A336568 Numbers that are not a product of two numbers each having distinct prime multiplicities.

30, 42, 66, 70, 78, 102, 105, 110, 114, 130, 138, 154, 165, 170, 174, 182, 186, 190, 195, 210, 222, 230, 231, 238, 246, 255, 258, 266, 273, 282, 285, 286, 290, 310, 318, 322, 330, 345, 354, 357, 366, 370, 374, 385, 390, 399, 402, 406, 410, 418, 420, 426, 429

Offset: 1

Author

Gus Wiseman, Aug 06 2020

Keywords

• nonn

Comments

First differs from A007304 and A093599 in having 210.

First differs from A287483 in having 222.

First differs from A350352 in having 420.

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.

Examples

			Selected terms together with their prime indices:
   660: {1,1,2,3,5}
   798: {1,2,4,8}
   840: {1,1,1,2,3,4}
  3120: {1,1,1,1,2,3,6}
  9900: {1,1,2,2,3,3,5}
		

Crossrefs

A336500 has zeros at these positions.

A007425 counts divisors of divisors.

A056924 counts divisors greater than their quotient.

A074206 counts strict chains of divisors from n to 1.

A130091 lists numbers with distinct prime multiplicities.

A181796 counts divisors with distinct prime multiplicities.

A336424 counts factorizations using A130091.

A336422 counts divisible pairs of divisors, both in A130091.

A327498 is the maximum divisor with distinct prime multiplicities.

A336423 counts chains in A130091, with maximal version A336569.

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

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

Programs

Mathematica

strsig[n_]:=UnsameQ@@Last/@FactorInteger[n]
Select[Range[100],Function[n,Select[Divisors[n],strsig[#]&&strsig[n/#]&]=={}]]