A014612 -id:A014612 - cp's OEIS frontend

A126436 Number of composites between successive values of A014612.

2, 3, 0, 5, 0, 0, 8, 0, 0, 3, 1, 7, 2, 0, 1, 2, 0, 1, 10, 4, 0, 1, 1, 2, 2, 1, 0, 6, 0, 3, 5, 7, 0, 2, 0, 7, 0, 3, 0, 0, 0, 0, 4, 3, 1, 1, 2, 9, 3, 9, 4, 0, 3, 1, 1, 1, 0, 0, 7, 1, 2, 3, 1, 2, 1, 2, 1, 0, 0, 0, 3, 1

Offset: 1

Jonathan Vos Post, Mar 12 2007

			a(1) = 2 because there are two composites {9,10} between A014612(1)=8 and A014612(2)=12.
a(2) = 3 because there are two composites {14, 15, 16} between A014612(2)=12 and A014612(3)=18.
a(3) = 0 because there are no composites between A014612(3)=18 and A014612(4)=20, only the prime 19.
a(7) = 8 because {32,33,34,35,36,38,39,40} between A014612(7)=30 and A014612(8)=42.

3-almost prime analog of A046933 = number of composites between successive primes.

Cf. A002808, A014612, A046933, A070552, A114403.

isA014612 :=proc(n) if numtheory[bigomega](n) = 3 then true ; else false ; fi ; end: isA002808 := proc(n) RETURN(not isprime(n) and n <> 1 ); end: A126436 := proc(nmax) local a ; a := -1 ; for n from 1 to nmax do if isA014612(n) then if a >= 0 then printf("%d,",a) ; fi ; a := 0 ; elif isA002808(n) and a>= 0 then a := a+1 ; fi ; od : end: A126436(300) : # R. J. Mathar, Apr 03 2007

nmax = 72;
S = Select[Range[300](* increase range if a(n) unevaluated *), PrimeOmega[#] == 3&];
a[n_ /; n+1 <= Length[S]] := Count[Range[S[[n]]+1, S[[n+1]]-1], _?CompositeQ];
Table[a[n], {n, 1, nmax}] (* Jean-François Alcover, Oct 26 2023 *)

a(n) <= A114403(n) - 1.

More terms from R. J. Mathar, Apr 03 2007

A226768 Least prime in a string of exactly n consecutive primes all differing by 3-almost primes (A014612).

Original entry on oeis.org

2, 89, 199, 4679, 1499, 24593, 118801, 318569, 1055741, 2253311, 3695513, 4559983, 22790969, 29625383, 85780433, 108547289, 1092631789, 520956851, 23522240041, 11229387053, 50466176303, 325923513479, 57958489439, 537724352057, 932937059897, 1970378236613, 12522644028841

Offset: 1

Giovanni Resta, Jun 17 2013

nonn

a(28) > 4*10^13.

			a(3) = 199 since (199, 211, 223) is the first triple (which cannot be extended to a quadruplet) of consecutive primes whose first differences are 3-almost primes, 211-199 = 223-211 = 2^2*3.

Cf. A220697, A226769, A226840.

A361073 Lexicographically least increasing sequence of triprimes (A014612) a(n) such that a(n) - a(n-1) and a(n) + a(n-1) are also triprimes.

Original entry on oeis.org

8, 20, 50, 125, 279, 426, 531, 539, 814, 822, 897, 1002, 1010, 1076, 1146, 1209, 1325, 1353, 1398, 1406, 1516, 1558, 1868, 1898, 1948, 1978, 1986, 2013, 2225, 2233, 2397, 2527, 2547, 2575, 2763, 2783, 2810, 2908, 2938, 2946, 3009, 3054, 3081, 3414, 3422, 3452, 3522, 3567, 3714, 3759, 3786, 3813

Offset: 1

Zak Seidov and Robert Israel, Apr 09 2023

nonn

			a(3) = 50 because 50 = 2^2*5, 50 - a(2) = 30 = 2*3*5 and 50 + a(2) = 70 = 2*5*7 are all products of 3 (not necessarily distinct) primes, and 50 is the least number that works.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A014612, A361611, A361215.

A[1]:= 8:
for i from 2 to 100 do
  for x from A[i-1]+8 do
    if numtheory:-bigomega(x) = 3 and numtheory:-bigomega(x-A[i-1]) = 3 and numtheory:-bigomega(x+A[i-1]) = 3 then
       A[i]:= x; break
    fi
od od:
seq(A[i],i=1..100);

s = {m = 8}; Do[p = m + 8; While[{3, 3, 3} != PrimeOmega[{p, m + p,
p - m}], p++];  AppendTo[s, m = p], {50}]; s

A114425 Product of the first n 3-almost primes (A014612).

