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 1-7 of 7 results.

A077092 Fixed points of iteration in A077091.

Original entry on oeis.org

0, 4, 24, 8064, 34944, 35520, 38880, 69480, 268560, 420096, 1054944, 2946560, 3054080, 5660160, 6621120, 9768960, 10264320, 25885760, 29062656, 33933312, 36484992, 38707200, 78532608, 163418112, 260601600, 458987520, 4044349440
Offset: 1

Views

Author

Labos Elemer, Oct 31 2002

Keywords

Comments

When iteration of f(k) = phi(sigma(k)-phi(k)) is started at various initial values, not ending in cycles and converging, it ends at these fixed points.
a(28) <= 11435212800. a(29) <= 15083274240. a(30) <= 90215424000. - Donovan Johnson, Dec 14 2009

Examples

			n=30: FixedPointList={30,32,46,20,16,22,12,8,10,6,4},end=4; n=94: FixedPointList={94,42,24},end=24. n=41708: FixedPointList={41708,26064,32352,21216,15232,8064},end=8064; n=12100: FixedPointList={12100,24000,34944},end=34944.

Crossrefs

Cf. A077090, A077091, A077093, A077094, A077095, A077096, A077099, A077100.

Programs

  • Mathematica
    f[x_] := EulerPhi[DivisorSigma[1, x]-EulerPhi[x]] Do[s=NestList[f, n, 100]; s1=Part[s, 99]; s2=Part[s, 100]; If[Equal[s1, s2]&&!PrimeQ[n], Print[{n, s1}]], {n, 1, 1000}]

Extensions

a(9) corrected and a(11)-a(27) from Donovan Johnson, Dec 14 2009

A073815 Least number x such that gcd(phi(x), sigma(x)) = n.

Original entry on oeis.org

1, 3, 18, 12, 200, 14, 3364, 15, 722, 328, 9801, 42, 25281, 116, 1800, 165, 36992, 810, 4414201, 88, 196, 29161, 541696, 35, 2928200, 1413, 103968, 172, 98942809, 488, 1547536, 336, 19602, 17536, 814088, 370, 49042009, 55297, 1521, 319, 3150464641
Offset: 1

Views

Author

Labos Elemer, Nov 12 2002

Keywords

Comments

Values are frequently identical to terms of A077102. Since gcd(a,b) and gcd(a+b,a-b) may differ, so may the smallest solutions. A077102(m) and a(m) differ at m = 1, 2, 4, 8, 16, 28, 32, 40, etc.

Links

Crossrefs

Cf. A000203, A000010, A009223, A055008, A077099-A077102, A051612, A065387, A023897, A020492.

Programs

  • Mathematica
    f[x_] := Apply[GCD, {DivisorSigma[1, x], EulerPhi[x]}] t=Table[0, {100}]; Do[s=f[n]; If[s<101&&t[[s]]==0, t[[s]]=n], {n, 1, 10^13}];
  • PARI
    a(n)=my(x=n);while(gcd(eulerphi(x),sigma(x))!=n, x++); x \\ Charles R Greathouse IV, Dec 09 2013

Formula

a(n) = Min{x; A055008(x)=n}. a(n)=Min{x; gcd(A000203(x), A000010(x))=n}
a(n) = Min{x: A023897(x)= n}, smallest balanced number (A020492) for which the quotient equals n.

A077093 Smallest initial values leading to fixed points listed in A077092.

Original entry on oeis.org

1, 4, 24, 7530, 12100, 32784, 34950, 69480, 121104, 420096, 1018980, 2846484, 2946560, 4160040, 5387484, 5512800, 5729520, 13108800, 23524452, 24799080, 25885760, 31382952, 53344272, 100929036, 150057300, 352636452
Offset: 1

Views

Author

Labos Elemer, Oct 31 2002

Keywords

Examples

			n=12100: FixedPointList={12100,24000,34944},end=34944; n=121104: FixedPointList={121104,268560}, end=268560.

Crossrefs

Cf. A077090, A077091, A077092, A077094, A077095, A077096, A077099, A077100.

Programs

  • Mathematica
    f[x_] := EulerPhi[DivisorSigma[1, x]-EulerPhi[x]]; Do[s=NestList[f, n, 100]; s1=Part[s, 99]; s2=Part[s, 100]; If[Equal[s1, s2]&&!PrimeQ[n], Print[{n, s1}]], {n, 1, 1000}] (*n=site if fixed point appears; s1=fixed point*)

Extensions

a(11)-a(26) from Sean A. Irvine, May 06 2025

A077100 a(n) = lcm(A051612(n), A065387(n)), where A051612(n) = sigma(n) - phi(n) and A065387(n) = sigma(n) + phi(n).

Original entry on oeis.org

0, 4, 6, 45, 10, 70, 14, 209, 133, 154, 22, 96, 26, 90, 32, 897, 34, 495, 38, 850, 220, 598, 46, 884, 561, 270, 638, 748, 58, 320, 62, 3713, 476, 1330, 72, 8137, 74, 546, 160, 3922, 82, 756, 86, 832, 918, 2350, 94, 3780, 495, 8249, 520, 4514, 106, 2346, 224
Offset: 1

Views

Author

Labos Elemer, Nov 06 2002

Keywords

Links

Crossrefs

Cf. A000010, A000203, A051612, A065387, A077099, A077101.

