A137794 -id:A137794 - cp's OEIS frontend

A073491 Numbers having no prime gaps in their factorization.

1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 13, 15, 16, 17, 18, 19, 23, 24, 25, 27, 29, 30, 31, 32, 35, 36, 37, 41, 43, 45, 47, 48, 49, 53, 54, 59, 60, 61, 64, 67, 71, 72, 73, 75, 77, 79, 81, 83, 89, 90, 96, 97, 101, 103, 105, 107, 108, 109, 113, 120, 121, 125, 127, 128, 131, 135

Offset: 1

Reinhard Zumkeller, Aug 03 2002

A073490(a(n)) = 0; subsequences are: A000040, A000961, A006094, A002110, A000142, A073485.
A137721(n) = number of terms not greater than n; A137794(a(n))=1; complement of A073492. - Reinhard Zumkeller, Feb 11 2008
Essentially the same as A066311. - R. J. Mathar, Sep 23 2008
The Heinz numbers of the partitions that have no gaps. The Heinz number of a partition p = [p_1, p_2, ..., p_r] is defined as Product_{j=1..r} (p_j-th prime) (concept used by Alois P. Heinz in A215366 as an "encoding" of a partition). Example: (i) 18 (= 2*3*3) is in the sequence because it is the Heinz number of the partition [1,2,2]; (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

			360 is a term, as 360 = 2*2*2*3*3*5 with consecutive prime factors.

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

Cf. A137791, A137792, A137793, A137895.

a073491 n = a073491_list !! (n-1)
a073491_list = filter ((== 0) . a073490) [1..]
-- Reinhard Zumkeller, Dec 20 2013

ok[n_] := (p = FactorInteger[n][[All, 1]]; PrimePi[Last@p] - PrimePi[First@p] == Length[p] - 1); Select[Range[135], ok] (* Jean-François Alcover, Apr 29 2011 *)
npgQ[n_]:=Module[{f=Transpose[FactorInteger[n]][[1]]},f==Prime[Range[ PrimePi[ f[[1]]], PrimePi[f[[-1]]]]]]; Join[{1},Select[Range[2,200],npgQ]] (* Harvey P. Dale, Apr 12 2013 *)

is(n)=my(f=factor(n)[,1]); for(i=2,#f,if(precprime(f[i]-1)>f[i-1], return(0))); 1 \\ Charles R Greathouse IV, Apr 28 2015

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

Original entry on oeis.org

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 *)

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

A137721 Number of numbers not greater than n with no prime gaps in their factorization.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 9, 10, 11, 12, 12, 13, 14, 15, 16, 17, 17, 17, 17, 18, 19, 20, 20, 21, 21, 22, 23, 24, 25, 25, 25, 26, 27, 28, 28, 28, 28, 29, 29, 30, 30, 31, 31, 32, 33, 34, 34, 34, 34, 35, 36, 36, 36, 36, 36, 37, 38, 39, 39, 39, 40, 40, 40, 41, 41, 41, 41, 42, 43, 44

Offset: 1

Reinhard Zumkeller, Feb 09 2008

nonn

a(n) > a(n-1) iff A073490(n) = 0;
a(n) > A137722(n) for n < 134;
a(n) < A137722(n) for n > 140;
a(A137723(n) + n) = a(A137723(n)) + 1.
Partial sums of A137794. - Reinhard Zumkeller, Feb 11 2008

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Cf. A073490, A073491, A137722, A137723, A137794.

b[n_] := With[{pp = PrimePi @ FactorInteger[ n ][[All, 1]]},
     Boole[pp[[-1]] - pp[[1]] + 1 == Length[pp]]]; (* b is A137794 *)
Array[b, 105] // Accumulate (* Jean-François Alcover, Dec 09 2021 *)

a(n) = Sum_{k=1..n} 0^A073490(k).

A137795 Smallest positive m such that m*n is free of prime gaps in canonical factorization.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 15, 1, 1, 1, 1, 1, 3, 5, 105, 1, 1, 1, 1155, 1, 15, 1, 1, 1, 1, 35, 15015, 1, 1, 1, 255255, 385, 3, 1, 5, 1, 105, 1, 4849845, 1, 1, 1, 3, 5005, 1155, 1, 1, 7, 15, 85085, 111546435, 1, 1, 1, 3234846615, 5, 1, 77, 35, 1, 15015, 1616615, 3, 1, 1

Offset: 1

Reinhard Zumkeller, Feb 11 2008

nonn

			n=42: A073490(42) = A073490([2*3]*[7]) = 1,
the gap is filled by a(42) = 5: A073490(42*5) = 0.

Antti Karttunen, Table of n, a(n) for n = 1..4096
Index entries for sequences computed from indices in prime factorization

Cf. A020639, A034386, A073490, A073491, A073492, A083720, A137794.

A137795(n) = if(1==n,1, my(f = factor(n), p = f[1, 1], gpf = f[#f~, 1], m = 1); while(pAntti Karttunen, Sep 06 2018

A073490(n*a(n)) = 0; A137794(n*a(n)) = 1.
For m < a(n), A073490(n*m) > 0 and A137794(n*m) = 0.
a(A073491(n)) = 1; a(A073492(n)) > 1.
a(n) = A083720(n) / A034386(A020639(n)-1). - Peter Munn, Feb 24 2024

A302049 a(n) = 1 if n = prime(k)*prime(1+k) for some k, otherwise 0.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Antti Karttunen, Apr 24 2018

nonn

Characteristic function for products of 2 successive primes (A006094).

Cf. A006094, A078898, A137794, A246277, A280710, A302047, A302048.

A302049(n) = if(n<=1,0,my(p=precprime(sqrtint(n))); p>1 && 0==(n%p) && isprime(n/p) && (nextprime(p+1)==n/p)); \\ After code in A006094

first(n) = my(res = vector(n), p = 2); forprime(q = 3, , if(p * q > n, return(res)); res[p * q]++; p = q) \\ David A. Corneth, Apr 24 2018

a(n) = A137794(n) * A280710(n).

a(n) = A185013(A246277(n)) = A185013(A078898(n)).

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A073491 Numbers having no prime gaps in their factorization.

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

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A073490 Number of prime gaps in factorization of n.

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A137721 Number of numbers not greater than n with no prime gaps in their factorization.

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A137795 Smallest positive m such that m*n is free of prime gaps in canonical factorization.

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A302049 a(n) = 1 if n = prime(k)*prime(1+k) for some k, otherwise 0.

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