A343340 Numbers with a prime index dividing all the other prime indices, but with no prime index divisible by all the other prime indices.

30, 60, 66, 70, 90, 102, 110, 120, 132, 138, 140, 150, 154, 170, 180, 182, 186, 190, 198, 204, 210, 220, 238, 240, 246, 264, 270, 273, 276, 280, 282, 286, 290, 300, 306, 308, 310, 322, 330, 340, 350, 354, 360, 364, 372, 374, 380, 396, 402, 406, 408, 410, 414

Offset: 1

Author

Gus Wiseman, Apr 15 2021

Keywords

• nonn

Comments

Alternative name: Numbers > 1 whose smallest prime index divides all the other prime indices, but whose greatest prime index is not divisible by all the other prime indices.

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.

Also Heinz numbers of partitions with greatest part not divisible by all the others, but smallest part dividing all the others (counted by A343345). The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), giving a bijective correspondence between positive integers and integer partitions.

Examples

			The sequence of terms together with their prime indices begins:
     30: {1,2,3}        182: {1,4,6}          282: {1,2,15}
     60: {1,1,2,3}      186: {1,2,11}         286: {1,5,6}
     66: {1,2,5}        190: {1,3,8}          290: {1,3,10}
     70: {1,3,4}        198: {1,2,2,5}        300: {1,1,2,3,3}
     90: {1,2,2,3}      204: {1,1,2,7}        306: {1,2,2,7}
    102: {1,2,7}        210: {1,2,3,4}        308: {1,1,4,5}
    110: {1,3,5}        220: {1,1,3,5}        310: {1,3,11}
    120: {1,1,1,2,3}    238: {1,4,7}          322: {1,4,9}
    132: {1,1,2,5}      240: {1,1,1,1,2,3}    330: {1,2,3,5}
    138: {1,2,9}        246: {1,2,13}         340: {1,1,3,7}
    140: {1,1,3,4}      264: {1,1,1,2,5}      350: {1,3,3,4}
    150: {1,2,3,3}      270: {1,2,2,2,3}      354: {1,2,17}
    154: {1,4,5}        273: {2,4,6}          360: {1,1,1,2,2,3}
    170: {1,3,7}        276: {1,1,2,9}        364: {1,1,4,6}
    180: {1,1,2,2,3}    280: {1,1,1,3,4}      372: {1,1,2,11}
		

Crossrefs

The first condition alone gives the complement of A342193.

The second condition alone gives A343337.

The partitions with these Heinz numbers are counted by A343345.

A000005 counts divisors.

A000070 counts partitions with a selected part.

A001055 counts factorizations.

A056239 adds up prime indices, row sums of A112798.

A067824 counts strict chains of divisors starting with n.

A253249 counts strict chains of divisors.

Cf. A083710, A083711, A097986, A098965, A130714, A339562, A339563, A343341, A343377, A343381.

Programs

Mathematica

Select[Range[2,100],With[{p=PrimePi/@First/@FactorInteger[#]},!And@@IntegerQ/@(Max@@p/p)&&And@@IntegerQ/@(p/Min@@p)]&]

Formula

Complement of A342193 in A343337.