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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A109806 Admirable numbers such that the subtracted divisor is square.

Original entry on oeis.org

20, 56, 104, 368, 464, 650, 836, 992, 1952, 8415, 11096, 16256, 17816, 28544, 31815, 45356, 77744, 83312, 91388, 98048, 113072, 122624, 128768, 130304, 254012, 351351, 388076, 507392, 522752, 537248, 698528, 780975, 791264, 1081568
Offset: 1

Views

Author

Jason Earls, Aug 16 2005

Keywords

Examples

			a(2)=56 because 1+2+7+8+14+28-4 = 56 and the subtracted divisor is square.

Links

Crossrefs

Cf. A111592.

Programs

  • Mathematica
    aQ[n_] := (d = DivisorSigma[1, n] - 2n) > 0 && EvenQ[d] && Mod[n, d/2] == 0 && d < 2n && IntegerQ @ Sqrt[d/2]; Select[Range[10^5], aQ] (* Amiram Eldar, Sep 22 2019 *)

A135502 Admirable numbers in the middle of twin primes.

Original entry on oeis.org

12, 30, 42, 102, 138, 270, 282, 618, 642, 822, 1488, 1698, 1878, 2082, 2238, 2382, 2658, 2802, 3462, 3558, 3918, 4638, 4722, 5442, 6198, 6702, 8538, 8598, 9678, 10938, 12162, 12378, 12822, 12918, 13218, 13722, 13758, 13998, 14082, 16062, 17418
Offset: 1

Views

Author

Olaf Voß, Feb 09 2008

Keywords

Comments

Numbers n such that n is admirable, n-1 is prime and n+1 is prime.

Examples

			30 is in the sequence, as 29=30-1 and 31=30+1 are a pair of twin primes.

Links

Crossrefs

Cf. A111592, A001359, A006512.

Programs

  • Mathematica
    Select[Range[18000],MemberQ[Most[Divisors[#]],(DivisorSigma[1,#]-2#)/2]&&AllTrue[#+{1,-1},PrimeQ]&] (* Harvey P. Dale, Sep 11 2023 *)

Formula

A014574 INTERSECT A111592. - R. J. Mathar, Feb 10 2008

A282754 Admirable numbers such that the subtracted divisor is a Fibonacci number.

Original entry on oeis.org

12, 20, 40, 70, 88, 104, 464, 650, 1504, 1888, 1952, 4030, 5830, 7192, 7912, 8925, 9555, 10792, 13736, 17272, 30555, 30592, 32128, 32445, 78975, 130304, 442365, 521728, 522752, 1713592, 1848964, 4526272, 8353792, 8378368, 8382464, 9928792, 11547352, 17999992
Offset: 1

Views

Author

Michel Lagneau, Feb 21 2017

Keywords

Comments

Subsequence of A111592.
The corresponding Fibonacci numbers are given by the sequence {b(n)} = 2, 1, 5, 2, 2, 1, 1, 1, 8, 2, 1, 2, 2, 8, 8, 3, 21, 8, 34, 8, 21, 8, 2, 3, 13, 1, 3, 2, 1, ....

Examples

			40 is in the sequence because sigma(40) - 2*5 = 90 - 10 = 80 = 2*40, where 5 is a Fibonacci number, or 1 + 2 + 4 + 8 + 10 + 20 - 5 = 40 where the subtracted divisor is 5.

Links

Crossrefs

Cf. A000045, A010056, A111592.

Programs

  • Maple
    with(numtheory):
   for n from 1 to 20000 do:
     x:=divisors(n):n0:=nops(x):
       for i from 1 to n0 do:
         u:=sqrt(5*x[i]^2-4):v:=sqrt(5*x[i]^2+4):
          if (floor(u)=u or floor(v)=v) and sigma(n)-2*x[i]=2*n
             then
             printf(`%d %d \n`,n, x[i]):
             else
            fi:
         od:
   od:
  • Mathematica
    With[{nn = 10^6}, Function[s, Flatten@ Position[#, 1] &@ Table[Total@ Boole@ Map[MemberQ[s, #] &, Select[Most@ Divisors@ n, Function[d, DivisorSigma[1, n] - 2 d == 2 n]]], {n, nn}]]@ Fibonacci@ Range[2 + Floor@ Log[GoldenRatio, nn]]] (* Michael De Vlieger, Feb 24 2017 *) (* or *)
fibQ[n_] := IntegerQ@ Sqrt[5 n^2 + 4] || IntegerQ@ Sqrt[5 n^2 - 4]; ok[n_] := Block[{d = DivisorSigma[1, n] - 2 n}, d>0 && EvenQ@d && Mod[n, d/2] == 0 && fibQ[d/2]]; Select[Range[10^6], ok] (* faster, Giovanni Resta, Mar 10 2017 *)
  • PARI
    isadmirable(n)=if(issquare(n)||issquare(n/2), 0, my(d=sigma(n)/2-n); (d>0 && d!=n && n%d==0)*d);
isfib(n) = my(k=n^2); k+=(k+1)<<2; issquare(k) || (n>0 && issquare(k-8))
isok(n) = (d=isadmirable(n)) && isfib(d); \\ Michel Marcus, Mar 10 2017

