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

%I A353834 #9 May 27 2022 21:13:50
%S A353834 1,4,8,9,12,16,25,27,32,40,49,63,64,81,112,121,125,128,144,169,243,
%T A353834 256,289,325,343,351,352,361,512,529,625,675,729,832,841,931,961,1008,
%U A353834 1024,1331,1369,1539,1600,1681,1728,1849,2048,2176,2187,2197,2209,2401
%N A353834 Nonprime numbers whose prime indices have all equal run-sums.
%C A353834 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.
%C A353834 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).
%e A353834 The terms together with their prime indices begin:
%e A353834      1: {}
%e A353834      4: {1,1}
%e A353834      8: {1,1,1}
%e A353834      9: {2,2}
%e A353834     12: {1,1,2}
%e A353834     16: {1,1,1,1}
%e A353834     25: {3,3}
%e A353834     27: {2,2,2}
%e A353834     32: {1,1,1,1,1}
%e A353834     40: {1,1,1,3}
%e A353834     49: {4,4}
%e A353834     63: {2,2,4}
%e A353834     64: {1,1,1,1,1,1}
%e A353834     81: {2,2,2,2}
%e A353834    112: {1,1,1,1,4}
%e A353834    121: {5,5}
%e A353834    125: {3,3,3}
%e A353834    128: {1,1,1,1,1,1,1}
%e A353834 For example, 675 is in the sequence because its prime indices {2,2,2,3,3} have run-sums (6,6).
%t A353834 Select[Range[100],!PrimeQ[#]&&SameQ@@Cases[FactorInteger[#],{p_,k_}:>PrimePi[p]*k]&]
%o A353834 (Python)
%o A353834 from itertools import count, islice
%o A353834 from sympy import factorint, primepi
%o A353834 def A353848_gen(startvalue=1): # generator of terms >= startvalue
%o A353834     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)))
%o A353834 A353848_list = list(islice(A353848_gen(),30)) # _Chai Wah Wu_, May 27 2022
%Y A353834 For equal run-lengths we have A072774\A000040, counted by A047966(n)-1.
%Y A353834 These partitions are counted by A304442(n) - 1.
%Y A353834 These are the nonprime positions of prime powers in A353832.
%Y A353834 Including the primes gives A353833.
%Y A353834 For distinct run-sums we have A353838\A000040, counted by A353837(n)-1.
%Y A353834 For compositions we have A353848\A000079, counted by A353851(n)-1.
%Y A353834 A001222 counts prime factors, distinct A001221.
%Y A353834 A005811 counts runs in binary expansion, distinct run-lengths A165413.
%Y A353834 A056239 adds up prime indices, row sums of A112798 and A296150.
%Y A353834 A124010 gives prime signature, sorted A118914.
%Y A353834 A300273 ranks collapsible partitions, counted by A275870.
%Y A353834 A353835 counts distinct run-sums of prime indices, weak A353861.
%Y A353834 A353840-A353846 pertain to partition run-sum trajectory.
%Y A353834 A353862 gives greatest run-sum of prime indices, least A353931.
%Y A353834 A353866 ranks rucksack partitions, counted by A353864.
%Y A353834 Cf. A000005, A000961, A007947, A025475, A071625, A073093, A323014, A353839.
%K A353834 nonn
%O A353834 1,2
%A A353834 _Gus Wiseman_, May 26 2022