STATUS
proposed
approved
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Revision History for A371169
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Positive integers with at least as many prime factors (A001222) as distinct divisors of prime indices (A370820).
(history; published version)
(history; published version)
STATUS
editing
proposed
CROSSREFS
Other equalitiesChoosable partitions: A371165 A239312 (A368110), A355740 (A370320), A370592 (A371172A368100), A371177 A370593 (A371178A355529).
Other inequalities: A371166, A371167.
A239312 counts divisor-choosable partitions, ranks A368110.
A355740 counts non-divisor-choosable partitions, ranks A370320.
Cf. `A000792, A003963, `A355737, A355739, `A355741, ~A368100, A370808, A371127.
NAME
DATA
1, 2, 4, 6, 8, 9, 10, 12, 16, 18, 20, 22, 24, 25, 27, 28, 30, 32, 34, 36, 40, 42, 44, 45, 48, 50, 54, 56, 60, 62, 63, 64, 66, 68, 72, 75, 80, 81, 82, 84, 88, 90, 92, 96, 98, 99, 100, 102, 104, 108, 110, 112, 118, 120, 121, 124, 125, 126, 128, 132, 134, 135
OFFSET
1,2
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.
EXAMPLE
The terms together with their prime indices begin:
1: {}
2: {1}
4: {1,1}
6: {1,2}
8: {1,1,1}
9: {2,2}
10: {1,3}
12: {1,1,2}
16: {1,1,1,1}
18: {1,2,2}
20: {1,1,3}
22: {1,5}
24: {1,1,1,2}
25: {3,3}
27: {2,2,2}
28: {1,1,4}
30: {1,2,3}
32: {1,1,1,1,1}
34: {1,7}
36: {1,1,2,2}
MATHEMATICA
Select[Range[100], PrimeOmega[#]>=Length[Union @@ Divisors/@PrimePi/@First/@If[#==1, {}, FactorInteger[#]]]&]
CROSSREFS
The strict version is A370348 counted by A371171.
The case of equality is A370802, counted by A371130, strict A371128.
The RHS is A370820, for prime factors instead of divisors A303975.
The complement is A371168, counted by A371173.
The opposite version is A371170.
The version for prime factors instead of divisors on the RHS is A319899.
Other equalities: A371165 (A371172), A371177 (A371178).
Other inequalities: A371166, A371167.
A000005 counts divisors.
A001221 counts distinct prime factors.
A027746 lists prime factors, indices A112798, length A001222.
A239312 counts divisor-choosable partitions, ranks A368110.
A355731 counts choices of a divisor of each prime index, firsts A355732.
A355740 counts non-divisor-choosable partitions, ranks A370320.
A370814 counts divisor-choosable factorizations, complement A370813.
Cf. `A000792, A003963, `A355737, A355739, `A355741, ~A368100, A370808, A371127.
KEYWORD
allocated
nonn
AUTHOR
Gus Wiseman, Mar 16 2024
STATUS
approved
editing
NAME
allocated for Gus Wiseman
KEYWORD
allocated
STATUS
approved