A353834 Nonprime numbers whose prime indices have all equal run-sums.

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

Author

Gus Wiseman, May 26 2022

Keywords

• nonn

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).

Examples

			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).
		

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.

Cf. A000005, A000961, A007947, A025475, A071625, A073093, A323014, A353839.

Programs

Mathematica

Select[Range[100],!PrimeQ[#]&&SameQ@@Cases[FactorInteger[#],{p_,k_}:>PrimePi[p]*k]&]

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