Programs

  • Mathematica
    Table[LCM[#1 - #2, #1 + #2] & @@ {DivisorSigma[1, n], EulerPhi@ n}, {n, 55}] (* Michael De Vlieger, Dec 17 2016 *)
  • PARI
    a(n)=my(f=factor(n),e=eulerphi(f),s=sigma(f)); lcm(s+e,s-e) \\ Charles R Greathouse IV, Nov 27 2013

Formula

If p is prime, then a(p) = 2*p.

Extensions

Edited by Dean Hickerson, Nov 07 2002

A077101 a(n) = A051612(n)*A065387(n) = sigma(n)^2-phi(n)^2, where A051612(n) = sigma(n) - phi(n) and A065387(n) = sigma(n) + phi(n).

Original entry on oeis.org

0, 8, 12, 45, 20, 140, 28, 209, 133, 308, 44, 768, 52, 540, 512, 897, 68, 1485, 76, 1700, 880, 1196, 92, 3536, 561, 1620, 1276, 2992, 116, 5120, 124, 3713, 1904, 2660, 1728, 8137, 148, 3276, 2560, 7844, 164, 9072, 172, 6656, 5508, 4700, 188, 15120, 1485
Offset: 1

Views

Author

Labos Elemer, Nov 06 2002

Keywords

Comments

If n is prime, then a(n) = 4n.

Links

Crossrefs

Cf. A000010, A000203, A002117, A051612, A062354, A065387, A065464, A072861, A077099, A077100, A127473.

Programs

Formula

a(n) = A077099(n) * A077100(n). - Antti Karttunen, May 26 2017
From Amiram Eldar, Dec 04 2023: (Start)
a(n) = A072861(n) - A127473(n).
Sum_{k=1..n} a(k) ~ c * n^3 / 3, where c = 5*zeta(3)/2 - Product_{p prime}(1 - (2*p-1)/p^3) = (5/2)*A002117 - A065464 = 2.576892... . (End)

Extensions

Edited by Dean Hickerson, Nov 07 2002

A077102 Smallest number m such that GCD(a+b,a-b) = n, where a = sigma(m) and b = phi(m).

Original entry on oeis.org

4, 1, 18, 21, 200, 14, 3364, 12, 722, 328, 9801, 42, 25281, 116, 1800, 15, 36992, 810, 4414201, 88, 196, 29161, 541696, 35, 2928200, 1413, 103968, 284, 98942809, 488, 1547536, 364, 19602, 17536, 814088, 370, 49042009, 55297, 1521, 440, 3150464641
Offset: 1

Views

Author

Labos Elemer, Nov 12 2002

Keywords

Examples

			For n = 10, a(10) = 328, sigma(328) = 630, phi(328) = 160, sigma(328) + phi(328) = 790, sigma(328) - phi(328) = 470, GCD(790,470) = 10.
For n = odd number, a(n) should be either a square or twice a square and so faster search for large values is possible, like e.g., for n = 97: a(97) = 435979^2 is the smallest solution.

Links

Crossrefs

Cf. A000010, A000203, A051612, A065387, A077099, A077100, A077101.

Programs

  • Mathematica
    f[x_] := Apply[GCD, {DivisorSigma[1, x]+EulerPhi[x], DivisorSigma[1, x]-EulerPhi[x]}]; t=Table[0, {100}]; Do[s=f[n]; If[s<101&&t[[s]]==0, t[[s]]=n], {n, 1, 10^13}]; t
  • PARI
    lista(len) = {my(v = vector(len), c = 0, k = 1, a, b, i); while(c < len, f = factor(k); a = sigma(f); b = eulerphi(f); i = gcd(a+b,a-b); if(i <= len && v[i] == 0, c++; v[i] = k); k++); v;} \\ Amiram Eldar, Nov 14 2024

Formula

a(n) = Min{x; A077099(x) = n}.

A077103 Numbers n such that gcd(a,b) is not equal to gcd(a+b,a-b), where a=sigma(n)=A000203(n) and b=phi(n)=A000010(n).

Original entry on oeis.org

1, 2, 12, 15, 30, 39, 44, 55, 56, 76, 78, 87, 95, 99, 110, 111, 125, 140, 143, 147, 159, 171, 172, 174, 175, 183, 184, 190, 198, 215, 216, 222, 236, 247, 250, 252, 264, 268, 286, 287, 294, 295, 303, 315, 318, 319, 327, 332, 335, 336, 342, 350, 357, 363, 364
Offset: 1

Views

Author

Labos Elemer, Nov 12 2002

Keywords

Examples

			n=76: a=sigma(76)=140, b=phi(76)=36, a+b=176, a-b=104, gcd(a,b) = gcd(140,36) = 4 < gcd(a+b,a-b) = gcd(176,104) = 8.

Crossrefs

Cf. A000010, A000203, A051612, A065387, A055008, A077099.

Programs

  • Mathematica
    Do[s=GCD[a=DivisorSigma[1, n], b=EulerPhi[n]]; s1=GCD[a+b, a-b]; If[ !Equal[s, s1], Print[{n, a, b, a+b, a-b, s, s1, s1/s}]], {n, 1, 1000}]

Formula

gcd(A000010(n), A000203(n)) is not equal to gcd(A065387(n), A051612(n)); or A055008(n) is not equal to A077099(n).
Showing 1-7 of 7 results.

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