A338552 Non-powers of primes whose prime indices have a common divisor > 1.

21, 39, 57, 63, 65, 87, 91, 111, 115, 117, 129, 133, 147, 159, 171, 183, 185, 189, 203, 213, 235, 237, 247, 259, 261, 267, 273, 299, 301, 303, 305, 319, 321, 325, 333, 339, 351, 365, 371, 377, 387, 393, 399, 417, 427, 441, 445, 453, 477, 481, 489, 497, 507

Offset

1,1

Comments

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 non-constant, non-relatively prime partitions. 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

Equals A024619 /\ A318978.

Complement of A000961 \/ A289509.

Example

The sequence of terms together with their prime indices begins:
     21: {2,4}      183: {2,18}       305: {3,18}
     39: {2,6}      185: {3,12}       319: {5,10}
     57: {2,8}      189: {2,2,2,4}    321: {2,28}
     63: {2,2,4}    203: {4,10}       325: {3,3,6}
     65: {3,6}      213: {2,20}       333: {2,2,12}
     87: {2,10}     235: {3,15}       339: {2,30}
     91: {4,6}      237: {2,22}       351: {2,2,2,6}
    111: {2,12}     247: {6,8}        365: {3,21}
    115: {3,9}      259: {4,12}       371: {4,16}
    117: {2,2,6}    261: {2,2,10}     377: {6,10}
    129: {2,14}     267: {2,24}       387: {2,2,14}
    133: {4,8}      273: {2,4,6}      393: {2,32}
    147: {2,4,4}    299: {6,9}        399: {2,4,8}
    159: {2,16}     301: {4,14}       417: {2,34}
    171: {2,2,8}    303: {2,26}       427: {4,18}

Mathematica

Select[Range[100], !(#==1||PrimePowerQ[#]||GCD@@PrimePi/@First/@FactorInteger[#]==1)&]

Crossrefs

A318978 allows prime powers, counted by A018783, with complement A289509.

A327685 allows nonprime prime powers.

A338330 is the coprime instead of relatively prime version.

A338554 counts the partitions with these Heinz numbers.

A338555 is the complement.

A000740 counts relatively prime compositions.

A000961 lists powers of primes, with complement A024619.

A051424 counts pairwise coprime or singleton partitions.

A108572 counts nontrivial periodic partitions, with Heinz numbers A001597.

A291166 ranks relatively prime compositions, with complement A291165.

A302696 gives the Heinz numbers of pairwise coprime partitions.

A327516 counts pairwise coprime partitions, with Heinz numbers A302696.

Cf. A000005, A000837, A007916, A056239, A112798, A289508, A302569, A302796, A318716, A327658, A328867, A328677, A338331, A338553.

Sequence in context: A102478 A221048 A182297 * A339002 A339004 A270667

Adjacent sequences: A338549 A338550 A338551 * A338553 A338554 A338555

Keyword

nonn

Author

Gus Wiseman, Nov 03 2020

Status

approved