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.

Previous Showing 31-38 of 38 results.

A065555 Numbers n such that phi(phi(n)) = phi(sigma(n)) where phi is Euler's totient and sigma is the multiplicative sum-of-divisors function.

Original entry on oeis.org

1, 5, 11, 71, 145, 319, 323, 377, 779, 865, 911, 1007, 1073, 1167, 1195, 1343, 1441, 1585, 1609, 1691, 1903, 2117, 2147, 2249, 2591, 2629, 2723, 2987, 3013, 3107, 3239, 3247, 3265, 3383, 3487, 3569, 3777, 3791, 3827, 4121, 4199, 4339, 5249, 5455, 5597
Offset: 1

Views

Author

Walter Nissen, Nov 28 2001

Keywords

Examples

			5 is in the sequence because phi(5) = 4, sigma(5) = 6, phi(4) = 2 = phi(6).

Links

Crossrefs

Cf. A033632, A000010, A000203.

Programs

  • PARI
    { n=0; for (m=1, 10^9, if (eulerphi(eulerphi(m)) == eulerphi(sigma(m)), write("b065555.txt", n++, " ", m); if (n==1000, return)) ) } \\ Harry J. Smith, Oct 22 2009

A065556 Numbers n such that sigma (phi ( n ) ) = sigma (sigma (n ) ) where phi is Euler's totient and sigma is the multiplicative sum-of-divisors function.

Original entry on oeis.org

1, 367, 919, 1967, 3641, 4379, 5143, 7379, 11843, 12767, 13493, 15293, 21797, 26039, 28381, 29807, 30263, 30593, 30599, 30887, 37523, 40199, 48559, 49781, 51101, 51397, 55277, 62573, 67223, 72433, 73979, 87047, 89255, 89851, 95393
Offset: 1

Views

Author

Walter Nissen, Nov 28 2001

Keywords

Examples

			367 is in the sequence because phi(367) = 366, sigma(367) = 368, sigma(366) = 744 = sigma(368).

Links

Crossrefs

Cf. A033632, A000010, A000203.

Programs

  • Mathematica
    Select[Range[100000],DivisorSigma[1,EulerPhi[#]]==DivisorSigma[ 1, DivisorSigma[1,#]]&] (* Harvey P. Dale, Jun 23 2013 *)
  • PARI
    { n=0; for (m=1, 10^9, if (sigma(eulerphi(m)) == sigma(sigma(m)), write("b065556.txt", n++, " ", m); if (n==1000, return)) ) } \\ Harry J. Smith, Oct 22 2009

Formula

sigma(phi(n)) = sigma(sigma(n)).

A228434 Primes expressible as sigma(n) + sigma(sigma(n)), in order of their occurrence.

Original entry on oeis.org

2, 7, 11, 23, 127, 167, 137, 269, 547, 547, 383, 547, 269, 431, 547, 547, 293, 383, 431, 1171, 1039, 1171, 641, 1039, 1103, 1171, 887, 1361, 2551, 1861, 3001, 2753, 1193, 2963, 1499, 2153, 2753, 2551, 2963, 4327, 5281, 1823, 2963, 4219, 4327, 3593, 3583, 6763
Offset: 1

Views

Author

K. D. Bajpai, Nov 10 2013

Keywords

Examples

			a(6)= 167: sigma(32)+sigma(sigma(32))= 63+104= 167, which is prime.
a(11)= 383: sigma(93)+sigma(sigma(93))= 128+255= 383, which is prime.

Links

Crossrefs

Cf. A000203 (sigma(n): sum of divisors of n).
Cf. A019279 (superperfect numbers: sigma(sigma(n))).
Cf. A033632 (numbers n: sigma(n)is prime).
Cf. A051027 (a(n)= sigma(sigma(n))).

Programs

  • Maple
    with(numtheory):KD := proc() local a; a:= sigma(n)+sigma(sigma(n));if isprime(a) then RETURN (a);fi;end:seq(KD(),n=1..5000);

A243996 Numbers n such that phi(sigma*(n)) = sigma*(phi(n)), where sigma*(n) is the sum of anti-divisors of n and phi(n) is the Euler totient function.

