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 1-8 of 8 results.

A097393 Emirpimes: numbers n such that n and its reversal are distinct semiprimes.

Original entry on oeis.org

15, 26, 39, 49, 51, 58, 62, 85, 93, 94, 115, 122, 123, 129, 143, 155, 158, 159, 169, 177, 178, 183, 185, 187, 203, 205, 221, 226, 265, 289, 302, 314, 319, 321, 326, 327, 329, 335, 339, 341, 355, 381, 394, 398, 413, 415, 437, 493, 497, 502, 511, 514, 533
Offset: 1

Views

Author

Jonathan Vos Post

Keywords

Comments

Computed by Eric W. Weisstein, Aug 13 2004.

Examples

			26 is a semiprime, as it is 2 * 13, and so is 62 = 2 * 31. 26 and 62 are therefore both in the sequence.

Links

Crossrefs

Cf. A001358, A097394.
Equals A085751 \ A046328.

Programs

  • Maple
    isA097393 := proc(n)
    local R ;
    R := digrev(n) ;
    if R <> n then
        if numtheory[bigomega](R) = 2 and numtheory[bigomega](n) = 2 then
            return true;
        else
            false;
        end if;
      else
        false;
    end if;
end proc:
for n from 1 to 500 do
    if isA097393(n) then
        printf("%d,",n) ;
    end if;
end do: # R. J. Mathar, Apr 05 2012
  • Mathematica
    Cases[{#, IntegerReverse@#} & /@ DeleteCases[Range@5000, ?PalindromeQ], {?(PrimeOmega@# == 2 &) ..}][[All,1]] (* Hans Rudolf Widmer, Jan 07 2024 *)
  • PARI
    rev(n)=subst(Polrev(digits(n)),'x,10)
issemi(n)=bigomega(n)==2
list(lim)=my(v=List(),r);forprime(p=2,lim\2,forprime(q=2,min(lim\p,p),r=rev(p*q);if(issemi(r)&&r!=p*q,listput(v,p*q))));Set(v) \\ Charles R Greathouse IV, Jan 27 2015
  • Python
    from sympy import factorint
from itertools import islice
def sp(n): f = factorint(n); return sum(f[p] for p in f) == 2
def ok(n): r = int(str(n)[::-1]); return r != n and sp(n) and sp(r)
print([k for k in range(534) if ok(k)]) # Michael S. Branicky, Jul 03 2022

A115656 Both n and the reverse of n are powerful(1) numbers (A001694).

Original entry on oeis.org

1, 4, 8, 9, 27, 72, 100, 121, 144, 169, 343, 400, 441, 484, 576, 675, 676, 800, 900, 961, 1000, 1089, 1331, 1800, 2700, 3087, 4000, 7200, 7803, 8000, 9000, 9801, 10000, 10201, 10404, 10609, 12100, 12321, 12544, 12769, 14400, 14641, 14884
Offset: 1

Views

Author

Giovanni Resta, Jan 28 2006

Keywords

Examples

			9801=3^4*11^2 and 1089=3^2*11^2.

Links

Crossrefs

Cf. A001694, A115655, A085751.

Programs

  • PARI
    is(n)=ispowerful(n) && ispowerful(subst(Polrev(digits(n)),'x,10)) \\ Charles R Greathouse IV, Sep 16 2014
  • PARI
    has(n)=ispowerful(subst(Polrev(digits(n)),'x,10))
list(lim)=my(v=List(),t,t2); for(m=1,lim^(1/3), t=m^3; for(n=1,sqrtint(lim\t), if(has(t2=t*n^2), listput(v,t2)))); Set(v) \\ Charles R Greathouse IV, Sep 16 2014

A263106 Semiprimes such that the leftward cyclic permutation of its decimal digits is also semiprime.