Original entry on oeis.org

8, 96, 1728, 34560, 933120, 26127360, 783820800, 32920473600, 1448500838400, 65182537728000, 3259126886400000, 169474598092800000, 10676899679846400000, 704675378869862400000, 47917925763150643200000

Offset: 1

Jonathan Vos Post, Feb 13 2006

3-almost prime analog of primorial (A002110). The semiprime analog of primorial is A112141. Equivalent for product of what A086062 is for sum. Bigomega(a(n)) = the number of not necessarily distinct prime factors of a(n) = A001222(a(n)) = A008585(n) = 3*n.

			a(5) = 933120 = 8 * 12 * 18 * 20 * 27 = the product of the first 5 values of the 3-almost primes = 2^8 * 3^6 * 5, which has 3*5 = 15 prime factors (with multiplicity).
a(20) = 137199755075271237225676800000000 = 8 * 12 * 18 * 20 * 27 * 28 * 30 * 42 * 44 * 45 * 50 * 52 * 63 * 66 * 68 * 70 * 75 * 76 * 78 * 92 = 2^26 * 3^15 * 5^8 * 7^4 * 11, which has 20*3 = 60 prime factors (with multiplicity).

Harvey P. Dale, Table of n, a(n) for n = 1..364

Cf. A002110, A008585, A014612, A086062, A112141.

FoldList[Times,Select[Range[70],PrimeOmega[#]==3&]] (* Harvey P. Dale, Apr 26 2020 *)

a(n) = Prod[from i = 1 to n] A014612(i).

A209323 Values of omega(n) (A001221) as n runs through the triprimes (A014612).

Original entry on oeis.org

1, 2, 2, 2, 1, 2, 3, 3, 2, 2, 2, 2, 2, 3, 2, 3, 2, 2, 3, 2, 2, 2, 3, 3, 3, 3, 2, 2, 2, 1, 3, 3, 2, 2, 2, 3, 2, 3, 3, 2, 2, 3, 2, 3, 3, 2, 3, 3, 2, 2, 3, 3, 3, 2, 3, 2, 2, 2, 3, 3, 3, 2, 3, 2, 3, 2, 2, 3, 2, 3, 3, 3, 2, 3, 2, 3, 3, 2, 2, 2, 2, 1, 3, 3, 2, 3, 2

Offset: 1

Wesley Ivan Hurt, Jan 19 2013

nonn

omega(n) is the number of distinct prime divisors of n. A number is triprime if it has exactly 3 (not necessarily distinct) prime divisors.

a(n) = omega(A014612(n)).

			First triprime = 8, has 1 distinct prime divisor (2) so a(1) = 1.

T. D. Noe, Table of n, a(n) for n = 1..1000

Cf. A001221, A014612.

t = {}; Do[f = FactorInteger[n]; If[Total[Transpose[f][[2]]] == 3, AppendTo[t, Length[f]]], {n, 400}]; t (* T. D. Noe, Jan 23 2013 *)

A217297 Triprimes (numbers that are a product of exactly three primes: A014612) that become cubes when their central digit or central pair of digits is deleted.

Original entry on oeis.org

207, 604, 654, 2007, 2037, 2057, 2067, 2097, 2107, 2197, 2247, 2337, 2367, 2387, 2397, 2527, 2547, 2597, 2607, 2637, 2667, 2697, 2717, 2737, 2817, 2847, 2877, 2937, 2967, 6014, 6034, 6044, 6054, 6094, 6114, 6124, 6154, 6194, 6214, 6234, 6254, 6284, 6294, 6394

Offset: 1

Kevin L. Schwartz and Christian N. K. Anderson, May 03 2013

If a number (a product of exactly three primes) has an odd number of digits, only its central digit is deleted to test for status as a cube; if such a number has an even number of digits, its two central digits are deleted to test whether that's a cube. - Harvey P. Dale, Dec 19 2020
In theory, a cube with an even number of digits could be represented in the sequence by up to 110 numbers by inserting {0,1,...,9} and {00,01,...,99}. In the first 10000 terms, 1079^3 has a record 46 representatives, though it is unlikely that this is a global record.
The cubes of 10, 20 and 48 are the first three cubes not represented in the sequence.
It would be nice to have a proof that this sequence is infinite. - N. J. A. Sloane, Dec 19 2020

			207 = 3*3*23 is a term: it becomes the cube 27 when the central digit is deleted.
2007 = 3*3*223 is a term: it becomes the cube 27 when the two central digits are deleted.
Here is a larger example taken at random from the b-file:
             4178131923 = (3)  (7)  (198958663)
Delete the central pair of digits and we get a cube: 41781923 = 347^3. - _N. J. A. Sloane_, Dec 19 2020

