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.

A075474 Duplicate of A069225.

Original entry on oeis.org

1, 65, 2771, 10920, 158804, 2413091, 16636354, 117650080
Offset: 1

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Author

Keywords

A230012 Numbers n such that phi(n) = sigma(n) - reversal(sigma(n)).

Original entry on oeis.org

19, 199, 437, 603, 1999, 12834, 16348, 21293, 22183, 23383, 25273, 27263, 44377, 46367, 199999, 670661, 691351, 803851, 845321, 1425650, 2103643, 2111191, 2123893, 2174143, 2543773, 2552723, 2753713, 3791659, 4003997, 4034347, 4133332, 4223887, 4244287, 4492429, 4663667
Offset: 1

Views

Author

Farideh Firoozbakht, Feb 04 2014

Keywords

Comments

A prime p is in the sequence iff p is of the form 2*10^m-1. 19, 199, 1999 are such terms of the sequence.
If both numbers p=2*10^m-1 and q=2*10^m+3 are prime then n=p*q is in the sequence. For m=1, 3, 5, 7 both p and q are prime.
If both numbers p=2*10^m-1 and q=(7*10^m-1)/3 are prime then n=p*q is in the sequence. For m=1, 2, 3, 53 both p and q are prime.

Links

Crossrefs

Cf. A000010, A000203, A004086, A069225, A237257.

Programs

  • Mathematica
    r[n_]:=FromDigits[Reverse[IntegerDigits[n]]];Do[If[DivisorSigma[1,n]- r[DivisorSigma[1,n]]==EulerPhi[n],Print[n]],{n,15000000}]

A075475 Numbers n such that reverse(phi(n+1)) = phi(n).

Original entry on oeis.org

1, 3, 15, 129, 270, 833, 5044, 59815, 9633693, 26427528, 4642779313, 26594615202, 62935243460, 91770096041, 1648209018135
Offset: 1

Views

Author

Labos Elemer, Sep 20 2002

Keywords

Comments

a(12) > 25*10^9. - Donovan Johnson, Jun 09 2011
a(16) > 2*10^12. - Giovanni Resta, Aug 12 2019

Examples

			n=59815: phi(n) = 40992, phi(n+1) = 29904 = rev(phi(n)).

Crossrefs

Cf. A004086, A069225, A074241, A074242.

Programs

  • Mathematica
    Select[Range[60000], IntegerReverse[EulerPhi[# + 1]] == EulerPhi[#] &] (* Giovanni Resta, Aug 12 2019 *)
  • PARI
    isok(n) = eulerphi(n) == fromdigits(Vecrev(digits(eulerphi(n+1)))); \\ Michel Marcus, Aug 12 2019

Extensions

a(9)-a(11) from Donovan Johnson, Jun 09 2011
a(12)-a(15) from Giovanni Resta, Aug 12 2019

A247080 Numbers whose Euler totient is the reverse of the sum of its aliquot parts.

Original entry on oeis.org

2, 735, 7665, 11505, 42630, 64578, 3440409, 11263073973
Offset: 1

Views

Author

Paolo P. Lava, Nov 17 2014

Keywords

Comments

Value of x such that phi(x) = Rev(sigma(x) - x).
a(9) > 2*10^11. - Hiroaki Yamanouchi, Nov 22 2014
a(9) > 10^13. - Giovanni Resta, Aug 26 2019

Examples

			phi(2) = 1 and sigma(2) - 2 = 1.
phi(735) = 336 and sigma(735) - 735 = 633.
phi(7665) = 3456 and sigma(7665) - 7665 = 6543.

Crossrefs

Cf. A000010, A001065, A069225, A254320.

Programs

  • Maple
    with(numtheory): T:=proc(w) local x,y,z; x:=w; y:=0;
for z from 1 to ilog10(x)+1 do
y:=10*y+(x mod 10); x:=trunc(x/10); od; y; end:
P:=proc(q) local n; for n from 1 to q do
if phi(n)=T(sigma(n)-n) then print(n); fi; od; end: P(10^9);
  • Mathematica
    Select[Range[10^6], EulerPhi[#] == FromDigits[Reverse[IntegerDigits[DivisorSigma[1, #] - #]]] &] (* Michael De Vlieger, Jan 29 2015 *)
  • PARI
    rev(n) = subst(Polrev(digits(n)), x, 10);
isok(n) = rev(sigma(n)-n) == eulerphi(n); \\ Michel Marcus, Jan 29 2015

Extensions

a(7)-a(8) from Hiroaki Yamanouchi, Nov 22 2014

A252255 Numbers n such that sigma(Rev(phi(n))) = phi(Rev(sigma(n))), where sigma is the sum of divisors and phi the Euler totient function.

