A073486 -id:A073486 - cp's OEIS frontend

A073492 Numbers having at least one prime gap in their factorization.

10, 14, 20, 21, 22, 26, 28, 33, 34, 38, 39, 40, 42, 44, 46, 50, 51, 52, 55, 56, 57, 58, 62, 63, 65, 66, 68, 69, 70, 74, 76, 78, 80, 82, 84, 85, 86, 87, 88, 91, 92, 93, 94, 95, 98, 99, 100, 102, 104, 106, 110, 111, 112, 114, 115, 116, 117, 118, 119, 122, 123, 124, 126

Offset: 1

Reinhard Zumkeller, Aug 03 2002

nonn

A073490(a(n)) > 0.

A137794(a(n))=0, complement of A073491. - Reinhard Zumkeller, Feb 11 2008

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A005117, A073493, A073494, A073495, A073491, A073486.

a073492 n = a073492_list !! (n-1)
a073492_list = filter ((> 0) . a073490) [1..]
-- Reinhard Zumkeller, Dec 20 2013

pa[n_, k_] := If[k == NextPrime[n], 0, 1]; Select[Range[126],Total[pa @@@ Partition[First /@ FactorInteger[#], 2, 1]] > 0 &] (* Jayanta Basu, Jul 01 2013 *)

A073485 Product of any number of consecutive primes; squarefree numbers with no gaps in their prime factorization.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 11, 13, 15, 17, 19, 23, 29, 30, 31, 35, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 77, 79, 83, 89, 97, 101, 103, 105, 107, 109, 113, 127, 131, 137, 139, 143, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 210, 211, 221, 223, 227, 229, 233

Offset: 1

Reinhard Zumkeller, Aug 03 2002

A073484(a(n)) = 0 and A073483(a(n)) = 1;
See A097889 for composite terms. - Reinhard Zumkeller, Mar 30 2010
A169829 is a subsequence. - Reinhard Zumkeller, May 31 2010
a(A192280(n)) = 1: complement of A193166.
Also fixed points of A053590: a(n) = A053590(a(n)). - Reinhard Zumkeller, May 28 2012
The Heinz numbers of the partitions into distinct consecutive integers. The Heinz number of a partition p = [p_1, p_2, ..., p_r] is defined as Product_{j=1..r} prime(p_j) (concept used by Alois P. Heinz in A215366 as an "encoding" of a partition). Example: (i) 15 (= 3*5) is in the sequence because it is the Heinz number of the partition [2,3]; (ii) 10 (= 2*5) is not in the sequence because it is the Heinz number of the partition [1,3]. - Emeric Deutsch, Oct 02 2015
Except for the term 1, each term can uniquely represented as A002110(k)/A002110(m) for k > m >= 0; 1 = A002110(k)/A002110(k) for all k. - Michel Marcus and Jianing Song, Jun 19 2019

			105 is a term, as 105 = 3*5*7 with consecutive prime factors.

Alois P. Heinz, Table of n, a(n) for n = 1..20000 (first 1000 terms from T. D. Noe)

Complement: A193166.
Intersection of A005117 and A073491.
Subsequence of A277417.
Cf. A053590, A073483, A073484, A073486, A192280.
Cf. A000040, A006094, A002110, A097889, A169829 (subsequences).
Cf. A096334.

a073485 n = a073485_list !! (n-1)
a073485_list = filter ((== 1) . a192280) [1..]
-- Reinhard Zumkeller, May 28 2012, Aug 26 2011

isA073485 := proc(n)
    local plist,p,i ;
    plist := sort(convert(numtheory[factorset](n),list)) ;
    for i from 1 to nops(plist) do
        p := op(i,plist) ;
        if modp(n,p^2) = 0 then
            return false;
        end if;
        if i > 1 then
            if nextprime(op(i-1,plist)) <> p then
                return false;
            end if;
        end if;
    end do:
    true;
end proc:
for n from 1 to 1000 do
    if isA073485(n) then
        printf("%d,",n);
    end if;
end do: # R. J. Mathar, Jan 12 2016
# second Maple program:
q:= proc(n) uses numtheory; n=1 or issqrfree(n) and (s->
      nops(s)=1+pi(max(s))-pi(min(s)))(factorset(n))
    end:
select(q, [$1..288])[];  # Alois P. Heinz, Jan 27 2022

