A073495 -id:A073495 - cp's OEIS frontend

A356227 Smallest size of a maximal gapless submultiset of the prime indices of n.

0, 1, 1, 2, 1, 2, 1, 3, 2, 1, 1, 3, 1, 1, 2, 4, 1, 3, 1, 1, 1, 1, 1, 4, 2, 1, 3, 1, 1, 3, 1, 5, 1, 1, 2, 4, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 5, 2, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 4, 1, 1, 1, 6, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 3, 1, 2, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1

Offset: 1

Gus Wiseman, Aug 13 2022

nonn

A sequence is gapless if it covers an unbroken interval of positive integers. For example, the multiset {2,3,5,5,6,9} has three maximal gapless submultisets: {2,3}, {5,5,6}, {9}.

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

			The prime indices of 18564 are {1,1,2,4,6,7}, with maximal gapless submultisets {1,1,2}, {4}, {6,7}, so a(18564) = 1.

Positions of first appearances are A000079.
The maximal gapless submultisets are counted by A287170, firsts A066205.
These are the row-minima of A356226, firsts A356232.
The greatest instead of smallest size is A356228.
A001221 counts distinct prime factors, with sum A001414.
A001222 counts prime factors with multiplicity.
A001223 lists the prime gaps, reduced A028334.
A003963 multiplies together the prime indices of n.
A056239 adds up prime indices, row sums of A112798.
A073491 lists numbers with gapless prime indices, cf. A073492-A073495.
A356224 counts even gapless divisors, complement A356225.
Cf. A000005, A055874, A060680-A060683, A132747, A132881, A137921, A193829, A286470, A356229.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[If[n==1,0,Min@@Length/@Split[primeMS[n],#1>=#2-1&]],{n,100}]

a(n) = A333768(A356230(n)).

a(n) = A055396(A356231(n)).

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.

A073494 Numbers having exactly two prime gaps in their factorization.

Original entry on oeis.org

110, 130, 170, 182, 190, 220, 230, 238, 260, 266, 273, 290, 310, 322, 340, 357, 364, 370, 374, 380, 399, 406, 410, 418, 430, 434, 440, 460, 470, 476, 483, 494, 506, 518, 520, 530, 532, 546, 550, 561, 574, 580, 590, 598, 602, 609, 610, 620, 627, 638, 644

Offset: 1

Reinhard Zumkeller, Aug 03 2002

nonn

A073490(a(n)) = 2.

			220 is a term, as 220 = 2*2*5*11 with two gaps: between 2 and 5 and between 5 and 11.

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

Cf. A005117, A073492, A073493, A073495, A073488.

a073494 n = a073494_list !! (n-1)
a073494_list = filter ((== 2) . a073490) [1..]
-- Reinhard Zumkeller, Dec 20 2013

pa[n_, k_] := If[k == NextPrime[n], 0, 1]; Select[Range[645], Total[pa @@@ Partition[First /@ FactorInteger[#], 2, 1]] == 2 &] (* Jayanta Basu, Jul 01 2013 *)

cp's OEIS Frontend

A356227 Smallest size of a maximal gapless submultiset of the prime indices of n.

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A073489 Squarefree numbers having exactly three prime gaps.

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A073494 Numbers having exactly two prime gaps in their factorization.

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