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Numbers that are not a product of two numbers each having distinct prime multiplicities.
23

%I #17 Jul 08 2022 16:06:51

%S 30,42,66,70,78,102,105,110,114,130,138,154,165,170,174,182,186,190,

%T 195,210,222,230,231,238,246,255,258,266,273,282,285,286,290,310,318,

%U 322,330,345,354,357,366,370,374,385,390,399,402,406,410,418,420,426,429

%N Numbers that are not a product of two numbers each having distinct prime multiplicities.

%C First differs from A007304 and A093599 in having 210.

%C First differs from A287483 in having 222.

%C First differs from A350352 in having 420.

%C A number's prime signature (row n of A124010) is the sequence of positive exponents in its prime factorization, so a number has distinct prime multiplicities iff all the exponents in its prime signature are distinct.

%e Selected terms together with their prime indices:

%e 660: {1,1,2,3,5}

%e 798: {1,2,4,8}

%e 840: {1,1,1,2,3,4}

%e 3120: {1,1,1,1,2,3,6}

%e 9900: {1,1,2,2,3,3,5}

%t strsig[n_]:=UnsameQ@@Last/@FactorInteger[n]

%t Select[Range[100],Function[n,Select[Divisors[n],strsig[#]&&strsig[n/#]&]=={}]]

%Y A336500 has zeros at these positions.

%Y A007425 counts divisors of divisors.

%Y A056924 counts divisors greater than their quotient.

%Y A074206 counts strict chains of divisors from n to 1.

%Y A130091 lists numbers with distinct prime multiplicities.

%Y A181796 counts divisors with distinct prime multiplicities.

%Y A336424 counts factorizations using A130091.

%Y A336422 counts divisible pairs of divisors, both in A130091.

%Y A327498 is the maximum divisor with distinct prime multiplicities.

%Y A336423 counts chains in A130091, with maximal version A336569.

%Y A336571 counts divisor sets using A130091, with maximal version A336570.

%Y Cf. A001055, A002033, A098859, A124010, A167865, A253249, A336420.

%K nonn

%O 1,1

%A _Gus Wiseman_, Aug 06 2020