Christian N. K. Anderson, Table of n, a(n) for n = 1..10000

Cf. A014612 ("triprimes"), A225082, A080603, A000578, A339578.

cdn[n_]:=Module[{idn=IntegerDigits[n],len},len=Length[idn];If[OddQ[ len],FromDigits[ Drop[idn,{(len+1)/2}]],FromDigits[Drop[idn,{len/2,len/2+1}]]]]; Select[Range[100,100000],PrimeOmega[#]==3 && IntegerQ[ Surd[ cdn[#],3]]&]  (* Harvey P. Dale, Dec 19 2020 *)

library(gmp);
removecentraldigit<-function(x) { s=as.character(x); n=nchar(s);
    as.bigz(paste(substr(s,1,ifelse(n%%2==0,n/2-1,(n-1)/2)), substr(s,ifelse(n%%2==0,n/2+2,(n+3)/2),n),sep=""))};
istriprime=function(x) ifelse(as.bigz(x)<8,F,length(factorize(x))==3);
iscube<-function(x) ifelse(as.bigz(x)<2,T,all(table(as.numeric(factorize(x)))%%3==0));
which(sapply(201:6400, function(x) istriprime(x) & iscube(removecentraldigit(x))))+200

Edited by N. J. A. Sloane, Dec 19 2020

A339578 Numbers k such that k^3 can be obtained by taking a number m that is the product of exactly three primes (cf. A014612) and deleting the central digit if m has an odd number of digits, or by deleting the central pair of digits if m has an even number of digits.

Original entry on oeis.org

3, 4, 11, 12, 13, 14, 15, 16, 17, 18, 19, 21, 47, 49, 51, 53, 57, 59, 61, 63, 67, 69, 71, 73, 77, 79, 81, 83, 87, 89, 91, 93, 97, 99, 217, 219, 221, 223, 227, 229, 231, 233, 237, 239, 241, 243, 247, 249, 251, 253, 257, 259, 261, 263, 267, 269, 271, 273, 277, 279, 281, 283, 287, 289

Offset: 1

Harvey P. Dale and N. J. A. Sloane, Dec 20 2020

This is a kind of inverse to A217297.
From David A. Corneth, Dec 20 2020: (Start)
If a(n) > 100 then gcd(a(n), 10) = 1. Proof: Suppose gcd(a(n), 10) > 1 then a(n) is divisible by 2 or 5. As the last three digits of m (as defined in name) will have the last three digits such that m is divisible by 2^3 = 8 or 5^3 = 125 we have a number that is a product of strictly more than three primes. Contradiction.
Furthermore, a(n)^3 has an even number of digits as after deletion of the middle digits there always is an even number of digits left. (End)

			3 is a term because 3^3 = 27 can be obtained by deleting the central digit of 3*3*23 = 207.
4 is a term because 4^3 = 64 can be obtained by deleting the central pair of digits of 2*31*97 = 6014.
20 is not a term because 20^3 = 8000, and any candidate for m will end in 00, and therefore will have at least four prime factors.

David A. Corneth, Table of n, a(n) for n = 1..10000

Cf. A014612, A217297.

p3oQ[x_]:=AnyTrue[Table[With[{td=TakeDrop[IntegerDigits[x],IntegerLength[ x]/2]},FromDigits[ Flatten[ Join[ {td[[1]],{v},td[[2]]}]]]],{v,0,9}],PrimeOmega[#] == 3&]; p3eQ[x_]:= AnyTrue[ Table[With[ {td=TakeDrop[ IntegerDigits[x],IntegerLength[ x]/2]},FromDigits[Flatten[ Join[ {td[[1]],PadLeft[{w},2],td[[2]]}]]]],{w,0,99}],PrimeOmega[#]==3&]; p3Q[m_]:=p3oQ[m] || p3eQ[m]; Module[{rng={Ceiling[Reduce[k^3>#,k][[-1]]],Floor[Reduce[k^3<10#,k][[-1]]] }&/@ Table[10^(2n-1),{n,4}]},Surd[#,3]&/@Select[Flatten[Table[w^3,{w,#[[1]],#[[2]]}]&/@ rng],p3Q]] // Quiet (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Dec 20 2020 *)

is(n) = { my(d = digits(n^3), half); if(#d % 2 == 1, return(0)); half = #d \ 2; left = vector(half, i, d[i]); right = vector(half, i, d[i + half]); for(j = 1, 2, c = fromdigits(left) * 10^(half + j) + fromdigits(right); for(i = 0, 10^j - 1, if(bigomega(c + i*10^half) == 3, print(c + i*10^half); return(1) ) ) ); 0 } \\ David A. Corneth, Dec 20 2020

A358240 Consider all invertible residues mod n. For each residue, find the smallest product of three primes (A014612) which is in that residue class mod n. a(n) is the greatest of these.

