A105999 -id:A105999 - cp's OEIS frontend

A105997 Semiprime function n -> A001358(n) applied three times to n.

26, 39, 74, 77, 118, 119, 178, 194, 219, 235, 299, 301, 329, 377, 381, 454, 471, 502, 535, 565, 566, 634, 679, 703, 721, 779, 842, 886, 893, 914, 973, 995, 998, 1006, 1126, 1174, 1227, 1282, 1294, 1317, 1337, 1343, 1389, 1418, 1457, 1563, 1577, 1623, 1642

Offset: 1

Jonathan Vos Post, Apr 29 2005

			a(1) = semiprime(semiprime(semiprime(1))) = semiprime(semiprime(4)) = semiprime(10) = 26.

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Cf. A001358, A007097, A091022, A105998, A105999.

issp:= n-> not isprime(n) and numtheory[bigomega](n)=2:
sp:= proc(n) option remember; local k; if n=1 then 4 else
       for k from 1+sp(n-1) while not issp(k) do od; k fi end:
a:= n-> (sp@@3)(n):
seq(a(n), n=1..49);  # Alois P. Heinz, Aug 16 2024

f[n_] := Plus @@ Flatten[ Table[ # [[2]], {1}] & /@ FactorInteger[ n]]; t = Select[ Range[ 1700], f[ # ] == 2 &]; Table[ Nest[ t[[ # ]] &, n, 3], {n, 50}] (* Robert G. Wilson v, Apr 30 2005 *)

from math import isqrt
from sympy import primepi, primerange
def A105997(n):
    def f(x,n): return int(n+x+((t:=primepi(s:=isqrt(x)))*(t-1)>>1)-sum(primepi(x//k) for k in primerange(1, s+1)))
    def A001358(n):
        m, k = n, f(n,n)
        while m != k:
            m, k = k, f(k,n)
        return m
    return A001358(A001358(A001358(n))) # Chai Wah Wu, Aug 16 2024

a(n) = A001358(A001358(A001358(n))).

Corrected and extended by Robert G. Wilson v, Apr 30 2005

A105998 Semiprime function n -> A001358(n) applied four times to n.

Original entry on oeis.org

77, 119, 219, 235, 377, 381, 566, 634, 721, 779, 998, 1006, 1126, 1282, 1294, 1563, 1642, 1745, 1853, 1959, 1961, 2209, 2402, 2483, 2554, 2785, 3005, 3149, 3173, 3242, 3481, 3574, 3587, 3622, 4101, 4282, 4471, 4681, 4714, 4798, 4859, 4882, 5095, 5201

Offset: 1

Jonathan Vos Post, Apr 29 2005

			a(1) = semiprime(semiprime(semiprime(semiprime(1)))) = semiprime(semiprime(semiprime(4))) = semiprime(semiprime(10)) = semiprime(26) = 77.

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Cf. A001358, A007097, A091022, A105997, A105999.

issp:= n-> not isprime(n) and numtheory[bigomega](n)=2:
sp:= proc(n) option remember; local k; if n=1 then 4 else
       for k from 1+sp(n-1) while not issp(k) do od; k fi end:
a:= n-> (sp@@4)(n):
seq(a(n), n=1..44);  # Alois P. Heinz, Aug 16 2024

f[n_] := Plus @@ Flatten[ Table[ # [[2]], {1}] & /@ FactorInteger[ n]]; t = Select[ Range[ 5210], f[ # ] == 2 &]; Table[ Nest[ t[[ # ]] &, n, 4], {n, 45}] (* Robert G. Wilson v, Apr 30 2005 *)

from math import isqrt
from sympy import primepi, primerange
def A105998(n):
    def f(x): return int(x+((t:=primepi(s:=isqrt(x)))*(t-1)>>1)-sum(primepi(x//k) for k in primerange(1, s+1)))
    def A001358(n):
        m, k = n, f(n)+n
        while m != k:
            m, k = k, f(k)+n
        return m
    return A001358(A001358(A001358(A001358(n)))) # Chai Wah Wu, Aug 16 2024

a(n) = A001358(A001358(A001358(A001358(n)))).

More terms from Robert G. Wilson v, Apr 30 2005

A119684 Ternary emirpimes.

