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A353834
Nonprime numbers whose prime indices have all equal run-sums.
25
1, 4, 8, 9, 12, 16, 25, 27, 32, 40, 49, 63, 64, 81, 112, 121, 125, 128, 144, 169, 243, 256, 289, 325, 343, 351, 352, 361, 512, 529, 625, 675, 729, 832, 841, 931, 961, 1008, 1024, 1331, 1369, 1539, 1600, 1681, 1728, 1849, 2048, 2176, 2187, 2197, 2209, 2401
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.
The sequence of runs of a sequence consists of its maximal consecutive constant subsequences when read left-to-right. For example, the runs of (2,2,1,1,1,3,2,2) are (2,2), (1,1,1), (3), (2,2), with sums (4,3,3,4).
EXAMPLE
The terms together with their prime indices begin:
1: {}
4: {1,1}
8: {1,1,1}
9: {2,2}
12: {1,1,2}
16: {1,1,1,1}
25: {3,3}
27: {2,2,2}
32: {1,1,1,1,1}
40: {1,1,1,3}
49: {4,4}
63: {2,2,4}
64: {1,1,1,1,1,1}
81: {2,2,2,2}
112: {1,1,1,1,4}
121: {5,5}
125: {3,3,3}
128: {1,1,1,1,1,1,1}
For example, 675 is in the sequence because its prime indices {2,2,2,3,3} have run-sums (6,6).
MATHEMATICA
Select[Range[100], !PrimeQ[#]&&SameQ@@Cases[FactorInteger[#], {p_, k_}:>PrimePi[p]*k]&]
PROG
(Python)
from itertools import count, islice
from sympy import factorint, primepi
def A353848_gen(startvalue=1): # generator of terms >= startvalue
return filter(lambda n: n == 1 or (sum((f:=factorint(n)).values()) > 1 and len(set(primepi(p)*e for p, e in f.items())) <= 1), count(max(startvalue, 1)))
A353848_list = list(islice(A353848_gen(), 30)) # Chai Wah Wu, May 27 2022
CROSSREFS
For equal run-lengths we have A072774\A000040, counted by A047966(n)-1.
These partitions are counted by A304442(n) - 1.
These are the nonprime positions of prime powers in A353832.
Including the primes gives A353833.
For distinct run-sums we have A353838\A000040, counted by A353837(n)-1.
For compositions we have A353848\A000079, counted by A353851(n)-1.
A001222 counts prime factors, distinct A001221.
A005811 counts runs in binary expansion, distinct run-lengths A165413.
A056239 adds up prime indices, row sums of A112798 and A296150.
A124010 gives prime signature, sorted A118914.
A300273 ranks collapsible partitions, counted by A275870.
A353835 counts distinct run-sums of prime indices, weak A353861.
A353840-A353846 pertain to partition run-sum trajectory.
A353862 gives greatest run-sum of prime indices, least A353931.
A353866 ranks rucksack partitions, counted by A353864.
Sequence in context: A102211 A244032 A321454 * A319302 A119025 A167903
KEYWORD
nonn
AUTHOR
Gus Wiseman, May 26 2022
STATUS
approved