A024556 -id:A024556 - cp's OEIS frontend

A321644 Squarefree odd composite numbers whose factors are all twin primes (not necessarily from the same pair).

15, 21, 33, 35, 39, 51, 55, 57, 65, 77, 85, 87, 91, 93, 95, 105, 119, 123, 129, 133, 143, 145, 155, 165, 177, 183, 187, 195, 203, 205, 209, 213, 215, 217, 219, 221, 231, 247, 255, 273, 285, 287, 295, 301, 303, 305, 309, 319, 321, 323, 327, 341, 355, 357, 365

Offset: 1

Dimitris Valianatos, Nov 15 2018

This sequence has infinitely many terms if and only if the twin prime conjecture is true.

			a(3) = 33 = 3 * 11; 3 and 11 are both twin primes, but not from the same pair.

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

Subsequence of A024556, and hence of A056911, A061346, and A071904.

Cf. A001097.

N:= 1000: # to get all terms <= N
P:= select(isprime, {seq(i,i=3..(N+6)/3,2)}):
TP:= P intersect map(`-`,P,2):
TP:= TP union map(`+`,TP,2):
Agenda:= map(t -> [t],TP): Res:= NULL:
while Agenda <> {} do
   Agenda:= map(proc(t) local s; seq([op(t),s], s = select(s -> s > t[-1] and s*convert(t,`*`) <= N , TP)) end proc, Agenda);
   Res:= Res, op(map(convert,Agenda,`*`));
od:
sort([Res]); # Robert Israel, Jan 27 2019

seqQ[n_] := CompositeQ[n] && SquareFreeQ[n] && Module[{f = FactorInteger[n][[;;, 1]]}, Length[Select[f, PrimeQ[# - 2] || PrimeQ[# + 2] &]] == Length[f]]; Select[ Range[1, 365, 2], seqQ] (* Amiram Eldar, Nov 15 2018 *)

{forcomposite(n=3, 1000, if(moebius(n) <> 0, v = factor(n)~; i = 0;for(k = 1, #v,p=v[1,k]; if(isprime(p-2)||isprime(p+2), i++));if(i==#v,print1(n", "))))}

A322493 Start of first occurrence of n consecutive odd squarefree composite numbers.

Original entry on oeis.org

15, 33, 91, 213, 213, 893, 893, 1343

Offset: 1

Hugo Pfoertner, Dec 19 2018

The sequence is finite because among 9 or more consecutive odd numbers there is always a multiple of 3*3. - Rémy Sigrist, Dec 19 2018

			a(1) = 15 = A024556(1).
a(2) = 33 because 33 = 3*11 and 35 = 5*7 is the smallest pair of odd squarefree composite numbers. 31 and 37 are prime.
a(8) = 1343: 1343=17*79, 1345=5*269, 1347=3*449, 1349=19*71, 1351=7*193, 1353=3*11*41, 1355=5*271, 1357=23*59, whereas 1341=3^2*149 and 1359=3^2*151 are not squarefree.

Cf. A024556, A321617.

a[n_] := For[k = 1, True, k = k+2, If[AllTrue[Range[k, k+2(n-1), 2], CompositeQ[#] && SquareFreeQ[#]&], Return[k]]];
Array[a, 8] (* Jean-François Alcover, Dec 31 2018 *)

A382856 Numbers whose prime indices do not have a mode of 1.

Original entry on oeis.org

1, 3, 5, 7, 9, 11, 13, 15, 17, 18, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49, 50, 51, 53, 54, 55, 57, 59, 61, 63, 65, 67, 69, 71, 73, 75, 77, 79, 81, 83, 85, 87, 89, 90, 91, 93, 95, 97, 98, 99, 101, 103, 105, 107, 108, 109, 111, 113, 115

Offset: 1

Gus Wiseman, Apr 07 2025

nonn

			The terms together with their prime indices begin:
   1: {}
   3: {2}
   5: {3}
   7: {4}
   9: {2,2}
  11: {5}
  13: {6}
  15: {2,3}
  17: {7}
  18: {1,2,2}
  19: {8}
  21: {2,4}
  23: {9}
  25: {3,3}
  27: {2,2,2}

The case of non-unique mode is A024556.
The complement is A360015 except first.
Partitions of this type are are counted by A382526 except first, complement A241131.
A091602 counts partitions by the greatest multiplicity, rank statistic A051903.
A112798 lists prime indices, length A001222, sum A056239.
A116598 counts ones in partitions, rank statistic A007814.
A240312 counts partitions with max part = max multiplicity, ranks A381542.
A362611 counts modes in prime indices, triangle A362614.
For co-mode see A359178, A362613, A364061 (A364062), A364158 (A364159).
Cf. A000265, A002865, A106529, A327473, A327476, A362605, A363486, A356862, A360013, A360014, A381437.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],FreeQ[Commonest[prix[#]],1]&]

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A321644 Squarefree odd composite numbers whose factors are all twin primes (not necessarily from the same pair).

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A322493 Start of first occurrence of n consecutive odd squarefree composite numbers.

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A382856 Numbers whose prime indices do not have a mode of 1.

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