cp's OEIS Frontend

A379313 Positive integers whose prime indices are not all composite.

%I A379313 #5 Dec 28 2024 19:35:57
%S A379313 2,3,4,5,6,8,9,10,11,12,14,15,16,17,18,20,21,22,24,25,26,27,28,30,31,
%T A379313 32,33,34,35,36,38,39,40,41,42,44,45,46,48,50,51,52,54,55,56,57,58,59,
%U A379313 60,62,63,64,65,66,67,68,69,70,72,74,75,76,77,78,80,81,82
%N A379313 Positive integers whose prime indices are not all composite.
%C A379313 Or, positive integers whose prime indices include at least one 1 or prime number.
%C A379313 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.
%e A379313 The terms together with their prime indices begin:
%e A379313      2: {1}
%e A379313      3: {2}
%e A379313      4: {1,1}
%e A379313      5: {3}
%e A379313      6: {1,2}
%e A379313      8: {1,1,1}
%e A379313      9: {2,2}
%e A379313     10: {1,3}
%e A379313     11: {5}
%e A379313     12: {1,1,2}
%e A379313     14: {1,4}
%e A379313     15: {2,3}
%e A379313     16: {1,1,1,1}
%e A379313     17: {7}
%e A379313     18: {1,2,2}
%e A379313     20: {1,1,3}
%e A379313     21: {2,4}
%e A379313     22: {1,5}
%e A379313     24: {1,1,1,2}
%t A379313 prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A379313 Select[Range[100],!And@@CompositeQ/@prix[#]&]
%Y A379313 Partitions of this type are counted by A000041 - A023895.
%Y A379313 The "old" primes are listed by A008578.
%Y A379313 For no composite parts we have A302540, counted by A034891 (strict A036497).
%Y A379313 The complement is A320629, counted by A023895 (strict A204389).
%Y A379313 For a unique prime we have A331915, counted by A379304 (strict A379305).
%Y A379313 Positions of nonzeros in A379311.
%Y A379313 For a unique 1 or prime we have A379312, counted by A379314 (strict A379315).
%Y A379313 A000040 lists the prime numbers, differences A001223.
%Y A379313 A002808 lists the composite numbers, nonprimes A018252, differences A073783 or A065310.
%Y A379313 A055396 gives least prime index, greatest A061395.
%Y A379313 A056239 adds up prime indices, row sums of A112798, counted by A001222.
%Y A379313 A080339 is the characteristic function for the old prime numbers.
%Y A379313 A376682 gives k-th differences of old prime numbers, see A030016, A075526.
%Y A379313 A377033 gives k-th differences of composite numbers, see A073445, A377034.
%Y A379313 Other counts of prime indices:
%Y A379313 - A257991 odd, see A000041, A000070, A066207, A349158.
%Y A379313 - A257992 even, see A000009, A038348, A066208, A379317.
%Y A379313 - A257994 prime, see A000586, A000607, A076610, A331386.
%Y A379313 - A330944 nonprime, see A002095, A096258, A320628, A330945.
%Y A379313 - A379306 squarefree, see A302478, A379308, A379309, A379316.
%Y A379313 - A379310 nonsquarefree, see A114374, A256012, A379307.
%Y A379313 `Cf. A038550, A087436, A173390, A379300, A379301.
%K A379313 nonn
%O A379313 1,1
%A A379313 _Gus Wiseman_, Dec 28 2024