A367227 - OEIS

A367227

Numbers m whose prime indices have no nonnegative linear combination equal to bigomega(m).

3, 5, 7, 11, 13, 17, 19, 23, 25, 27, 29, 31, 35, 37, 41, 43, 47, 49, 53, 55, 59, 61, 63, 65, 67, 71, 73, 77, 79, 83, 85, 89, 91, 95, 97, 99, 101, 103, 107, 109, 113, 115, 117, 119, 121, 127, 131, 133, 137, 139, 143, 145, 147, 149, 151, 153, 155, 157, 161, 163

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OFFSET

1,1

COMMENTS

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.

These are the Heinz numbers of the partitions counted by A367219.

All terms are odd. Includes all odd primes and semiprimes coprime to 6. - Robert Israel, Jul 16 2026

LINKS

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

EXAMPLE

The prime indices of 24 are {1,1,1,2} with (1+1+1+1) = 4 or (1+1)+(2) = 4 or (2+2) = 4, so 24 is not in the sequence.

The terms together with their prime indices begin:

3: {2} 43: {14} 85: {3,7}

5: {3} 47: {15} 89: {24}

7: {4} 49: {4,4} 91: {4,6}

11: {5} 53: {16} 95: {3,8}

13: {6} 55: {3,5} 97: {25}

17: {7} 59: {17} 99: {2,2,5}

19: {8} 61: {18} 101: {26}

23: {9} 63: {2,2,4} 103: {27}

25: {3,3} 65: {3,6} 107: {28}

27: {2,2,2} 67: {19} 109: {29}

29: {10} 71: {20} 113: {30}

31: {11} 73: {21} 115: {3,9}

35: {3,4} 77: {4,5} 117: {2,2,6}

37: {12} 79: {22} 119: {4,7}

41: {13} 83: {23} 121: {5,5}

MAPLE

filter:= proc(n) local F, W, w, S, j, k;

F:= ifactors(n)[2];

W:= map(t -> NumberTheory:-pi(t[1]), F);

w:= convert(F[.., 2], `+`);

S:= {0};

for j from 1 to nops(F) do

S:= map(s -> seq(s+k*W[j], k=0..(w-s)/W[j]), S);

if member(w, S) then return false fi

od;

true

end proc:

select(filter, [$2..200]); # Robert Israel, Jul 16 2026

MATHEMATICA

prix[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];

combs[n_, y_]:=With[{s=Table[{k, i}, {k, y}, {i, 0, Floor[n/k]}]}, Select[Tuples[s], Total[Times@@@#]==n&]];

Select[Range[100], combs[PrimeOmega[#], Union[prix[#]]]=={}&]

CROSSREFS

The following sequences count and rank integer partitions and finite sets according to whether their length is a subset-sum or linear combination of the parts. The current sequence is starred.

sum-full sum-free comb-full comb-free

-------------------------------------------

partitions: A367212 A367213 A367218 A367219

strict: A367214 A367215 A367220 A367221

subsets: A367216 A367217 A367222 A367223

ranks: A367224 A367225 A367226 A367227*

A000700 counts self-conjugate partitions, ranks A088902.

A112798 lists prime indices, reverse A296150, length A001222, sum A056239.

A124506 appears to count combination-free subsets, differences of A326083.

A229816 counts partitions whose length is not a part, ranks A367107.

A304792 counts subset-sums of partitions, strict A365925.

A365046 counts combination-full subsets, differences of A364914.