A353833 - OEIS

%I #9 Jun 26 2022 09:45:16

%S 1,2,3,4,5,7,8,9,11,12,13,16,17,19,23,25,27,29,31,32,37,40,41,43,47,

%T 49,53,59,61,63,64,67,71,73,79,81,83,89,97,101,103,107,109,112,113,

%U 121,125,127,128,131,137,139,144,149,151,157,163,167,169,173,179

%N Numbers whose multiset of prime indices has all equal run-sums.

%C 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 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).

%H Mathematics Stack Exchange, <a href="https://math.stackexchange.com/q/87559">What is a sequence run? (answered 2011-12-01)</a>

%e The prime indices of 12 are {1,1,2}, with run-sums (2,2), so 12 is in the sequence.

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

%Y For parts instead of run-sums we have A000961, counted by A000005.

%Y For run-lengths instead of run-sums we have A072774, counted by A047966.

%Y These partitions are counted by A304442.

%Y These are the positions of powers of primes in A353832.

%Y The restriction to nonprimes is A353834.

%Y For distinct instead of equal run-sums we have A353838, counted by A353837.

%Y The version for compositions is A353848, counted by A353851.

%Y A001222 counts prime factors, distinct A001221.

%Y A005811 counts runs in binary expansion, distinct run-lengths A165413.

%Y A056239 adds up prime indices, row sums of A112798 and A296150.

%Y A124010 gives prime signature, sorted A118914.

%Y A300273 ranks collapsible partitions, counted by A275870.

%Y A353835 counts distinct run-sums of prime indices, weak A353861.

%Y A353840-A353846 deal with iterated run-sums for partitions.

%Y A353862 gives greatest run-sum of prime indices, least A353931.

%Y A353866 ranks rucksack partitions, counted by A353864.

%Y Cf. A007947, A071625, A073093, A181819, A238279, A304660, A323014, A333755, A353839.

%K nonn

%O 1,2

%A _Gus Wiseman_, May 23 2022