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.

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A348266 k-digit numbers whose digit(s) are the number of distinct prime factors in each of the preceding k integers.

Original entry on oeis.org

22, 313, 2232, 2323, 2333, 32215, 432152, 2434332, 4222423, 43332543, 332325334, 2535332433, 4532543535234, 5435433351423
Offset: 1

Views

Author

Metin Sariyar, Oct 09 2021

Keywords

Comments

a(12) <= 2535332433. - David A. Corneth, Oct 10 2021
a(12) >= 10^9. - Michel Marcus, Oct 11 2021

Examples

			22 is a term because omega(20) = 2 and omega(21) = 2, whose concatenation is 22.
313 is a term because preceding it omega(310) = 3, omega(311) = 1 and omega(312) = 3, and their concatenation is 313.
32215 is a term because, the number of distinct prime divisors of 32210, 32211, 32212, 32213 and 32214 are 3, 2, 2, 1, 5 and their ordered concatenation gives the next number 32215.

Crossrefs

Cf. A001221, A000961, A007774, A033992, A033993, A051270, A074969, A176655.
Cf. A323083.

Programs

  • Mathematica
    Select[Range[33000], FromDigits[PrimeNu /@ (# - Range[IntegerLength[#], 1, -1])] == # &] (* Amiram Eldar, Oct 09 2021 *)
  • PARI
    isok(m) = {my(s="", k=m, i=1); while(1, s = concat(s, Str(omega(k))); if (eval(s) == m+i, return (i)); if (eval(s) > m+i, return(0)); k++; i++;);}
lista(nn) = my(nb); for(n=1, nn, if (nb=isok(n), print1(n+nb, ", "))); \\ Michel Marcus, Oct 09 2021

Extensions

a(8)-a(9) from Amiram Eldar, Oct 09 2021
a(10)-a(11) from Michel Marcus, Oct 10 2021
a(12) confirmed by Martin Ehrenstein, Oct 28 2021
a(13)-a(14) from Martin Ehrenstein, Oct 30 2021

A348882 Numbers that are expressible as the product of the number of distinct prime factors of preceding integers.

Original entry on oeis.org

16, 48, 72, 96, 144, 432, 576, 1296, 2592, 5184, 20736, 32805, 221184, 1555200, 11197440, 55987200, 95551488, 268738560, 302330880, 382205952, 524880000, 671846400, 6718464000, 34012224000, 155520000000, 403107840000, 6856864358400, 107495424000000, 110075314176000
Offset: 1

Views

Author

Metin Sariyar, Nov 02 2021

Keywords

Examples

			The number of distinct prime factors of the numbers 15, 14, 13, 12, 11, 10 are respectively 2, 2, 1, 2, 1, 2 and 2*2*1*2*1*2 = 16, hence 16 is a term.

Crossrefs

Cf. A001221, A000961, A007774, A033992, A033993, A051270, A074969, A176655, A348266.

Programs

  • Mathematica
    om[n_] := om[n] = PrimeNu[n]; q[n_] := Module[{m = n, k = n - 1}, While[k > 1 && Divisible[m, om[k]], m /= om[k]; k--]; m == 1]; Select[Range[2, 10^6], q] (* Amiram Eldar, Nov 02 2021 *)

Extensions

a(13)-a(17) from Amiram Eldar, Nov 02 2021
More terms from David A. Corneth, Nov 02 2021

A368832 Integers not of one of the 5 forms p^k, p*q^k, 2*p*q^k, p*q*r or 2*p*q*r with p, q, r distinct primes and k>=0.

Original entry on oeis.org

36, 60, 72, 84, 100, 108, 120, 132, 140, 144, 156, 168, 180, 196, 200, 204, 216, 220, 225, 228, 240, 252, 260, 264, 276, 280, 288, 300, 308, 312, 315, 324, 336, 340, 348, 360, 364, 372, 380, 392, 396, 400, 408, 420, 432, 440, 441, 444, 450, 456, 460, 468, 476, 480, 484, 492, 495, 500, 504, 516, 520, 525, 528, 532, 540, 552
Offset: 1

Views

Author

R. J. Mathar, Jan 07 2024

Keywords

Comments

Cyclic groups of these orders cannot be Schur groups, see the Theorem by [Evdokimov et al.].

Links

Crossrefs

Cf. A051270 (subsequence), A036785 (subsequence), A074969 (subsequence).

Programs

  • Maple
    isA007304 := proc(n)
    if bigomega(n) = 3 and A001221(n) =3 then
        true;
    else
        false ;
    end if;
end proc:
# list of prime exponents
pexp := proc(n)
    local e,pe ;
    e := [] ;
    for pe in ifactors(n)[2] do
        e := [op(e),op(2,pe)] ;
    end do:
    e ;
end proc:
isCycSchGr := proc(n)
    local om,nhalf ,pe;
    om := A001221(n) ;
    if  om > 4 then
        return false;
    elif om = 4 then
        # require 2*p*q*r
        if type(n,'even') and type(n/2,'odd') then
            nhalf := n/2 ;
            # require nhalf =p*q*r in A007304
            return isA007304(nhalf) ;
        else
            false;
        end if;
    elif om = 3 then
        # require p*q*r or 2*p*q^k
        if type(n,'even') and type(n/2,'odd') then
            nhalf := n/2 ;
            # require nhalf =p*q^k
            pe := pexp(nhalf) ;
            if nops(pe) =2 and 1 in convert(pe,set) then
                true;
            else
                false ;
            end if;
        elif type(n,'odd') then
            # require n =p*q*r
            if isA007304(n) then
                true;
            else
                false ;
            end if;
        else
            false;
        end if;
    elif om = 2 then
        # require p*q^k
        pe := pexp(n) ;
        if 1 in convert(pe,set) then
            true;
        else
            false;
        end if;
    else
        # p^k, k>=0
        true ;
    end if;
end proc:
for n from 1 to 3000 do
    if not isCycSchGr(n) then
        printf("%d,",n) ;
    end if;
end do:
Previous Showing 11-13 of 13 results.

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