Extensions

More terms from Michel Marcus, Mar 10 2017

A290703 Solutions to x + d(x) = sigma(x)/2, where d(x) is the number of divisors of x.

Original entry on oeis.org

176, 2205, 7544, 10184, 28544, 503296, 1020568, 2051072, 6019264, 10001848, 15317696, 35019968, 354375344, 535871488, 9175990784, 115917884416
Offset: 1

Views

Author

Paolo P. Lava, Aug 09 2017

Keywords

Examples

			d(176) = 10, sigma(176) = 372 and 176 + 10 = 186 = 372/2.

Crossrefs

Cf. A000005, A099650, A111592, A290704.

Programs

  • Maple
    with(numtheory): P:=proc(q) local n; for n from 1 to q do if
n+tau(n)=sigma(n)/2 then print(n); fi; od; end: P(10^9);
  • Mathematica
    Select[Range[10^7], # + DivisorSigma[0, #] == DivisorSigma[1, #]/2 &] (* Michael De Vlieger, Aug 26 2017 *)
  • PARI
    isok(x) = x + numdiv(x) == sigma(x)/2; \\ Michel Marcus, Aug 25 2017

Extensions

a(11)-a(12) from Michel Marcus, Aug 25 2017
a(13)-a(16) from Giovanni Resta, Aug 25 2017

A290704 Solutions to 2*x - phi(x) = sigma(x)/2, where phi(x) is the Euler totient function of x.

Original entry on oeis.org

1680, 4200, 27000, 175392, 282960, 707400, 1668480, 3344544, 5658480, 14146200, 48644064, 90008880, 130110624, 225022200, 357994728, 460763160, 607281696, 1926458352, 3830537880, 5857651296, 7840881216, 8414628480, 8704032876, 8843224500, 14279194512, 29522053080
Offset: 1

Views

Author

Paolo P. Lava, Aug 09 2017

Keywords

Examples

			phi(1680) = 384, sigma(1680) = 5952 and 2*1680 - 384 = 2976 = 5952/2.

References

  • Like A099650 but with totient phi(x) replaced by cototient x - phi(x).

Crossrefs

Cf. A000010, A051953, A099650, A111592, A290703.

Programs

  • Maple
    with(numtheory): P:=proc(q) local n; for n from 1 to q do
if 2*n-phi(n)=sigma(n)/2 then print(n); fi; od; end: P(10^9);

Extensions

a(10)-a(26) from Giovanni Resta, Aug 25 2017

A335121 Admirable totient numbers: numbers that are equal to the sum of their iterated phi, with one of them taken with a minus sign.

Original entry on oeis.org

5, 7, 33, 35, 55, 87, 95, 175, 201, 215, 219, 245, 531, 747, 927, 939, 1047, 1295, 1463, 1473, 1551, 1855, 2015, 2103, 2421, 2431, 2547, 2619, 2631, 2765, 3535, 4833, 5067, 5215, 7655, 7743, 7851, 10503, 11127, 11307, 13055, 13707, 16247, 16593, 17805, 18471
Offset: 1

Views

Author

Amiram Eldar, May 24 2020

Keywords

Comments

Analogous to A111592 (admirable numbers) as A082897 (perfect totient numbers) is analogous to A000396 (perfect numbers).

Examples

			5 is a term since the values of the iterated phi of 5 are 4, 2 and 1 and 5 = 4 + 2 - 1.

Links

Crossrefs

Subsequence of A286265.
Cf. A000010, A053478, A082897, A111592, A288452.