Original entry on oeis.org

4, 6, 9, 15, 22, 26, 33, 39, 49, 51, 55, 58, 62, 77, 85, 93, 94, 111, 122, 129, 134, 141, 145, 155, 158, 159, 161, 177, 178, 183, 185, 187, 202, 206, 214, 226, 254, 262, 298, 303, 309, 314, 321, 339, 341, 355, 358, 362, 371, 381, 391, 393, 394, 403, 407, 413
Offset: 1

Views

Author

Zak Seidov, Oct 09 2015

Keywords

Comments

First 18 terms are also in A085751.

Examples

			15 = 3 * 5, 51 = 3 * 17; 889 = 7 * 17, 898 = 2 * 449.

Links

Crossrefs

Cf. A001292, A001358, A085751.

Programs

  • Mathematica
    Select[Range[4, 1000], 2 == PrimeOmega[#] == PrimeOmega[FromDigits[RotateLeft[IntegerDigits[#]]]] &]
  • PARI
    shl(n)=if(n<10,return(n)); my(d=digits(n)); fromdigits(concat(d[2..#d], d[1]))
is(n)=bigomega(n)==2 && bigomega(shl(n))==2 \\ Charles R Greathouse IV, Oct 12 2015

A242592 Squarefree semiprimes, n=p*q, where reversal(n) is semiprime and reversal(n) = reversal(p)*reversal(q).

Original entry on oeis.org

6, 22, 26, 33, 39, 55, 62, 77, 93, 143, 187, 202, 226, 262, 303, 339, 341, 393, 505, 622, 626, 707, 781, 933, 939, 1111, 1177, 1243, 1313, 1441, 1469, 1661, 1717, 1991, 2042, 2062, 2066, 2206, 2402, 2426, 2446, 2462, 2602, 2642, 3063, 3093, 3099, 3131, 3309
Offset: 1

Views

Author

Michel Lagneau, May 18 2014

Keywords

Comments

Subsequence of A085751.

Examples

			1469 = 13*113 is in the sequence because reversal(1469) = 9641 = 31*311 where 31 = reversal(13) and 311 = reversal(113).

Links

Crossrefs

Cf. A001358, A006881, A085751, A210616.

Programs

  • Maple
    for n from 6 to 4000 do :
  x:=factorset(n):n1:=nops(x):
   if bigomega(n)= 2 and n1>1
   then                 y:=convert(n,base,10):n2:=nops(y):p:=x[1]:q:=x[2]:xp1:=convert(p,base,10):nxp1:=nops(xp1):xq1:=convert(q,base,10):nxq1:=nops(xq1):sp:=sum('xp1[i]*10^(nxp1-i)', 'i'=1..nxp1):sq:=sum('xq1[i]*10^(nxq1-i)', 'i'=1..nxq1):lst:={sp} union {sq}:s:=sum('y[i]*10^(n2-i)', 'i'=1..n2):x1:=factorset(s):nn1:=nops(x1):
    if bigomega(s)=2 and nn1>1
    then
    z:=convert(s,base,10):n3:=nops(z): p1:=x1[1]:q1:=x1[2]:
    lst1:={p1} union  {q1}:s1:=sum('z[i]*10^(n3-i)','i'=1..n3):
       if lst = lst1
        then
        printf(`%d, `,n):
        else
       fi:
     fi:
    fi:
  od:

A115655 Both n and the reverse of n are brilliant numbers (A078972).

Original entry on oeis.org

4, 6, 9, 121, 143, 169, 187, 319, 323, 341, 737, 767, 781, 913, 949, 961, 979, 989, 1273, 1343, 1691, 1843, 1961, 3431, 3481, 3721, 10201, 10807, 11413, 12769, 13231, 15049, 15151, 15251, 15347, 15707, 15857, 16171, 16837, 16867, 17161
Offset: 1

Views

Author

Giovanni Resta, Jan 28 2006

Keywords

Examples

			11413=101*113 and 31411=101*311.

Crossrefs

Cf. A078972, A115656, A085751.

A263108 Semiprimes m such that the leftward cyclic permutation of its decimal digits is a larger semiprime.

