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.

A046034 Numbers whose digits are primes.

Original entry on oeis.org

2, 3, 5, 7, 22, 23, 25, 27, 32, 33, 35, 37, 52, 53, 55, 57, 72, 73, 75, 77, 222, 223, 225, 227, 232, 233, 235, 237, 252, 253, 255, 257, 272, 273, 275, 277, 322, 323, 325, 327, 332, 333, 335, 337, 352, 353, 355, 357, 372, 373, 375, 377, 522, 523, 525, 527, 532
Offset: 1

Views

Author

Eric W. Weisstein

Keywords

Comments

If n is represented as a zerofree base-4 number (see A084544) according to n=d(m)d(m-1)...d(3)d(2)d(1)d(0) then a(n) = Sum_{j=0..m} c(d(j))*10^j, where c(k)=2,3,5,7 for k=1..4. - Hieronymus Fischer, May 30 2012
According to A153025, it seems that 5, 235 and 72335 are the only terms whose square is also a term, i.e., which are also in the sequence A275971 of square roots of the terms which are squares, listed in A191486. - M. F. Hasler, Sep 16 2016

Examples

			a(100)   = 2277,
a(10^3)  = 55327,
a(9881)  = 3233232,
a(10^4)  = 3235757,
a(10922) = 3333333,
a(10^5)  = 227233257.

Links

Crossrefs

Cf. A046035, A118950, A019546 (primes), A203263, A035232, A039996, A085823, A052382, A084544, A084984, A017042, A001743, A001744, A014261, A014263, A153025, A191486, A193238, A202267, A202268, A211681, A365471 (complement).

Programs

  • Haskell
    a046034 n = a046034_list !! (n-1)
a046034_list = filter (all (`elem` "2357") . show ) [0..]
-- Reinhard Zumkeller, Jul 19 2011
  • Magma
    [n: n in [2..532] | Set(Intseq(n)) subset [2, 3, 5, 7]];  // Bruno Berselli, Jul 19 2011
  • Mathematica
    Table[FromDigits /@ Tuples[{2, 3, 5, 7}, n], {n, 3}] // Flatten (* Michael De Vlieger, Sep 19 2016 *)
  • PARI
    is_A046034(n)=Set(isprime(digits(n)))==[1] \\ M. F. Hasler, Oct 12 2013
  • Python
    def A046034(n):
    m = (3*n+1).bit_length()-1>>1
    return int(''.join(('2357'[(3*n+1-(1<<(m<<1)))//(3<<((m-1-j)<<1))&3] for j in range(m)))) # Chai Wah Wu, Feb 08 2023

Formula

A055642(a(n)) = A193238(a(n)). - Reinhard Zumkeller, Jul 19 2011
From Hieronymus Fischer, Apr 20, May 30 and Jun 25 2012: (Start)
a(n) = Sum_{j=0..m-1} ((2*b(j)+1) mod 8 + floor(b(j)/4) - floor((b(j)-1)/4))*10^j, where m = floor(log_4(3*n+1)), b(j) = floor((3*n+1-4^m)/(3*4^j)).
a(n) = Sum_{j=0..m-1} A010877(A005408(b(j)) + A002265(b(j)) - A002265(b(j)-1))*10^j.
Special values:
a(1*(4^n-1)/3) = 2*(10^n-1)/9.
a(2*(4^n-1)/3) = 1*(10^n-1)/3.
a(3*(4^n-1)/3) = 5*(10^n-1)/9.
a(4*(4^n-1)/3) = 7*(10^n-1)/9.
Inequalities:
a(n) <= 2*(10^log_4(3*n+1)-1)/9, equality holds for n = (4^k-1)/3, k>0.
a(n) <= 2*A084544(n), equality holds iff all digits of A084544(n) are 1.
a(n) > A084544(n).
Lower and upper limits:
lim inf a(n)/10^log_4(n) = (7/90)*10^log_4(3) = 0.48232167706987..., for n -> oo.
lim sup a(n)/10^log_4(n) = (2/9)*10^log_4(3) = 1.378061934485343..., for n -> oo.
where 10^log_4(n) = n^1.66096404744...
G.f.: g(x) = (x^(1/3)*(1-x))^(-1) Sum_{j>=0} 10^j*z(j)^(4/3)*(2 + z(j) + 2*z(j)^2 + 2*z(j)^3 - 7*z(j)^4)/(1-z(j)^4), where z(j) = x^4^j.
Also g(x) = (x^(1/3)*(1-x))^(-1) Sum_{j>=0} 10^j*z(j)^(4/3)*(1-z(j))*(2 + 3*z(j) + 5*z(j)^2 + 7*z(j)^3)/(1-z(j)^4), where z(j)=x^4^j.
Also: g(x) = (1/(1-x))*(2*h_(4,0)(x) + h_(4,1)(x) + 2*h_(4,2)(x) + 2*h_(4,3)(x) - 7*h_(4,4)(x)), where h_(4,k)(x) = Sum_{j>=0} 10^j*x^((4^(j+1)-1)/3)*x^(k*4^j)/(1-x^4^(j+1)). (End)
Sum_{n>=1} 1/a(n) = 1.857333779940977502574887651449435985318556794733869779170825138954093657197... (calculated using Baillie and Schmelzer's kempnerSums.nb, see Links). - Amiram Eldar, Feb 15 2024