f[n_] := FoldList[ Times, 1, Prime[ Range[n, n + 3]]]; lst = {}; k = 1; While[k < 55, AppendTo[lst, f@k]; k++ ]; Take[ Union@ Flatten@ lst, 65] (* Robert G. Wilson v, Jun 11 2010 *)

list(lim)=my(v=List(primes(primepi(lim))),p,t);for(e=2, log(lim+.5)\log(2), p=1; t=prod(i=1,e-1,prime(i)); forprime(q=prime(e), lim, t*=q/p; if(t>lim,next(2)); listput(v,t); p=nextprime(p+1))); vecsort(Vec(v)) \\ Charles R Greathouse IV, Oct 24 2012

a(n) ~ n log n. - Charles R Greathouse IV, Oct 24 2012

Alternative description added to the name by Antti Karttunen, Oct 29 2016

A073490 Number of prime gaps in factorization of n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 1, 1, 0, 0, 1, 1, 1, 1, 0, 0, 0, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 2, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0

Offset: 1

Reinhard Zumkeller, Aug 03 2002

A137723(n) is the smallest number of the first occurring set of exactly n consecutive numbers with at least one prime gap in their factorization: a(A137723(n)+k)>0 for 0<=kA137723(n)-1)=a(A137723(n)+n)=0. - Reinhard Zumkeller, Feb 09 2008

			84 = 2*2*3*7 with one gap between 3 and 7, therefore a(84) = 1;
110 = 2*5*11 with two gaps: between 2 and 5 and between 5 and 11, therefore a(110) = 2.

R. Zumkeller, Table of n, a(n) for n = 1..50000

Cf. A073491, A073492, A073493, A073494, A073495.

Cf. A137721, A137722, A027748, A049084.

a073490 1 = 0
a073490 n = length $ filter (> 1) $ zipWith (-) (tail ips) ips
   where ips = map a049084 $ a027748_row n
-- Reinhard Zumkeller, Jul 04 2012

A073490 := proc(n)
    local a,plist ;
    plist := sort(convert(numtheory[factorset](n),list)) ;
    a := 0 ;
    for i from 2 to nops(plist) do
        if op(i,plist) <> nextprime(op(i-1,plist)) then
            a := a+1 ;
        end if;
    end do:
    a;
end proc:
seq(A073490(n),n=1..110) ; # R. J. Mathar, Oct 27 2019

gaps[n_Integer/;n>0]:=If[n===1, 0, Complement[Prime[PrimePi[Rest[ # ]]-1], # ]&[First/@FactorInteger[n]]]; Table[Length[gaps[n]], {n, 1, 120}] (* Wouter Meeussen, Oct 30 2004 *)
pa[n_, k_] := If[k == NextPrime[n], 0, 1]; Table[Total[pa @@@ Partition[First /@ FactorInteger[n], 2, 1]], {n, 120}] (* Jayanta Basu, Jul 01 2013 *)

from sympy import primefactors, nextprime
def a(n):
    pf = primefactors(n)
    return sum(p2 != nextprime(p1) for p1, p2 in zip(pf[:-1], pf[1:]))
print([a(n) for n in range(1, 121)]) # Michael S. Branicky, Oct 14 2021

a(n) = A073484(A007947(n)).
a(A000040(n))=0; a(A000961(n))=0; a(A006094(n))=0; a(A002110(n))=0; a(A073485(n))=0.
a(A073486(n))>0; a(A073487(n)) = 1; a(A073488(n))=2; a(A073489(n))=3.
a(n)=0 iff A073483(n) = 1.
a(A097889(n)) = 0. - Reinhard Zumkeller, Nov 20 2004
0 <= a(m*n) <= a(m) + a(n) + 1. A137794(n) = 0^a(n). - Reinhard Zumkeller, Feb 11 2008

More terms from Franklin T. Adams-Watters, May 19 2006

A073484 Number of gaps in factors of the n-th squarefree number.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 1, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 0, 1, 1, 1, 0, 1, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 0, 0, 1, 0, 0, 1, 0, 0, 2, 1, 0, 1, 1, 1, 1, 1, 1, 0, 1, 2, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 2, 0

Offset: 1

Reinhard Zumkeller, Aug 03 2002

nonn

			The 69th squarefree number is 110=2*5*11, therefore a(69)=2, as there are two gaps: between 2 and 5 and between 5 and 11.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A005117, A073483.

gaps[n_] := SequenceCount[FactorInteger[n][[;; , 1]], {p1_, p2_} /; p2 != NextPrime[p1], Overlaps -> True]; gaps /@ Select[Range[200], SquareFreeQ] (* Amiram Eldar, Apr 10 2021 *)

a(A000040(n))=0; a(A006094(n))=0; a(A002110(n))=0; a(A073485(n))=0.
a(A073486(n))>0; a(A073487(n))=1; a(A073488(n))=2; a(A073489(n))=3.
a(n)=0 iff A073483(n)=1.

