A210615 -id:A210615 - cp's OEIS frontend

A343597 Numbers divisible by a 7-smooth composite number.

4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 24, 25, 27, 28, 30, 32, 35, 36, 40, 42, 44, 45, 48, 49, 50, 52, 54, 56, 60, 63, 64, 66, 68, 70, 72, 75, 76, 78, 80, 81, 84, 88, 90, 92, 96, 98, 99, 100, 102, 104, 105, 108, 110, 112, 114, 116, 117, 120, 124, 125, 126, 128, 130

Offset: 1

Peter Munn, Apr 21 2021

Numbers divisible by at least one of 4, 6, 9, 10, 14, 15, 21, 25, 35, 49.
Exactly half of the first 10, first 100 and first 600 positive integers are divisible by a 7-smooth composite number; the largest 7-smooth divisor of the remaining numbers is 1, 2, 3, 5 or 7.
Intervals extending to hundreds of integers with exactly 50% membership of this sequence are far from rare, some notable examples being [3000, 3999], [8000, 8999], [20000, 20999], [21000, 21999] and [23000, 23999]. This reflects the asymptotic density of the corresponding set being close to 0.5, precisely 1847 / 3675 = 0.50258503... (and membership of the set has a periodic pattern). See A343598 for further information.

			33 = 11 * 3 has divisors 1, 3, 11, 33, of which only 33 is composite. 33 is not 7-smooth, as its prime factors include 11, which is greater than 7. So 33 is not in the sequence.
52 = 13 * 2 * 2 is divisible by 4, which is composite and 7-smooth, so 52 is in the sequence.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Smooth Number.

Cf. A002473 (7-smooth numbers), A014673, A020639, A210615 (smallest smoothest composite divisor), A343598.

Subsequence of A080672.

Select[Range[130], Plus @@ IntegerExponent[#, {2, 3, 5, 7}] > 1 &] (* Amiram Eldar, May 04 2021 *)

{a(n)} = {k : k >= 1, 2 <= A014673(k) <= 7}, where A014673(k) = lpf(k/lpf(k)), where lpf(m) = A020639(m), the least prime factor of m.
For n >= 1, a(22164 + n) = 44100 + a(n).
For n < 22164, a(22164 - n) = 44100 - a(n).

A210625 Least semiprime dividing digit reversal of n, or 0 if no such factor.

Original entry on oeis.org

0, 0, 0, 4, 0, 6, 0, 4, 9, 0, 0, 21, 0, 0, 51, 0, 0, 9, 91, 0, 4, 22, 4, 6, 4, 62, 4, 82, 4, 0, 0, 0, 33, 0, 0, 9, 0, 0, 93, 4, 14, 4, 34, 4, 6, 4, 74, 4, 94, 0, 15, 25, 35, 9, 55, 65, 15, 85, 95, 6, 4, 26, 4, 46, 4, 6, 4, 86, 4, 0, 0, 9, 0, 0, 57, 0, 77, 87

Offset: 1

Jonathan Vos Post, Mar 24 2012

Roughly the analog of A209190 (least prime factor of reversal of digits), but with semiprimes (A001358) instead of primes (A000040).

			a(12) = min {k such that k|R(12) and k = p*q for primes p and q (not necessarily distinct)} = min {k, k|21 and k semiprime} = 21 = 3*7.
a(42) = min {k, k|24 and k semiprime} = min {4,6} = 4 = 2*2.

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A001358, A004086, A004087, A011557, A209190, A210615.

r:= proc(n) option remember; local q;
      `if`(n<10, n, irem(n, 10, 'q') *10^(length(n)-1)+r(q))
    end:
a:= proc(n) local m, k;
      m:= r(n);
      for k from 4 to m do
         if irem(m, k)=0 and not isprime(k) and
            add(i[2], i=ifactors(k)[2])=2 then return k fi
      od; 0
    end:
seq(a(n), n=1..100);  # Alois P. Heinz, Mar 26 2012

spd[n_]:=Module[{sps=Select[Divisors[FromDigits[Reverse[ IntegerDigits[n]]]], PrimeOmega[#] == 2&,1]},If[sps=={},0,First[sps]]]; Array[spd,80] (* Harvey P. Dale, Aug 12 2012 *)

a(n) = A210615(R(n)) = A210615(A004086(n)).

a(p) = 0 iff p in (A004087 union A011557). - Alois P. Heinz, Mar 28 2012

A210665 Least semiprime dividing digit reversal of n-th semiprime, or 0 if no such factor.

Original entry on oeis.org

4, 6, 9, 0, 0, 51, 4, 22, 4, 62, 33, 0, 0, 0, 93, 4, 94, 15, 55, 15, 85, 26, 4, 4, 0, 77, 4, 58, 4, 6, 0, 39, 49, 0, 0, 111, 511, 0, 0, 121, 221, 321, 921, 0, 0, 141, 0, 341, 0, 0, 551, 851, 951, 161, 0, 961, 771, 871, 381, 581, 781, 0, 6, 202, 302, 502, 14

Offset: 1

Jonathan Vos Post, Mar 28 2012

			a(4) = 0 because the 4th semiprime is 10, and R(10) = 1, which is not divisible by any semiprime.
a(6) = 51 because the 6th semiprime is 15, and R(15) = 51, which is itself semiprime.
a(7) = 4 because the 7th semiprime is 21, R(21) = 12, and 4 is the least semiprime divisor of 12.

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A001358, A210615, A210616.

r:= proc(n) option remember; local q;
      `if`(n<10, n, irem(n, 10, 'q') *10^(length(n)-1)+r(q))
    end:
b:= proc(n) option remember; local k;
      if n=0 then 0
    else for k from b(n-1)+1
           while isprime(k) or 2<>add (i[2], i=ifactors(k)[2])
         do od; k
      fi
    end:
a:= proc(n) option remember; local m, k;
      m:= r(b(n));
      for k from 4 to m do
         if irem(m, k)=0 and not isprime(k) and
            add(i[2], i=ifactors(k)[2])=2 then return k fi
      od; 0
    end:
seq(a(n), n=1..100);  # Alois P. Heinz, Mar 28 2012

a(n) = A210615(A210616(n)).

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A343597 Numbers divisible by a 7-smooth composite number.

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A210625 Least semiprime dividing digit reversal of n, or 0 if no such factor.

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A210665 Least semiprime dividing digit reversal of n-th semiprime, or 0 if no such factor.

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Maple

Formula