cp's OEIS Frontend

A379312 Positive integers whose prime indices include a unique 1 or prime number.

%I A379312 #7 Dec 28 2024 16:27:31
%S A379312 2,3,5,11,14,17,21,26,31,35,38,39,41,46,57,58,59,65,67,69,74,77,83,86,
%T A379312 87,94,95,98,106,109,111,115,119,122,127,129,141,142,143,145,146,147,
%U A379312 157,158,159,178,179,182,183,185,191,194,202,206,209,211,213,214
%N A379312 Positive integers whose prime indices include a unique 1 or prime number.
%C A379312 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 A379312 The terms together with their prime indices begin:
%e A379312     2: {1}
%e A379312     3: {2}
%e A379312     5: {3}
%e A379312    11: {5}
%e A379312    14: {1,4}
%e A379312    17: {7}
%e A379312    21: {2,4}
%e A379312    26: {1,6}
%e A379312    31: {11}
%e A379312    35: {3,4}
%e A379312    38: {1,8}
%e A379312    39: {2,6}
%e A379312    41: {13}
%e A379312    46: {1,9}
%e A379312    57: {2,8}
%e A379312    58: {1,10}
%e A379312    59: {17}
%e A379312    65: {3,6}
%e A379312    67: {19}
%e A379312    69: {2,9}
%e A379312    74: {1,12}
%e A379312    77: {4,5}
%t A379312 prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A379312 Select[Range[1000],Length[Select[prix[#],#==1||PrimeQ[#]&]]==1&]
%Y A379312 These "old" primes are listed by A008578.
%Y A379312 For no composite parts we have A302540, counted by A034891 (strict A036497).
%Y A379312 For all composite parts we have A320629, counted by A023895 (strict A204389).
%Y A379312 For a unique prime part we have A331915, counted by A379304 (strict A379305).
%Y A379312 Positions of ones in A379311, see A379313.
%Y A379312 Partitions of this type are counted by A379314, strict A379315.
%Y A379312 A000040 lists the prime numbers, differences A001223.
%Y A379312 A002808 lists the composite numbers, nonprimes A018252, differences A073783 or A065310.
%Y A379312 A055396 gives least prime index, greatest A061395.
%Y A379312 A056239 adds up prime indices, row sums of A112798, counted by A001222.
%Y A379312 A080339 is the characteristic function for the old prime numbers.
%Y A379312 A376682 gives k-th differences of old prime numbers, see A030016, A075526.
%Y A379312 Other counts of prime indices:
%Y A379312 - A257991 odd, see A000041, A000070, A066207, A349158.
%Y A379312 - A257992 even, see A000009, A038348, A066208, A379317.
%Y A379312 - A257994 prime, see A000586, A000607, A076610, A331386.
%Y A379312 - A330944 nonprime, see A002095, A096258, A320628, A330945.
%Y A379312 - A379306 squarefree, see A302478, A379308, A379309, A379316.
%Y A379312 - A379310 nonsquarefree, see A114374, A256012, A379307.
%Y A379312 Cf. A038550, A073445, A087436, A173390, A379300, A379301, A379302.
%K A379312 nonn
%O A379312 1,1
%A A379312 _Gus Wiseman_, Dec 28 2024