A073488 Squarefree numbers having exactly two prime gaps.

Original entry on oeis.org

110, 130, 170, 182, 190, 230, 238, 266, 273, 290, 310, 322, 357, 370, 374, 399, 406, 410, 418, 430, 434, 470, 483, 494, 506, 518, 530, 546, 561, 574, 590, 598, 602, 609, 610, 627, 638, 651, 658, 670, 682, 710, 714, 730, 741, 742, 754, 759, 777, 782, 790

Offset: 1

Reinhard Zumkeller, Aug 03 2002

nonn

			1430 is a term, as 1430 = 2*5*11*13 with two gaps: between 2 and 5 and between 5 and 11.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A005117, A073486, A073487, A073489, A073494.

q[n_] := SequenceCount[FactorInteger[n][[;; , 1]], {p1_, p2_} /; p2 != NextPrime[p1], Overlaps -> True] == 2; Select[Range[800], SquareFreeQ[#] && q[#] &] (* Amiram Eldar, Apr 10 2021 *)

A073484(a(n)) = 2.

A073489 Squarefree numbers having exactly three prime gaps.

Original entry on oeis.org

1870, 2090, 2470, 2530, 2990, 3190, 3410, 3458, 3770, 3910, 4030, 4070, 4186, 4510, 4730, 4810, 4930, 5170, 5187, 5270, 5278, 5330, 5474, 5510, 5590, 5642, 5830, 5890, 6110, 6279, 6290, 6490, 6710, 6734, 6890, 6902, 6970, 7030, 7130, 7310, 7370, 7378

Offset: 1

Reinhard Zumkeller, Aug 03 2002

nonn

			1870 is a term, as 1870 = 2*5*11*17 = with three gaps: between 2 and 5, between 5 and 11 and between 11 and 17.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A005117, A073484, A073486, A073488, A073495.

q[n_] := SequenceCount[FactorInteger[n][[;; , 1]], {p1_, p2_} /; p2 != NextPrime[p1], Overlaps -> True] == 3; Select[Range[7500], SquareFreeQ[#] && q[#] &] (* Amiram Eldar, Apr 10 2021 *)
sfQ[n_]:=SquareFreeQ[n]&&Total[Boole[NextPrime[#[[1]]]!=#[[2]]&/@ Partition[ FactorInteger[n][[All,1]],2,1]]]==3; Select[Range[7500],sfQ] (* Harvey P. Dale, Aug 29 2021 *)

A073484(a(n)) = 3.

A073487 Squarefree numbers having exactly one prime gap.

Original entry on oeis.org

10, 14, 21, 22, 26, 33, 34, 38, 39, 42, 46, 51, 55, 57, 58, 62, 65, 66, 69, 70, 74, 78, 82, 85, 86, 87, 91, 93, 94, 95, 102, 106, 111, 114, 115, 118, 119, 122, 123, 129, 133, 134, 138, 141, 142, 145, 146, 154, 155, 158, 159, 161, 165, 166, 174, 177, 178, 183, 185

Offset: 1

Reinhard Zumkeller, Aug 03 2002

nonn

A073484(a(n)) = 1.

			78 is a term, as 78 = 2*3*13 with one gap between 3 and 13.

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

Cf. A005117, A073486, A073488, A073489, A073493.