Programs

  • Mathematica
    admTotQ[n_] := Module[{s = Most @ Rest @ FixedPointList[EulerPhi, n]}, (ab = Plus @@ s - n) > 0 && EvenQ[ab] && ab/2 < n && MemberQ[s, ab/2]]; Select[Range[8000], admTotQ]

A335196 Nonunitary admirable numbers: numbers k such that there is a nonunitary divisor d of k such that nusigma(k) - 2*d = k, where nusigma is the sum of nonunitary divisors function (A048146).

Original entry on oeis.org

48, 80, 96, 108, 120, 160, 168, 180, 192, 216, 224, 252, 264, 280, 300, 312, 320, 336, 352, 360, 384, 396, 408, 416, 432, 448, 456, 468, 480, 504, 528, 540, 552, 560, 600, 612, 624, 640, 672, 684, 696, 704, 720, 744, 756, 768, 792, 816, 828, 832, 840, 864, 880
Offset: 1

Views

Author

Amiram Eldar, May 26 2020

Keywords

Comments

Equivalently, numbers that are equal to the sum of their nonunitary divisors, with one of them taken with a minus sign.

Examples

			48 is a term since 48 = 2 - 4 + 6 + 8 + 12 + 24 is the sum of its nonunitary divisors with one of them, 4, taken with a minus sign.

Links

Crossrefs

The nonunitary version of A111592.
Subsequence of A064597.
Similar sequences: A328328, A334972, A334974.
Cf. A048146.

Programs

  • Mathematica
    usigma[1] = 1; usigma[n_] := Times @@ (1 + Power @@@ FactorInteger[n]); nusigma[n_] := DivisorSigma[1, n] - usigma[n]; nuAdmQ[n_] := (ab = nusigma[n] - n) > 0 && EvenQ[ab] && ab/2 < n && !CoprimeQ[ab/2, 2*n/ab]; Select[Range[1000], nuAdmQ]

A364728 Numbers that are not the sum of admirable numbers (not necessarily distinct).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 18, 19, 21, 22, 23, 25, 26, 27, 28, 29, 31, 33, 34, 35, 37, 38, 39, 41, 43, 45, 46, 47, 48, 49, 51, 53, 55, 57, 58, 59, 61, 63, 65, 67, 69, 71, 73, 75, 77, 79, 81, 83, 85, 87, 89, 91, 93, 95, 97, 99, 101
Offset: 1

Views

Author

Amiram Eldar, Aug 05 2023

Keywords

Comments

First differs from A053460 at n = 39.
Giovanni Resta found that 1003 is the largest number that is not a sum of admirable numbers.

Links

Crossrefs

Cf. A053460, A111592.
Analogous sequence with abundant numbers: A283550.

Programs

  • Mathematica
    admQ[n_] := (ab = DivisorSigma[1, n] - 2 n) > 0 && EvenQ[ab] && ab/2 < n && Divisible[n, ab/2];
With[{adm = Select[Range[1200], admQ]}, Position[Rest[CoefficientList[Series[Product[(1 + x^adm[[k]]), {k, 1, Length[adm]}], {x, 0, adm[[-1]]}], x]], 0] // Flatten]

A109547 Admirable oblong numbers.

Original entry on oeis.org

12, 20, 30, 42, 56, 650, 812, 992, 16256, 67100672, 17179738112, 274877382656, 1125625079263232, 4611686016279904256
Offset: 1

Views

Author

Jason Earls, Aug 30 2005

Keywords

Comments

a(15) <= 18889393874090701094912. - Donovan Johnson, Jun 20 2011

Crossrefs

Intersection of A002378 and A111592.

Extensions

a(12)-a(13) from Donovan Johnson, Jul 15 2009
a(14) from Donovan Johnson, Jun 21 2011

A109727 Number of admirable numbers < 10^n.

Original entry on oeis.org

0, 13, 65, 379, 2549, 19480, 159001, 1345929, 11676912, 103136395
Offset: 1

Views

Author

Jason Earls, Aug 10 2005

Keywords

Crossrefs

Cf. A111592.

Programs

  • Mathematica
    fQ[n_] := Block[{d = Most[Divisors[n]], k = 1}, l = Length[d]; s = Plus @@ d; While[k < l && s - 2d[[k]] > n, k++ ]; If[k > l || s != n + 2d[[k]], False, True]]; c = 0; k = 1; Do[ While[k <= 10^n, If[ fQ[k], c++ ]; k++ ]; k++; Print[c], {n, 7}] (* Robert G. Wilson v *)

Extensions

a(7) from Robert G. Wilson v, Aug 13 2005
a(8)-a(10) from Donovan Johnson, Jan 25 2012
Previous Showing 31-40 of 44 results. Next

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