Original entry on oeis.org

1, 14, 61, 966, 1428, 9174, 15642, 19934, 22155, 27075, 36650, 48731, 51095, 54184, 57902, 59711, 61039, 89276, 98645, 113080, 126850, 140283, 142149, 154670, 165822, 190908, 197705, 198712, 202096, 203107, 247268, 274368, 274716, 307836, 311925, 331037, 366740
Offset: 1

Views

Author

Paolo P. Lava, Dec 16 2014

Keywords

Examples

			phi(14) = 6, Rev(6) = 6 and sigma(6) = 12;
sigma(14) = 24, Rev(24) = 42 and sigma(42) = 12.

Links

Crossrefs

Cf. A000010, A000203, A069225, A070835, A070856.

Programs

  • Maple
    with(numtheory): T:=proc(w) local x, y, z; x:=0; y:=w;
for z from 1 to ilog10(w)+1 do x:=10*x+(y mod 10); y:=trunc(y/10); od; x; end:
P:=proc(q) local a, b, k; global n; for n from 1 to q do
if sigma(T(phi(n)))=phi(T(sigma(n))) then print(n); fi; od; end: P(10^12);
  • Mathematica
    Select[Range[400000],DivisorSigma[1,IntegerReverse[EulerPhi[#]]] == EulerPhi[ IntegerReverse[ DivisorSigma[ 1,#]]]&] (* Requires Mathematica version 10 or later *)  (* Harvey P. Dale, Apr 15 2017 *)

A254320 Numbers k such that the reversal of phi(k) is sigma(k)-k.

Original entry on oeis.org

2, 11, 101, 735, 7665, 11505, 16459, 64578, 378871, 541033, 3440409, 5639353, 5230000213, 5762782573, 5828558173, 8130408803, 8275586723, 9738107377, 11263073973, 37057275961, 38914628453, 58285958173, 231243884637, 350649946051, 380047486211, 516420024613, 547083380743, 576216622573
Offset: 1

Views

Author

Paolo P. Lava, Jan 28 2015

Keywords

Examples

			sigma(2) - 2 = 1; rev(1) = 1 = phi(2).
sigma(735) - 735 = 633; rev(633) = 336 = phi(735).

Crossrefs

Cf. A000010, A000203, A001065, A069225, A247080.

Programs

  • Maple
    with(numtheory):T:=proc(w) local x,y,z; x:=w; y:=0;
for z from 1 to ilog10(x)+1 do y:=10*y+(x mod 10); x:=trunc(x/10); od; y; end:
P:=proc(q) local n; for n from 1 to q do
if T(phi(n))=sigma(n)-n then print(n); fi; od; end: P(10^7);
  • Mathematica
    Select[Range[564*10^4],IntegerReverse[EulerPhi[#]]==DivisorSigma[1,#]-#&] (* The program generates the first 12 terms of the sequence. *) (* Harvey P. Dale, Jul 03 2024 *)
  • PARI
    rev(n) = subst(Polrev(digits(n)), x, 10);
isok(n) = (sigma(n)-n) == rev(eulerphi(n)); \\ Michel Marcus, Jan 29 2015

Extensions

a(12) from Michel Marcus, Jan 29 2015
a(13)-a(28) from Giovanni Resta, May 08 2015

A098215 Numbers n such that reverse(sigma(n)) = n - phi(n) = cototient(n).

Original entry on oeis.org

136, 261, 295, 493, 1218, 1576, 15976, 159976, 715381, 717181, 746341, 987109, 1585576, 5989003, 13390567, 14400136, 19326607, 78753211, 144144136, 145454536, 158401576, 275675661, 713791681, 737098261, 1440000136, 1441584136, 1454414536, 1455998536
Offset: 1

Views

Author

Labos Elemer, Oct 22 2004

Keywords

Examples

			n=5989003: sigma(n) = 5994000 and n - phi(n) = 4995

Crossrefs

Cf. A069225, A000203, A000010.

Programs

  • Mathematica
    Do[ If[ FromDigits[ Reverse[ IntegerDigits[ DivisorSigma[1, n]]]] == n - EulerPhi[n], Print[n]], {n, 10^8}] (* Robert G. Wilson v, Oct 23 2004 *)

Extensions

More terms from Robert G. Wilson v, Oct 27 2004
a(19)-a(28) from Donovan Johnson, Jan 14 2011
Showing 1-7 of 7 results.

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