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 1-3 of 3 results.

A069215 Numbers n such that phi(n) = reversal(n).

Original entry on oeis.org

1, 21, 63, 270, 291, 2991, 6102, 46676013, 69460293, 2346534651, 6313047393, 23400000651, 80050617822, 234065340651, 234659934651, 2340000000651, 2530227348360, 2934000006591
Offset: 1

Views

Author

Joseph L. Pe, Apr 11 2002

Keywords

Comments

If 10^n-3 is prime (n is in the sequence A089765) and m=3*(10^n-3) then m is in this sequence, for example 299999999999999991 is a term of this sequence because 299999999999999991=3*(10^17-3) and 17 is in the sequence A089675. So 3*(10^A089675-3) is a subsequence of this sequence, A101700 is this subsequence. - Farideh Firoozbakht, Dec 26 2004
A072395 is a subsequence of this sequence. If m is in the sequence and 10 doesn't divide m then reversal(m) is in the sequence A085331, so see Comments on A085331. - Farideh Firoozbakht, Jan 09 2005
If p=(79*10^(4n+1)-83)/101 is prime then 3p is in the sequence. The proof is easy. 21, 2346534651 & 3*(79*10^2697-83)/101 are the first three such terms. - Farideh Firoozbakht, Apr 22 2008, Aug 16 2008
a(19) > 10^13. - Giovanni Resta, Aug 07 2019

Examples

			phi(291) = 192.
phi(6102) = 2016 = reversal(6102), so 6102 belongs to the sequence.

Crossrefs

Cf. A101700, A004086, A000010, A085331, A072395, A101700, A102278.

Programs

  • Mathematica
    Do[If[EulerPhi[n] == FromDigits[Reverse[IntegerDigits[n]]], Print[n]], {n, 1, 10^5}]
  • PARI
    for( n=1,1e9, A004086(n)==eulerphi(n) & print1(n","))

Extensions

More terms from Farideh Firoozbakht, Aug 31 2004
One more term from Farideh Firoozbakht, Jan 09 2005
a(11)-a(13) from Donovan Johnson, Feb 03 2012
a(14)-a(15) from Giovanni Resta, Oct 28 2012
a(16)-a(18) from Giovanni Resta, Aug 07 2019

A089764 Primes of the form k! followed by a 1.

Original entry on oeis.org

11, 61, 241, 1201, 258520167388849766400001, 2631308369336935301672180121600000001, 203978820811974433586402817399028973568000000001, 5075802138772247988008568121766252272260045289880360030994059394809856000000000000001
Offset: 1

Views

Author

Amarnath Murthy, Nov 23 2003

Keywords

Comments

The next term has 739 digits. - Harvey P. Dale, Jun 08 2014

Examples

			1201 is a prime obtained as 5! followed by a 1.

Crossrefs

Cf. A089765, A126896.

Programs

  • Mathematica
    Select[Table[FromDigits[Join[IntegerDigits[n!],{1}]],{n,50}],PrimeQ] (* Harvey P. Dale, Jun 08 2014 *)

Extensions

More terms from Lior Manor, May 10 2004
One additional term from Harvey P. Dale, Jun 08 2014

A271101 Squarefree semiprimes (A006881) whose average prime factor is prime.

Original entry on oeis.org

21, 33, 57, 69, 85, 93, 129, 133, 145, 177, 205, 213, 217, 237, 249, 253, 265, 309, 393, 417, 445, 469, 489, 493, 505, 517, 553, 565, 573, 597, 633, 669, 685, 697, 753, 781, 793, 813, 817, 865, 889, 913, 933, 949, 973, 985, 993, 1057, 1077, 1137, 1149, 1177, 1257, 1273, 1285, 1329
Offset: 1

Views

Author

Antonio Roldán, Mar 30 2016

Keywords

Comments

Sum of factors of a(n) if semiprime (product 2*p, with p prime).
This sequence is subsequence of A006881, A089765, A187073, A108633 and A213015.
This sequence is also subsequence of A045835, because sopfr(omega(a(n))) = omega(sopfr(a(n))): sopfr(omega(a(n)))=sopfr(2)=2, and omega(sopfr(a(n)))=omega(2*p)=2 (p prime, p>2, average prime factor).

Examples

			133 is in the sequence because 133 is a squarefree semiprime: 133=7*19, and (7+19)/2=13, a prime number.

Links

Crossrefs

Cf. A006881, A046315, A046388, A115585, A187073, A089765, A108633, A213015.

Programs

  • Maple
    N:= 10000: # for terms <= N
Primes:= select(isprime, [seq(i, i=3..N/3)]):
SP:= [seq(seq([p, q], q = select(`<=`, Primes, min(p-1, N/p))), p=Primes)]:
B:= select(t -> isprime((t[1]+t[2])/2), SP):
sort(map(t -> t[1]*t[2], B)); # Robert Israel, Dec 14 2019
  • Mathematica
    Select[Select[Range@ 1330, SquareFreeQ@ # && PrimeOmega@ # == 2 &], PrimeQ@ Mean[First /@ FactorInteger@ #] &] (* Michael De Vlieger, Mar 30 2016 *)
  • PARI
    sopf(n)= { local(f, s=0); f=factor(n); for(i=1, matsize(f)[1], s+=f[i, 1]); return(s) }
{for (n=6, 2*10^3,  if(bigomega(n)==2&&omega(n)==2, m=sopf(n)/2;if(m==truncate(m),if(isprime(m), print1(n, ", ")))))}
Showing 1-3 of 3 results.

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