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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A268582 Sphenic numbers having identical digits.

Original entry on oeis.org

66, 222, 555, 777, 2222, 3333, 5555, 7777, 22222, 33333, 55555, 77777, 2222222, 3333333, 5555555, 7777777, 22222222222, 33333333333, 55555555555, 77777777777, 1111111111111, 22222222222222222, 33333333333333333, 55555555555555555, 77777777777777777, 6666666666666666666
Offset: 1

Views

Author

Michel Lagneau, Feb 07 2016

Keywords

Comments

Subsequence of A007304 (sphenic numbers: products of 3 distinct primes).
a(1)= A087331(4).

Examples

			222 is in the sequence because 222 = 2*3*37, product of 3 distinct primes.

Links

Crossrefs

Cf. A007304, A046053, A087331, A095370, A196104.

Programs

  • Maple
    with(numtheory):
for n from 1 to 23 do:
  for b from 1 to 9 do:
    x:=(((10^n)- 1)/9)*b:y:=factorset(x):n1:=nops(y):
     if bigomega(x)=3 and n1=3
      then
      printf(`%d, `,x):
      else
     fi:
   od:
od:
  • Mathematica
    Select[Flatten@ Map[Map[Function[k, FromDigits@ Table[k, {#}]], Range[1, 9]] &, Range@ 20], Length@ # == 3 && Times @@ Last /@ # == 1 &@ FactorInteger@ # &] (* Michael De Vlieger, Feb 07 2016 *)

A375784 Array read by rows: T(n,k) is the first number with n prime factors (counted with multiplicity) and n occurrences of decimal digit k.

Original entry on oeis.org

101, 13, 2, 3, 41, 5, 61, 7, 83, 19, 1003, 115, 22, 33, 445, 55, 166, 77, 818, 299, 10002, 1113, 222, 333, 4244, 555, 2666, 777, 8828, 3999, 100002, 11011, 22122, 33332, 4444, 15555, 6666, 75777, 38888, 9999, 1000004, 1011112, 222220, 333330, 444244, 552555, 666366, 777770, 88888, 999996
Offset: 1

Views

Author

Robert Israel, Aug 28 2024

Keywords

Examples

			T(5,1) = 1011112 = 2^3 * 211 * 599 has 5 prime factors (counted with multiplicity) and 5 1's, and is the first such number.
Array starts
    101      13      2      3     41      5     61      7    83     19
   1003     115     22     33    445     55    166     77   818    299
  10002    1113    222    333   4244    555   2666    777  8828   3999
 100002   11011  22122  33332   4444  15555   6666  75777 38888   9999
1000004 1011112 222220 333330 444244 552555 666366 777770 88888 999996

Links

Crossrefs

Cf. A001222, A002275, A046053, A375760.

Programs

  • Maple
    F:= proc(v, x) local d, y, z, L, S, SS, Cands, t, i, k;
   for d from v do
     Cands:= NULL;
     if x = 0 then SS:= combinat:-choose([$1..d-1], v)
     else SS:= combinat:-choose([$1..d], v)
     fi;
     for S in SS do
       for y from 9^(d-v+1) to 9^(d-v+1)+9^(d-v)-1 do
         L:= convert(y, base, 9)[1..d-v+1];
         L:= map(proc(s) if s < x then s else s+1 fi end proc, L);
         i:= 1;
         t:= 0:
         for k from 1 to d do
           if member(k, S) then t:= t + x*10^(k-1)
           else t:= t + L[i]*10^(k-1); i:= i+1;
           fi;
         od;
         Cands:= Cands, t
     od od;
     Cands:= sort([Cands]);
     for t in Cands do if numtheory:-bigomega(t)=v then return t fi od;
   od
end proc:
for i from 1 to 10 do
  seq(F(i, x), x=0..9)
od;
  • Mathematica
    T[n_, k_]:=Module[{m=2}, While[PrimeOmega[m]!=n||Count[IntegerDigits[m], k]!=n, m++]; m]; Table[T[n, k], {n, 1, 5}, {k, 0, 9}]//Flatten (* Stefano Spezia, Aug 30 2024 *)

Formula

If n - A046053(n) is odd and >= 1, then T(n,k) <= k * A002275(n) * 10^((n - A046053(n) - 1)/2) for k = 2, 3, 5 and 7.
If n - A046053(n) is odd and >= 3, then T(n,8) <= 8 * A002275(n) * 10^((n - A046053(n) - 3)/2).
If n - A046053(n) is even and >= 0, then T(n,1) <= A002275(n) * 10^((n - A046053(n))/2).
If n - A046053(n) is even and >= 2, then T(n,k) <= k * A002275(n) * 10^((n - A046053(n) - 2)/2) for k = 4, 6 and 9.

A046421 Index of smallest repunit having exactly n prime factors (counted with multiplicity).

Original entry on oeis.org

1, 2, 3, 13, 8, 6, 15, 12, 28, 18, 24, 32, 36, 30, 54, 42, 78, 100, 72, 176, 60, 208, 84, 132, 160, 198, 120, 204, 216, 308, 168, 280, 306, 180, 210, 264, 270, 252, 378, 336, 300
Offset: 0

Views

Author

Patrick De Geest, Jul 15 1998

Keywords

Comments

a(40) = 300; all other subsequent terms are > 322. - Ray Chandler, Apr 23 2017
a(41) <= 684, a(42) <= 546, a(43) <= 528, a(44) <= 462, a(45) = 360, a(46) <= 576, a(47) <= 624, a(48) <= 768. - Daniel Suteu, Jan 21 2023

Examples

			For n = 5: R_6 = 111111 = 3*7*11*13*37 is the smallest repunit with five prime factors, so a(5) = 6.

Links

Crossrefs

Cf. A000042, A001222, A002275, A004022, A046053.
Initial terms of A004023, A046413, A046414, A046415, A046416, A046417, A046418, A046419.
Cf. A086565 (equivalent with distinct prime factors).

Programs

  • PARI
    a(n) = my(k=1); while(bigomega((10^k - 1)/9) !=n, k++); k; \\ Michel Marcus, Apr 23 2017

Extensions

a(1) = 2 inserted and a(19)-a(37) added by Ray Chandler, Apr 23 2017
a(38)-a(40) from Jinyuan Wang, Apr 17 2020
Name corrected by Felix Fröhlich, Jun 04 2022
Previous Showing 21-23 of 23 results.

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