A073494 -id:A073494 - 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 *)

A073493 Numbers having exactly 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, 111, 112, 114, 115, 116, 117, 118, 119, 122, 123, 124, 126, 129

Offset: 1

Reinhard Zumkeller, Aug 03 2002

nonn

			200 is a term, as 200 = 2*2*2*5*5 with one gap between 2 and 5.

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

Cf. A005117, A073490, A073492, A073494, A073495, A073487.

a073493 n = a073493_list !! (n-1)
a073493_list = filter ((== 1) . a073490) [1..]
-- Reinhard Zumkeller, Dec 20 2013

pa[n_, k_] := If[k == NextPrime[n], 0, 1]; Select[Range[130], Total[pa @@@ Partition[First /@ FactorInteger[#], 2, 1]] == 1 &] (* Jayanta Basu, Jul 01 2013 *)

from sympy import primefactors, nextprime
def ok(n):
    pf = primefactors(n)
    return sum(p2 != nextprime(p1) for p1, p2 in zip(pf[:-1], pf[1:])) == 1
print(list(filter(ok, range(1, 130)))) # Michael S. Branicky, Oct 14 2021

A073490(a(n)) = 1.

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

A073495 Numbers having exactly three prime gaps in their factorization.

Original entry on oeis.org

1870, 2090, 2470, 2530, 2990, 3190, 3410, 3458, 3740, 3770, 3910, 4030, 4070, 4180, 4186, 4510, 4730, 4810, 4930, 4940, 5060, 5170, 5187, 5270, 5278, 5330, 5474, 5510, 5590, 5642, 5830, 5890, 5980, 6110, 6279, 6290, 6380, 6490, 6710, 6734, 6820, 6890

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.

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

Cf. A005117, A073490, A073492, A073493, A073494, A073489.

a073495 n = a073495_list !! (n-1)
a073495_list = filter ((== 3) . a073490) [1..]
-- Reinhard Zumkeller, Dec 20 2013

q[n_] := SequenceCount[FactorInteger[n][[;; , 1]], {p1_, p2_} /; p2 != NextPrime[p1], Overlaps -> True] == 3; Select[Range[7000], q] (* Amiram Eldar, Apr 10 2021*)

A073490(a(n)) = 3.

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.

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

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A073493 Numbers having exactly one prime gap in their factorization.

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A073490 Number of prime gaps in factorization of n.

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A073495 Numbers having exactly three prime gaps in their factorization.

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A073488 Squarefree numbers having exactly two prime gaps.

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