Original entry on oeis.org

7, 9, 20, 25, 80, 143, 825, 3117, 3216, 22774, 52026, 55804, 138276, 187733, 228384, 265545, 320766, 549540, 830814, 839784, 901376, 1293552, 1315776, 2635866, 6771114, 11126800, 12087848, 24351460, 49382242, 52344292, 60063744, 65980038, 78279016, 97638080
Offset: 1

Views

Author

Paolo P. Lava, Jun 18 2014

Keywords

Comments

a(70) > 10^10. - Hiroaki Yamanouchi, Sep 28 2015

Examples

			sigma*(phi(25)) = sigma*(20) = 24, phi(sigma*(25)) = phi(39) = 24.

Links

Crossrefs

Cf. A000203, A066417, A230373, A033632.

Programs

  • Maple
    with(numtheory): P:=proc(q) local a,b,c,d,j,k,n;
for n from 1 to q do
k:=0; j:=n; while j mod 2<>1 do k:=k+1; j:=j/2; od;
a:=sigma(2*n+1)+sigma(2*n-1)+sigma(n/2^k)*2^(k+1)-6*n-2;
k:=0; c:=phi(n); j:=phi(n); while j mod 2<>1 do k:=k+1; j:=j/2; od;
b:=sigma(2*c+1)+sigma(2*c-1)+sigma(c/2^k)*2^(k+1)-6*c-2;
if b=phi(a) then print(n); fi; od; end: P(10^10);
  • Mathematica
    antiDivisors[n_] := Select[ Union[ Join[ Select[ Divisors[2 n - 1], OddQ[#] && # != 1 &], Select[ Divisors[ 2n + 1], OddQ[#] && # != 1 &], 2n/Select[ Divisors[ 2n], OddQ[#] && # != 1 &]]], # < n &]; fQ[n_] := EulerPhi@ Total@ antiDivisors@ n == Total@ antiDivisors@ EulerPhi@ n; k = 3; lst = {}; While[k < 10000001, If[ fQ@ k, AppendTo[lst, k]]; k++]; lst (* Robert G. Wilson v, Jun 21 2014 *)

Extensions

a(22)-a(25) from Robert G. Wilson v, Jun 21 2014
a(26)-a(34) from Hiroaki Yamanouchi, Sep 28 2015

A271633 Numbers k such that sigma(phi(k)) - phi(k) = phi(sigma(k)), where phi(k) is the Euler totient function of k and sigma(k) is the sum of the divisors of k.

Original entry on oeis.org

21, 350, 366, 532, 702, 1072, 5264, 7128, 23604, 24102, 30222, 30636, 32142, 32274, 34350, 47338, 70722, 78530, 113550, 137214, 197316, 235624, 292206, 357490, 367704, 398346, 406596, 453096, 453264, 464820, 479880, 485460, 504966, 509124, 512430, 519870, 539220
Offset: 1

Views

Author

Paolo P. Lava, Apr 19 2016

Keywords

Examples

			sigma(phi(21)) - phi(21) = 28 - 12 = 16 = phi(sigma(21)).

Links

Crossrefs

Cf. A000010, A000203, A001065, A033632, A062401.

Programs

  • Maple
    with(numtheory): P:= proc(q) local n; for n from 1 to q do
if sigma(phi(n))-phi(n)=phi(sigma(n)) then print(n); fi;
od; end: P(10^6);
  • Mathematica
    Select[Range[10^6], DivisorSigma[1, #] - # &@ EulerPhi@ # == EulerPhi@ DivisorSigma[1, #] &] (* Michael De Vlieger, Apr 21 2016 *)
  • PARI
    isok(k) = my(x=eulerphi(k)); sigma(x) - x == eulerphi(sigma(k)); \\ Michel Marcus, Jul 13 2019

A329730 Numbers k such that usigma(uphi(k)) = uphi(usigma(k)), where usigma is the sum of unitary divisors function (A034448) and uphi is the unitary totient function (A047994).