Original entry on oeis.org

8, 27, 28, 45, 66, 175, 45, 105, 76, 171, 102, 325, 165, 261, 124, 273, 230, 385, 188, 369, 268, 255, 175, 475, 284, 549, 436, 477, 285, 1309, 332, 385, 430, 927, 318, 1127, 442, 639, 610, 657, 595, 1075, 742, 805, 724, 637, 646, 1705, 642, 741, 670, 1005, 885, 1435, 801, 1705, 1105, 873, 1004, 2821, 938, 873, 844

Offset: 1

Charles R Greathouse IV, Jan 18 2023

nonn

			The least product of 3 primes = 1 mod 3 is 28, while the least = 2 mod 3 is 8, so a(2) = 28.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Ramachandran Balasubramanian, Olivier Ramaré, and Priyamvad Srivastav, Product of three primes in large arithmetic progressions, arXiv:2208.04031 [math.NT], 2022.

All terms are in A014612.

Cf. A038026, A085420.

firstTri(m)=my(mod=m.mod); forprime(p=2,, if(mod%p==0, next); forprime(q=2,p, if(mod%q==0, next); forprimestep(r=2,q,m/p/q, return(p*q*r))))
a(n)=my(r=8); for(k=1,n-1, if(gcd(k,n)>1, next); r=max(firstTri(Mod(k,n)),r)); r

A result of Balasubramanian, Ramaré, & Srivastav proves that a(n) < n^e for each e > 9/2 and large enough n depending on e.

Corrected by Charles R Greathouse IV, May 10 2023

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

A121906 Excess of n-th 3-almost prime A014612 over previous prime.

Original entry on oeis.org

1, 1, 1, 1, 4, 5, 1, 1, 1, 2, 3, 5, 2, 5, 1, 3, 2, 3, 5, 3, 1, 2, 1, 2, 1, 1, 3, 4, 11, 12, 3, 1, 8, 9, 2, 3, 1, 2, 3, 4, 5, 1, 2, 1, 5, 7, 9, 2, 8, 1, 11, 1, 2, 3, 5, 1, 3, 4, 5, 4, 1, 4, 3, 5, 2, 4, 2, 1, 1, 2, 3, 7, 9, 3, 3, 1, 5, 8, 1, 2

Offset: 1

Jonathan Vos Post, Sep 01 2006

nonn

a(n) = 1 iff n-th 3-almost prime is of form prime + 1 (not yet in OEIS). See also: A109067 3-almost primes of the form semiprime + 1.

			a(1) = 8 - 7 = 1; a(2) = 12 - 11 = 1; a(3) = 18 - 17 = 1;
a(4) = 20 - 19 = 1; a(5) = 27 - 23 = 4; a(6) = 28 - 23 = 5;
a(7) = 30 - 29 = 1.

Cf. A000040, A014612, A114403.

A014612 := proc(n) option remember; local a; if n = 1 then 8; else for a from procname(n-1)+1 do if numtheory[bigomega](a) = 3 then return a; end if; end do: end if; end proc:
A121906 := proc(n) local t; t := A014612(n) ; t-prevprime(t) ; end proc:
seq(A121906(n),n=1..80) ; # R. J. Mathar, Dec 22 2010

lim=335;p3=Select[Range[lim], PrimeOmega[#] == 3 &] ;l3=Length[p3];Table[p3[[n]]-NextPrime[p3[[n]],-1],{n,l3}] (* James C. McMahon, Oct 24 2024 *)

a(n) = Min{A014612(n) - p where p is in A000040 and 1

A014612(n)}.

a(13) corrected by R. J. Mathar, Dec 22 2010

cp's OEIS Frontend

A126436 Number of composites between successive values of A014612.

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A226768 Least prime in a string of exactly n consecutive primes all differing by 3-almost primes (A014612).

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A361073 Lexicographically least increasing sequence of triprimes (A014612) a(n) such that a(n) - a(n-1) and a(n) + a(n-1) are also triprimes.

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A114425 Product of the first n 3-almost primes (A014612).

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A209323 Values of omega(n) (A001221) as n runs through the triprimes (A014612).

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A217297 Triprimes (numbers that are a product of exactly three primes: A014612) that become cubes when their central digit or central pair of digits is deleted.

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A339578 Numbers k such that k^3 can be obtained by taking a number m that is the product of exactly three primes (cf. A014612) and deleting the central digit if m has an odd number of digits, or by deleting the central pair of digits if m has an even number of digits.

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A358240 Consider all invertible residues mod n. For each residue, find the smallest product of three primes (A014612) which is in that residue class mod n. a(n) is the greatest of these.

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