 2023

Keywords

Comments

Indices of records of A362485.
The corresponding numbers of solutions are 2, 4, 6, 10, 14, 18, 22, ... (A362488).

Links

Crossrefs

Cf. A091732, A362484, A362485, A362486, A362488.
Similar sequences: A097942, A100827, A361968, A362183.

Programs

  • Mathematica
    solnum[n_] := Length[invIPhi[n]]; seq[kmax_] := Module[{s = {}, solmax=0}, Do[sol = solnum[k]; If[sol > solmax, solmax = sol; AppendTo[s, k]], {k, 1, kmax}]; s]; seq[10^4] (* using the function invIPhi from A362484 *)

A362402 Positive numbers m such that a record number of numbers k have m as the sum of divisors of k that have a square factor (A162296).

Original entry on oeis.org

1, 4, 48, 72, 216, 288, 864, 1440, 1728, 2880, 3456, 4320, 5184, 5760, 8640, 12096, 17280, 25920, 34560, 48384, 51840, 69120, 103680, 120960, 155520, 181440, 207360, 241920, 311040, 362880, 483840, 622080, 725760, 967680, 1088640, 1209600, 1451520, 2177280, 2903040
Offset: 1

Views

Author

Amiram Eldar, Apr 18 2023

Keywords

Comments

The value 0 appears in the range of A162296 for all squarefree numbers (A005117) and therefore it is excluded from this sequence.
The corresponding record values are in A362403.
Except for 1, a subsequence of A362401.

Links

Crossrefs

Cf. A162296, A362401, A362403.
Similar sequences: A097942, A100827, A145899, A238895.

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)]; s[1] = 0; seq[max_] := Module[{v = Select[Array[s, max], 0 < # <= max &], sq = {1}, t, tmax = 0}, t = Sort[Tally[v]]; Do[If[t[[k]][[2]] > tmax, tmax = t[[k]][[2]]; AppendTo[sq, t[[k]][[1]]]], {k, 1, Length[t]}]; sq]; seq[10^5]
  • 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(v = vector(kmax), vmax = 0, i); for(k=1, kmax, i = s(k); if(i > 0 && i <= kmax, v[i]++)); print1(1, ", "); for(k=1, kmax, if(v[k] > vmax, vmax = v[k]; print1(k, ", "))); }

A209193 Safe primes that are also highly cototient numbers.

Original entry on oeis.org

23, 47, 59, 83, 167, 839, 2099, 5879, 6719, 23099, 27299, 180179, 414959, 420419, 628319, 660659, 840839, 1141139, 1681679, 2042039, 2277659, 4084079, 5525519, 5705699
Offset: 1

Views

Author

N. J. A. Sloane, Mar 05 2012

Keywords

Links

Crossrefs

A005385 intersect A100827.

Extensions

a(19)-a(24) calculated from the b-file at A100827 by Amiram Eldar, Apr 09 2020

A209194 Sophie Germain primes that are also highly cototient numbers.

Original entry on oeis.org

2, 23, 83, 89, 113, 419, 509, 659, 1049, 1889, 3359, 4409, 9029, 10709, 11549, 13649, 20789, 23099, 57329, 84629, 90089, 113189, 115499, 207479, 210209, 270269, 314159, 324869, 330329, 420419, 570569, 658349, 660659, 810809, 840839, 1021019, 1138829
Offset: 1

Views

Author

N. J. A. Sloane, Mar 05 2012

Keywords

Links

Crossrefs

A005384 INTERSECT A100827.
Showing 1-9 of 9 results.

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

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