N:= 1000: # to get all terms <= N
Res:= NULL:
for a from 1 to numtheory:-pi(isqrt(N)) do
  for b from a do
    p:= mul(ithprime(i),i=a..b);
    if p > N/ithprime(b+2) then break fi;
    for c from b+2 while p*ithprime(c) <= N do
      for d from c do
        q:= mul(ithprime(i),i=c..d);
        if p*q > N then break fi;
        Res:= Res, p*q;
      od
    od
  od
od:
sort([Res]); # Robert Israel, Apr 20 2017

okQ[n_] := SquareFreeQ[n] && Length[SequencePosition[FactorInteger[n][[All, 1]], {p_?PrimeQ, q_?PrimeQ} /; q != NextPrime[p]]] == 1;
Select[Range[200], okQ] (* Jean-François Alcover, Feb 28 2019 *)

A193166 Numbers that are not the product of consecutive primes.

Original entry on oeis.org

4, 8, 9, 10, 12, 14, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28, 32, 33, 34, 36, 38, 39, 40, 42, 44, 45, 46, 48, 49, 50, 51, 52, 54, 55, 56, 57, 58, 60, 62, 63, 64, 65, 66, 68, 69, 70, 72, 74, 75, 76, 78, 80, 81, 82, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95

Offset: 1

Reinhard Zumkeller, Aug 26 2011

nonn

A192280(a(n)) = 0: complement of A073485.

Union of A013929 and A073486.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A025475.

a193166 n = a193166_list !! (n-1)
a193166_list = filter ((== 0) . a192280) [1..]
-- Reinhard Zumkeller, May 28 2012, Aug 26 2011

A367768 Numbers k such that MMK(k) = MMK(i) for some i < k, where MMK is multiset multiplicity kernel A367580.

Original entry on oeis.org

4, 8, 9, 10, 14, 16, 18, 21, 22, 24, 25, 26, 27, 32, 33, 34, 36, 38, 39, 40, 42, 46, 48, 49, 50, 51, 54, 55, 56, 57, 58, 62, 64, 65, 66, 69, 70, 72, 74, 75, 78, 80, 81, 82, 84, 85, 86, 87, 88, 91, 93, 94, 95, 96, 98, 100, 102, 104, 106, 108, 110, 111, 112, 114

Offset: 1

Gus Wiseman, Dec 01 2023

nonn

We define the multiset multiplicity kernel (MMK) of a positive integer n to be the product of (least prime factor with exponent k)^(number of prime factors with exponent k) over all distinct exponents k appearing in the prime factorization of n. For example, 90 has prime factorization 2^1 * 3^2 * 5^1, so for k = 1 we have 2^2, and for k = 2 we have 3^1, so MMK(90) = 12. As an operation on multisets MMK is represented by A367579, and as an operation on their ranks it is represented by A367580.

			The terms together with their prime indices begin:
    4: {1,1}
    8: {1,1,1}
    9: {2,2}
   10: {1,3}
   14: {1,4}
   16: {1,1,1,1}
   18: {1,2,2}
   21: {2,4}
   22: {1,5}
   24: {1,1,1,2}
   25: {3,3}
   26: {1,6}
   27: {2,2,2}
   32: {1,1,1,1,1}
   33: {2,5}
   34: {1,7}
   36: {1,1,2,2}

The squarefree case is A073486, complement A073485.
The MMK triangle is A367579, sum A367581, min A055396, max A367583.
Sorted positions of non-first appearances in A367580.
The complement is A367585, sorted version of A367584.
A007947 gives squarefree kernel.
A027746 lists prime factors, length A001222, indices A112798.
A027748 lists distinct prime factors, length A001221, indices A304038.
A071625 counts distinct prime exponents.
A124010 gives prime signature, sorted A118914.
Cf. A051904, A072774, A130091, A181819, A246547, A367685.

nn=100;
mmk[q_]:=With[{mts=Length/@Split[q]}, Sort[Table[Min@@Select[q,Count[q,#]==i&], {i,mts}]]];
qq=Table[Times@@mmk[Join @@ ConstantArray@@@FactorInteger[n]],{n,nn}];
Select[Range[nn], MemberQ[Take[qq,#-1], qq[[#]]]&]

A367580(a(k)) = A367580(i) for some i < a(k).

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A073492 Numbers having at least one prime gap in their factorization.

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A073485 Product of any number of consecutive primes; squarefree numbers with no gaps in their prime factorization.

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A073490 Number of prime gaps in factorization of n.

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A073484 Number of gaps in factors of the n-th squarefree number.

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A073488 Squarefree numbers having exactly two prime gaps.

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A073489 Squarefree numbers having exactly three prime gaps.

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A073487 Squarefree numbers having exactly one prime gap.

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