cp's OEIS Frontend

A379307 Positive integers whose prime indices include no squarefree numbers.

%I A379307 #6 Dec 29 2024 14:08:46
%S A379307 1,7,19,23,37,49,53,61,71,89,97,103,107,131,133,151,161,173,193,197,
%T A379307 223,227,229,239,251,259,263,281,307,311,337,343,359,361,371,379,383,
%U A379307 409,419,427,433,437,457,463,479,497,503,521,523,529,541,569,593,613,623
%N A379307 Positive integers whose prime indices include no squarefree numbers.
%C A379307 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 A379307 The terms together with their prime indices begin:
%e A379307     1: {}
%e A379307     7: {4}
%e A379307    19: {8}
%e A379307    23: {9}
%e A379307    37: {12}
%e A379307    49: {4,4}
%e A379307    53: {16}
%e A379307    61: {18}
%e A379307    71: {20}
%e A379307    89: {24}
%e A379307    97: {25}
%e A379307   103: {27}
%e A379307   107: {28}
%e A379307   131: {32}
%e A379307   133: {4,8}
%e A379307   151: {36}
%e A379307   161: {4,9}
%e A379307   173: {40}
%t A379307 prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A379307 Select[Range[1000],Length[Select[prix[#],SquareFreeQ]]==0&]
%Y A379307 Partitions of this type are counted by A114374, strict A256012.
%Y A379307 Positions of zero in A379306.
%Y A379307 For a unique squarefree part we have A379316, counted by A379308 (strict A379309).
%Y A379307 A000040 lists the primes, differences A001223.
%Y A379307 A005117 lists the squarefree numbers, differences A076259.
%Y A379307 A008966 is the characteristic function for the squarefree numbers.
%Y A379307 A013929 lists the nonsquarefree numbers, differences A078147.
%Y A379307 A055396 gives least prime index, greatest A061395.
%Y A379307 A056239 adds up prime indices, row sums of A112798, counted by A001222.
%Y A379307 A061398 counts squarefree numbers between primes, zeros A068360.
%Y A379307 A377038 gives k-th differences of squarefree numbers.
%Y A379307 Other counts of prime indices:
%Y A379307 - A257994 prime, see A002095, A096258, A320628, A331386, A331915, A379304, A379305.
%Y A379307 - A257991 odd, see A000041, A000070, A066207, A349158.
%Y A379307 - A257992 even, see A000009, A038348, A066208, A379317.
%Y A379307 - A330944 nonprime, see A000586, A000607, A076610, A330945.
%Y A379307 - A379300 composite, see A023895, A034891, A036497, A302540, A379301.
%Y A379307 - A379310 nonsquarefree, see A302478.
%Y A379307 - A379311 old prime, see A204389, A320629, A379312-A379315.
%Y A379307 Cf. A000720, A013928, A038550, A057627, A068361, A070321, A071403, A072284, A087436, A112929, A377430, A378086.
%K A379307 nonn
%O A379307 1,2
%A A379307 _Gus Wiseman_, Dec 27 2024