Original entry on oeis.org

112, 211, 1021, 1102, 1201, 2011, 2022, 2202, 10111, 11101, 11112, 12102, 12202, 12212, 20121, 20212, 20221, 21111, 21202, 21221, 100102, 100201, 101011, 101122, 101221, 102001, 102002, 102012, 102022, 102122, 102222, 110101, 110102, 110122, 110211, 111102, 111202, 112011, 112121, 112122, 112202

Offset: 1

Jonathan Vos Post, Jun 08 2006

These are semiprimes when read as base 3 numbers and their reversals are different semiprimes when read as base 3 numbers. Base 10 these are: 14, 22, 34, 38, 46, 58, 62, 74, 94, 118, 122, 146, 155, 158, 178, 185, 187, ... See: A097393 Emirpimes: numbers n such that n and its reversal are distinct semiprimes. See: A004086 Read n backwards (referred to as R(n) in many sequences). See: A007089 Numbers in base 3.

Apparently numbers with trailing zeros (reversed with leading zeros), like 1220 and 10020, are not included. - R. J. Mathar, Dec 22 2010

			a(1) = 112 because 112 (base 3) = 14 (base 10) is semiprime and R(112) = 211, where 211 (base 3) = 22 (base 10) is a different semiprime.
a(13) = 12202 because 12202 (base 3) = 155 (base 10) is semiprime and R(12202) = 20221, where 20221 (base 3) = 187 (base 10) is a different semiprime.

Robert Israel, Table of n, a(n) for n = 1..10000
Eric Weisstein, Jonathan Vos Post, et al., Emirpimes.
Eric Weisstein, Vincenzo Origlio, et al., Ternary.

Cf. A001358, A004086, A007089, A097393.

R:= NULL: count:= 0:
for m from 2 while count < 100 do
for j from 1 to 2 while count < 100 do
  n:= 3*m+j;
  if numtheory:-bigomega(n) <> 2 then next fi;
  L:= convert(n,base,3);
  r:= add(L[-i]*3^(i-1),i=1..nops(L));
  if r <> n and numtheory:-bigomega(r) = 2 then
     count:= count+1; R:= R, add(L[i]*10^(i-1),i=1..nops(L))
  fi
od od:
R; # Robert Israel, Jun 07 2020

(* First run the program for A105999 *) SemiPrimeQ[n_Integer] := TrueQ[SemiPrimePi[n] > SemiPrimePi[n - 1]]; BaseForm[Select[Table[SemiPrime[n], {n, 100}], GCD[#, 3] == 1 && # != FromDigits[Reverse[IntegerDigits[#, 3]], 3] && SemiPrimeQ[FromDigits[Reverse[IntegerDigits[#, 3]], 3]] &], 3] (* From Alonso del Arte, Dec 22 2010 *)

a(n) = A007089(i) for some i in A001358 and R(a(n)) = A007089(j) for some j =/= i in A001358. a(n) = A007089(i) for some i in A001358 and A004086(a(n)) = A007089(j) for some j =/= i in A001358.

More terms from Robert Israel, Jun 07 2020

A119965 The 3-almost primeth recurrence: a(0) = 1, a(n+1) = 3-almostprime(a(n)) = A014612(a(n)).

Original entry on oeis.org

1, 8, 42, 174, 705, 2764, 10772, 41967, 164793, 654242, 2634801, 10787937, 44983894, 191249703, 829651874, 3673967785, 16612478231, 76708135651, 361707435767, 1741601413569, 8561660600005

Offset: 0

Jonathan Vos Post and Robert G. Wilson v, May 31 2006

nonn

3-almostprime equivalent of Wilson's primeth recurrence: A007097.

Cf. A007097, A105999, A119966.

ThreeAlmostPrimePi[n_] := Sum[PrimePi[n/(Prime@i*Prime@j)] - j + 1, {i, PrimePi[n^(1/3)]}, {j, i, PrimePi@Sqrt[n/Prime@i]}]; ThreeAlmostPrime[n_] := Block[{e = Floor[Log[2, n] + 1], a, b}, a = 2^e; Do[b = 2^p; While[ThreeAlmostPrimePi[a] < n, a = a + b]; a = a - b/2, {p, e, 0, -1}]; a + b/2]; NestList[ThreeAlmostPrime@# &, 1, 18]

a(19)-a(20) from Donovan Johnson, Sep 29 2010

A119966 The n-almost primeth recurrence: a(0) = 1, a(n) = n-almostprime(a(n-1)).

Original entry on oeis.org

1, 2, 6, 28, 220, 2565, 45846, 1268622, 55336336, 3876385680, 443603651136, 84205632289664

Offset: 0

Jonathan Vos Post and Robert G. Wilson v, May 31 2006

nonn

			a(0)=1, a(1) is the first prime 2, a(2) is the second semiprime 6, a(3) is the sixth 3-almost prime 28, etc.

