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

1,1

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.

Links

Table of n, a(n) for n=1..53.

Formula

Complement of A342193 in A343337.

Example

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}

Mathematica

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

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.

Sequence in context: A121960 A040870 A356176 * A280011 A175785 A051488

Adjacent sequences: A343337 A343338 A343339 * A343341 A343342 A343343

Keyword

nonn

Author

Gus Wiseman, Apr 15 2021

Status

approved