Extensions

More terms from Cino Hilliard, Aug 06 2006
Typo in second formula corrected by Hieronymus Fischer, May 12 2012
Two typos in example section corrected by Hieronymus Fischer, May 30 2012

A191486 Squares using only the prime digits (2,3,5,7).

Original entry on oeis.org

25, 225, 7225, 27225, 55225, 235225, 2772225, 3553225, 23377225, 33235225, 57532225, 227557225, 252333225, 277722225, 337273225, 357777225, 523723225, 735223225, 777573225, 2523555225, 3325252225, 3377353225, 5232352225, 7333353225
Offset: 1

Views

Author

Giovanni Teofilatto, Jun 03 2011

Keywords

Comments

a(n) = 225 mod 1000 for n > 1. - Charles R Greathouse IV, May 14 2013
The sequence is infinite: it contains A030485 as an infinite proper subsequence which in turn contains all numbers of the form ((5*10^n-5)/3)^2 as a proper subsequence. - M. F. Hasler, Sep 16 2016

Links

Crossrefs

Cf. A077676, A030485.

Programs

  • Magma
    [n^2: n in [5..5*10^5] | Set(Intseq(n^2)) subset {2,3,5,7}];  // Bruno Berselli, Jun 06 2011
  • Maple
    for b from 1 do convert(convert(b^2,base,10),set) ; if % minus {2,3,5,7} = {} then printf("%d,\n",b^2) ; end if; end do: # R. J. Mathar, Jun 03 2011
  • Mathematica
    w = Boole@! PrimeQ@ # & /@ RotateLeft@ Range[0, 9]; Select[Range[10^5]^2, Total@ Pick[DigitCount@ #, w, 1] == 0 &] (* Michael De Vlieger, Aug 15 2016 *)
  • PARI
    toprime(n,k)=n<<=2;sum(i=0,k-1,n>>=2;[2,3,5,7][bitand(n,3)+1]*10^i)
v=List([25]);for(k=0,9,for(n=0,4^k-1,t=1000*toprime(n,k)+225;if(issquare(t),listput(v,t)))); vecsort(Vec(v)) \\ Charles R Greathouse IV, May 14 2013
  • Python
    from math import isqrt
def aupto(limit):
  alst, rootlimit = [], isqrt(limit)
  for k in range(1, rootlimit+1):
    if set(str(k*k)) <= set("2357"): alst.append(k*k)
  return alst
print(aupto(7333353225)) # Michael S. Branicky, May 15 2021

Formula

a(n) = A275971(n)^2. - M. F. Hasler, Sep 16 2016

A153025 Numbers n with property that n and n^2 use only prime digits.

Original entry on oeis.org

5, 235, 72335
Offset: 1

Views

Author

Zak Seidov, Dec 17 2008

Keywords

Comments

Probably there are no other terms. No other terms up to 10^40.
Intersection of A046034 and A275971. - M. F. Hasler, Sep 16 2016

Examples

			The squares of 5, 235, 72335 are 25, 55225, 5232352225.

Crossrefs

Cf. A046034, A275971.

Programs

  • Mathematica
    Flatten[Table[Select[FromDigits/@Tuples[{2,3,5,7},n],AllTrue[ IntegerDigits[ #^2], PrimeQ]&],{n,5}]] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Oct 16 2014 *)

Extensions

Edited by David Wilson and N. J. A. Sloane, Jan 25 2009
Showing 1-3 of 3 results.

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