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Positions of 4 are A374249.
A007947 (squarefree kernel) represents run-compression of prime indices.
A008480 counts permutations of prime factors (or prime indices).
-A037201 lists run-compressed prime differences, halved A373947.
A304038 lists run-compression of prime indices, sums sum A066328.
A374251 gives run-compression of compresses standard compositions, sum A373953, rank A373948.
Cf. A001223, A001414, A116608, `A238130, `A285981, ~A300061, A316413, `A320924, `A333489, `A333627, `A340387, `A344415, ~`A373952, A373954, A374246, A374247.
Positions of 0 or 1 are A000079 (powers of two) except 1.
Positions of 0 or 2 are A000244 (powers of three) except 1.
Positions of 0 or 3 are {6} U A000351 (six or powers of five) except 1.
For run-count number of runs instead of sum of run-compression-sum we have A373957.
A003242 counts run-compressed compositions or , i.e., anti-runs.
A116861 counts partitions by compressed sum, by compressed length A116608 of run-compression.
A373948 encodes run-compression using compositions in standard order.
A374251 lists gives run-compression of standard compositions, sums sum A373953, rank A373948.
Cf. A001223, A001414, A116608, `A238130, `A285981, ~A300061, A316413, `A320924, `A333489, `A333627, `A340387, `A344415, ~A373952, A373954, A374246, A374247.
For length instead of maximum we have A008480 (number of permutations of prime indices).
Compression-sum of sorted prime indices is A066328.
`A007947 (squarefree kernel) represents run-compression of prime indices, [sum A066328].
A037201 gives compressed first differences A008480 counts permutations of prime numbers, halved A373947indices.
A037201 lists run-compressed prime differences, halved A373947.
`A114901 counts compositions with no isolated parts.
A124767 counts runs (also compressed length) in standard compositions, anti-runs A333381.
`A240085 counts compositions with no unique partsA304038 lists run-compression of prime indices, sums A066328.
A304038 lists run-compression of prime indices, ranks A007947 (squarefree kernel), row-sums A066328.
A333755 counts compositions by number of runs (also compressed length).
`A334201 adds up all prime indices except one instance of the greatest.
`A344291 lists numbers m with A001222(m) <= A056239(m)/2, counted by A110618.
`A344296 lists numbers m with A001222(m) >= A056239(m)/2, counted by A025065.
A373949 counts compositions by sum of run-compressed sum, compression, opposite A373951.
A373953 gives A374251 lists run-compressed sum compression of standard compositions, (difference A373954), row sums of A374251A373953.
Cf. A001223, A001414, `A238130, `A285981, ~A300061, A316413, `A320924, `A333489, `A333627, `A340387, `A344415, ~A373952, A373954, A374246, A374247.
allocated for Gus WisemanGreatest sum of run-compression of a permutation of the prime indices of n.
0, 1, 2, 1, 3, 3, 4, 1, 2, 4, 5, 4, 6, 5, 5, 1, 7, 5, 8, 5, 6, 6, 9, 4, 3, 7, 2, 6, 10, 6, 11, 1, 7, 8, 7, 6, 12, 9, 8, 5, 13, 7, 14, 7, 7, 10, 15, 4, 4, 7, 9, 8, 16, 5, 8, 6, 10, 11, 17, 7, 18, 12, 8, 1, 9, 8, 19, 9, 11, 8, 20, 7, 21, 13, 8, 10, 9, 9, 22, 5
1,3
We define the run-compression of a sequence to be the anti-run obtained by reducing each run of repeated parts to a single part. Alternatively, run-compression removes all parts equal to the part immediately to their left. For example, (1,1,2,2,1) has run-compression (1,2,1).
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.
The prime indices of 24 are {1,1,1,2}, with permutations such as (1,1,2,1) whose run-compression sums to 4, so a(24) = 4.
The prime indices of 216 are {1,1,1,2,2,2}, with permutations such as (1,2,1,2,1,2) whose run-compression sums to 9, so a(216) = 9.
prix[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
Table[Max@@(Total[First/@Split[#]]&/@Permutations[prix[n]]), {n, 100}]
Positions of first appearances are 1 followed by the primes A000040.
Positions of 0 or 1 are A000079 (powers of two).
Positions of 0 or 2 are A000244 (powers of three).
Positions of 0 or 3 are {6} U A000351 (six or powers of five).
For length instead of maximum we have A008480 (number of permutations of prime indices).
Compression-sum of sorted prime indices is A066328.
For run-count instead of compression-sum we have A373957.
Positions of 4 are A374249.
For prime factors instead of indices we have A374250.
A001221 counts distinct prime factors, A001222 with multiplicity.
A003242 counts compressed compositions or anti-runs.
`A007947 (squarefree kernel) represents compression of prime indices, [sum A066328].
A037201 gives compressed first differences of prime numbers, halved A373947.
A056239 adds up prime indices, row sums of A112798.
`A114901 counts compositions with no isolated parts.
A116861 counts partitions by compressed sum, by compressed length A116608.
A124767 counts runs (also compressed length) in standard compositions, anti-runs A333381.
`A240085 counts compositions with no unique parts.
A304038 lists run-compression of prime indices, ranks A007947 (squarefree kernel), row-sums A066328.
A333755 counts compositions by number of runs (also compressed length).
`A334201 adds up all prime indices except one instance of the greatest.
A335433 lists numbers whose prime indices are separable, complement A335448.
`A344291 lists numbers m with A001222(m) <= A056239(m)/2, counted by A110618.
`A344296 lists numbers m with A001222(m) >= A056239(m)/2, counted by A025065.
A373948 encodes run-compression using compositions in standard order.
A373949 counts compositions by run-compressed sum, opposite A373951.
A373953 gives run-compressed sum of standard compositions, (difference A373954), row sums of A374251.
Cf. A001223, A001414, `A238130, `A285981, ~A300061, A316413, `A320924, `A333489, `A333627, `A340387, `A344415, ~A373952, A374246, A374247.
allocated
nonn
Gus Wiseman, Jul 06 2024
approved
editing
allocated for Gus Wiseman
allocated
approved