Original entry on oeis.org

1, 3, 4, 8, 12, 24, 33, 91, 132, 201, 728, 812, 921, 1608, 1612, 2064, 2496, 2854, 3058, 3240, 3435, 3500, 4426, 5074, 5664, 5762, 6860, 7318, 7368, 8434, 9500, 9846, 10286, 11073, 12982, 13773, 14252, 14386, 17241, 17246, 18321, 18723, 18898, 19628, 21309, 21538
Offset: 1

Views

Author

Amiram Eldar, Nov 19 2019

Keywords

Comments

The unitary version of A033632.

Examples

			33 is the sequence since usigma(uphi(33)) = usigma(20) = 30 and uphi(usigma(33)) = uphi(48) = 30.

Links

Crossrefs

Cf. A033632, A034448, A047994.

Programs

  • Mathematica
    usigma[1]=1; usigma[n_] := Times @@ (1 + Power @@@ FactorInteger[n]); uphi[1] = 1; uphi[n_] := Times @@ (-1 + Power @@@ FactorInteger[n]); aQ[n_] := usigma[uphi[n]] == uphi[usigma[n]]; Select[Range[22000], aQ]

A159939 Odd solutions of phi(sigma(k)) = sigma(phi(k)).

Original entry on oeis.org

1, 9, 225, 729, 18225, 65025, 140625, 531441, 5267025, 11390625, 13286025, 18792225, 40640625, 87890625, 1522170225, 2197265625, 3291890625, 3839661225, 5430953025, 7119140625, 8303765625, 11745140625, 25400390625
Offset: 1

Views

Author

Walter Nissen, Apr 26 2009

Keywords

Comments

sigma is the multiplicative sum-of-divisors function.
phi is Euler's totient.
Complete through 25558816403.
All given here are products of powers of consecutive Fermat primes based on generalized repunit primes; see links.
It is conjectured (see links) that all odd solutions are of this form, for which at least 10130 solutions are known.
a(24) > 10^11, if it exists. - Amiram Eldar, Nov 21 2024

Examples

			sigma(9) = 13, phi(9) = 6, sigma(6) = phi(13) = 12, so 9 is in the sequence.

References

  • Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section B42, pp. 150-152.
  • Oystein Ore, Number Theory and Its History, 1948, reprinted 1988, Dover, ISBN-10: 0486656209, pp. 88 et seq., 109 et seq.

Links

Crossrefs

Cf. A000203, A000010, A033632, A019434.

Programs

  • PARI
    isok(n) = (n % 2) && (eulerphi(sigma(n)) == sigma(eulerphi(n))) \\ Michel Marcus, Jul 23 2013

Extensions

Edited by Charles R Greathouse IV, Oct 28 2009
a(1) = 1 inserted by Amiram Eldar, Nov 21 2024

A218006 Numbers n such that sigma(tau(phi(n))) = tau(phi(sigma(n))) = phi(sigma(tau(n))).

Original entry on oeis.org

1, 34, 36, 96, 128, 468, 1200, 21216, 102060, 110976, 117684, 211428, 331380, 366660, 437220, 511680, 530712, 706560, 710388, 726240, 732240, 759360, 838080, 845376, 875840, 911040, 975936, 1014016, 1041216, 1093440, 1110720, 1141440, 1167696, 1289280
Offset: 1

Views

Author

Jayanta Basu, Mar 26 2013

Keywords

Comments

Here phi denotes Euler's totient function, tau(n) denotes number of divisors of n and sigma(n) denotes sum of all divisors of n. Only cyclic rotation of operators is considered.

Crossrefs

Cf. A000005, A000010, A000203, A033632, A076361, A078148.

Programs

  • Mathematica
    Select[Range[1000000], DivisorSigma[1, DivisorSigma[0, EulerPhi[#]]] == DivisorSigma[0, EulerPhi[DivisorSigma[1, #]]] == EulerPhi[DivisorSigma[1, DivisorSigma[0, #]]] &]
Previous Showing 31-38 of 38 results.

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

Content is available under The OEIS End-User License Agreement.