Original entry on oeis.org

15, 26, 39, 49, 58, 122, 129, 134, 141, 145, 155, 158, 159, 161, 177, 178, 183, 185, 187, 226, 254, 262, 298, 339, 341, 355, 358, 362, 371, 381, 391, 393, 394, 445, 451, 469, 473, 493, 497, 565, 581, 583, 586, 589, 674, 781, 791, 889, 895, 899, 1114, 1119
Offset: 1

Views

Author

Zak Seidov, Oct 09 2015

Keywords

Comments

Subsequence of A263106.

Examples

			Permute the digits of 15 = 3 * 5 to get 51 = 3 * 17.
Permute the digits of 26 = 2 * 13 to get 62 = 2 * 31.
Permute the digits of 122 = 2 * 61 to get 221 = 13 * 17.
Permute the digits of 129 = 3 * 43 to get 291 = 3 * 97.

Links

Crossrefs

Cf. A001292, A001358, A085751, A263106.

Programs

  • Mathematica
    Select[Range[4, 1000000], 2 == PrimeOmega[#] == PrimeOmega[fd = FromDigits[RotateLeft[IntegerDigits[#]]]] && fd > # &] (* for b-file *)

A366430 Triprimes whose reversal is also a triprime.

Original entry on oeis.org

8, 44, 66, 99, 117, 147, 165, 171, 212, 222, 242, 244, 246, 282, 285, 286, 290, 292, 333, 338, 343, 363, 366, 369, 404, 406, 418, 425, 434, 435, 438, 442, 474, 475, 494, 498, 506, 507, 508, 524, 534, 539, 548, 555, 561, 574, 575, 582, 595, 604, 605, 606, 609, 628, 642, 646, 663, 670, 682, 705
Offset: 1

Views

Author

Zak Seidov and Robert Israel, Nov 06 2023

Keywords

Examples

			a(5) = 117 is a term because 117 = 3^2 * 13 has 3 prime factors, counted with multiplicity, and so does its reversal 711 = 3^2 * 79.

Links

Crossrefs

Cf. A014612, A085751. Contains A046329. Includes 10*k for k in A367151.

Programs

  • Maple
    rev:= proc(n) local L,i;
  L:= convert(n,base,10);
  add(L[-i]*10^(i-1),i=1..nops(L))
end proc:
select(t -> numtheory:-bigomega(t) = 3 and numtheory:-bigomega(rev(t))=3, [$1..10000]);
  • Mathematica
    Select[Range[710], PrimeOmega[#]==3&&PrimeOmega[FromDigits[Reverse[IntegerDigits[#]]]]==3&] (* Stefano Spezia, Nov 07 2023 *)
  • Python
    from sympy import factorint
def tp(n): return sum(factorint(n).values()) == 3
def ok(n): return tp(n) and tp(int(str(n)[::-1]))
print([k for k in range(10**3) if ok(k)]) # Michael S. Branicky, Nov 21 2023

A344468 Semiprimes k such that every permutation of the digits of k is a semiprime.

Original entry on oeis.org

4, 6, 9, 15, 22, 26, 33, 39, 49, 51, 55, 58, 62, 77, 85, 93, 94, 111, 155, 177, 178, 187, 226, 262, 339, 355, 393, 515, 535, 551, 553, 622, 717, 718, 771, 781, 817, 871, 899, 933, 989, 998, 1111, 3777, 4555, 5455, 5545, 5554, 5999, 7377, 7737, 7773, 7999, 9599
Offset: 1

Views

Author

Ctibor O. Zizka, May 20 2021

Keywords

Examples

			k = 15, A001222(15) = A001222(51) = 2, thus 15 and 51 are terms;
k = 178, A001222(178) = A001222(187) = A001222(718) = A001222(781) = A001222(817) = A001222(871) = 2, thus 178, 187, 718, 781, 817, 871 are terms.

Crossrefs

Cf. A000005, A001222, A001358, A003459.
Subsequence of A085751 and A263106.

Programs

  • Mathematica
    q[n_] := AllTrue[Permutations[IntegerDigits[n]], PrimeOmega[FromDigits[#]] == 2 &]; Select[Range[10^4], q] (* Amiram Eldar, May 20 2021 *)
Showing 1-8 of 8 results.

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