Cf. A007097, A105999, A119965.

AlmostPrimePi[k_Integer, n_] := Module[{a, i}, a[0] = 1; If[k == 1, PrimePi[n], Sum[PrimePi[n/Times @@ Prime[Array[a, k - 1]]] - a[k - 1] + 1, Evaluate[ Sequence @@ Table[{a[i], a[i - 1], PrimePi[(n/Times @@ Prime[Array[a, i - 1]])^(1/(k - i + 1))]}, {i, k - 1}]]]]]; (* Eric W. Weisstein, Feb 07 2006 *)
AlmostPrime[k_Integer,n] = Block[{e = Floor[ Log[2, n] + 1], a, b}, a = 2^e; Do[b = 2^p; While[AlmostPrimePi[k, a] < n, a = a + b]; a = a - b/2, {p, e, 0, -1}]; a + b/2]; a[n_] := a[n] = AlmostPrime[n, a[n - 1]]; a[0] = 1; Array[a, 11, 0] (* Robert G. Wilson v, Apr 14 2017 *)

a(10)-a(11) from Donovan Johnson, Sep 18 2010

A179939 Largest semiprime divisor of all composite numbers between semiprime(n) and semiprime(n+1), or 0 if there are none.

Original entry on oeis.org

0, 4, 0, 6, 0, 10, 0, 6, 0, 15, 0, 0, 9, 0, 22, 6, 25, 26, 14, 0, 15, 21, 34, 35, 38, 39, 21, 0, 0, 22, 46, 0, 0, 51, 55, 57, 58, 0, 15, 0, 0, 62, 65, 0, 69, 0, 0, 9, 0, 77, 39, 0, 10, 82, 21, 87, 0, 91, 46, 93, 95, 65, 0, 0, 51, 0, 69, 106

Offset: 1

Jonathan Vos Post, Jan 12 2011

This is to A052248 as semiprimes (A001358) are to primes (A000040). This defines a mapping f from semiprimes to semiprimes or 0 and f(s) < s holds for all semiprimes s. There is a block of k-1 consecutive 0's corresponding to each block of k consecutive semiprimes (i.e., a block of two consecutive 0's starting at the least of the triples in A115394).

			a(1) = 0 because there are no composite numbers between the 1st semiprime 4 and the 2nd semiprime 6.
a(2) = 4 because the composite numbers between the 2nd semiprime 6 and the 3rd semiprime 9 are {8} which is divisible by the semiprime 4=2*2.
a(10) = 15 because the composite numbers between the 10th semiprime 26 and the 11th semiprime 33 are {27, 28, 30, 32} of which the maximum is found for 30 which is divisible by the semiprime 15=3*5.

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

Cf. A001358, A006530, A052248, A115394, A179312.

b:= proc(n) option remember; local k;
      if n=1 then 4
             else for k from b(n-1)+1 while
                    isprime(k) or add(i[2], i=ifactors(k)[2])<>2
                  do od; k
      fi
    end:
a:= proc(n) option remember; local k, l;
      k, l:= b(n)+1, b(n+1)-1;
      max(0,seq(seq(`if`(irem(j, b(i))=0, b(i), NULL),
                     i=1..n), j=k..l))
    end:
seq(a(n), n=1..100);  # Alois P. Heinz, Jan 14 2011

(* First run the program for A105999 *) semiPrimeQ[x_] := TrueQ[Plus @@ Last /@ FactorInteger[x] == 2]; spGPF[start_, end_] := Module[{divList, spList}, divList = Union[Flatten[Table[Divisors[n], {n, start + 1, end - 1}]]]; spList = Select[divList, semiPrimeQ]; If[Length[spList] > 0, Return[Max[spList]], Return[0]]]; Table[spGPF[SemiPrime[n], SemiPrime[n + 1]], {n, 50}] (* Alonso del Arte, Jan 13 2011 *)

a(n) = max_{A001358(n) < k < A001358(n+1)} A179312(k).

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A105997 Semiprime function n -> A001358(n) applied three times to n.

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A105998 Semiprime function n -> A001358(n) applied four times to n.

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A119684 Ternary emirpimes.

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A119965 The 3-almost primeth recurrence: a(0) = 1, a(n+1) = 3-almostprime(a(n)) = A014612(a(n)).

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A119966 The n-almost primeth recurrence: a(0) = 1, a(n) = n-almostprime(a(n-1)).

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A179939 Largest semiprime divisor of all composite numbers between semiprime(n) and semiprime(n+1), or 0 if there are none.

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