Relationships between A244052, A294492, and A300156.
Michael Thomas De Vlieger, St. Louis, Missouri, 201802241100, revised 201802261730.

Concerns sequences:
A000005: Divisor counting function.
A000027: The positive integers.
A000079: Powers 2^k with k >= 0.
A000961: Prime powers p^e with e >= 0.
A001221: Number of distinct prime divisors p of n (little omega(n)).
A002182: Numbers that set records in A000005.
A010846: Number of numbers less than n that divide n^e with integer e >= 0.
A027750: Triangle read by rows in which row n lists the divisors of n. 
A162306: Irregular triangle in which row n contains the numbers <= n 
         whose prime factors are a subset of prime factors of n. 
A243822: Nondivisors m < n that divide n^e with integer e > 1.
A244052: Numbers that set records in A010846.
A272618: Irregular array read by rows: n-th row contains (in ascending order) 
         the nondivisors 1 <= k < n such that all the prime divisors p of k also divide n. 
A294492: Numbers that set records in A243822.
A299990: A243822(n) - 2 A000005(n). (as A010846(n) = A000005(n) + A243822(n)).
A300156: Numbers that set records in A299990.
A300157: Records in A299990.

Contents:

1. Table 2: Relationship of A244052, A294492, and A300156.
2. Observations regarding Table 2.

Notes:

consider a positive integer n and consider the range 1 <= m <= n.
Let A010846(n) be the number of m | n^e with e >= 0. All m are products of primes p | n or m = 1;
it might be proper rather to suggest all m are not products of primes q coprime to n.
These m appear in row n of A162306.
There are 2 possible species of m in row n of A162306: terms d | n^e with 0 <= e <= 1 
(i.e., divisors, see A027750), and "semidivisors" k | n^e with e > 1 (see A272618).
These semidivisors are nondivisors in the cototient of n and appear in A133995.
Observe that for n = p^e, i.e., n in A000961, A010846(n) = A000005(n) and row n in A162306 
is identical to row n in A000005. This is because all p^i for 0 <= i <= e divide p^e; only divisors
appear in the cototient of p^e.
The premise of this study concerns the difference between A243822(n) and A000005(n), the two species
in A010846(n).
 
Let A299990(n) = A243822(n) - A000005(n) = A010846(n) - 2*A000005(n), 
as A010846(n) = A000005(n) + A243822(n). We select this difference rather than A000005(n) - A243822(n) since
this latter difference equals A000005(n) for n in A000961(n).

I have computed A010846(n) for 1 <= n <= 36,000,000 over several days in October-November 2017 (446 Mb b-file)
and thus have calculated sequences A299990-2 and A300155-7 using this broad dataset.

This study yields six sequences: A299990, A299991, A299992, A300155, A300156, and A300157.
The tables relate the numbers that set records in A000005, A010846, and A243822 to A299990 and A300156.

(This table also appears in A299990 as Table 2.)

  +---------------------------------------------------------+
  | Table 1: Relationship of A244052, A294492, and A300156  |
  +---------------------------------------------------------+

n = index in this document.
A300156(n) = indices of records in A299990.
MN(A300156(n)) = A054841(A300156(n)), multiplicity notation of A300156(n).
b(n) = position of A300156(n) in A294492(n) = indices of records in A243822.
c(n) = position of A300156(n) in A244052(n) = indices of records in A010846.

  n    A300156(n)   A300157(n) MN(A300156(n))  b(n)     c(n)
------------------------------------------------------------
  1	       1	   -1	0           	 1	  1
  2	      30	    2	111         	 8	  9
  3	      42	    3	1101        	 9	 10
  4	      66	    6	11001       	11	   
  5	      78	    7	110001      	12	   
  6	      90	    8	121         	13	 13
  7	     102	    9	1100001     	14	   
  8	     114	   10	11000001    	15	   
  9	     138	   11	110000001   	17	   
 10	     150	   17	112         	18	 15
 11	     210	   36	1111        	19	 17
 12	     330	   45	11101       	20	 18
 13	     390	   48	111001      	21	 19
 14	     510	   54	1110001     	23	   
 15	     570	   56	11100001    	24	   
 16	     630	   67	1211        	25	 21
 17	     870	   69	1110000001  	  	   
 18	     990	   76	12101       	  	   
 19	    1050	   97	1121        	26	 23
 20	    1470	  118	1112        	27	 25
 21	    1890	  119	1311        	  	 27
 22	    2100	  120	2121        	  	 28
 23	    2310	  219	11111       	28	 29
 24	    2730	  231	111101      	  	 30
 25	    3570	  249	1111001     	  	 31
 26	    3990	  258	11110001    	  	 32
 27	    4620	  286	21111       	29	 33
 28	    5460	  299	211101      	  	 34
 29	    6510	  302	11110000001 	  	   
 30	    6930	  356	12111       	30	 35
 31	    8190	  367	121101      	  	 36
 32	    9240	  377	31111       	  	 37
 33	   10710	  392	1211001     	  	   
 34	   11550	  455	11211       	31	 39
 35	   13650	  471	112101      	  	 40
 36	   16170	  533	11121       	32	 42
 37	   19110	  547	111201      	  	   
 38	   20790	  563	13111       	  	 44
 39	   23100	  573	21211       	  	 45
 40	   24570	  576	131101      	  	   
 41	   25410	  647	11112       	33	 46
 42	   30030	 1033	111111      	34	 48
 43	   39270	 1096	1111101     	  	 49
 44	   43890	 1125	11111001    	  	 50
 45	   46410	 1129	1111011     	  	 51
 46	   51870	 1157	11110101    	  	 52
 47	   53130	 1178	111110001   	  	 53
 48	   60060	 1334	211111      	35	 54
 49	   78540	 1405	2111101     	  	 55
 50	   87780	 1439	21111001    	  	 56
 51	   90090	 1587	121111      	36	 57
 52	  117810	 1664	1211101     	  	 58
 53	  120120	 1721	311111      	  	 59
 54	  150150	 1952	112111      	37	 60
 55	  180180	 2006	221111      	  	 61
 56	  196350	 2035	1121101     	  	   
 57	  210210	 2228	111211      	38	 62
 58	  270270	 2389	131111      	  	 64
 59	  300300	 2455	212111      	  	 65
 60	  330330	 2644	111121      	39	 66
 61	  390390	 2809	111112      	40	 68
 62	  450450	 2865	122111      	  	 70
 63	  510510	 4587	1111111     	41	 72
 64	  570570	 4683	11111101    	  	 73
 65	  690690	 4863	111111001   	  	 74
 66	  746130	 4882	11111011    	  	 75
 67	  870870	 5108	1111110001  	  	 76
 68	  930930	 5180	11111100001 	  	 77
 69	 1021020	 5841	2111111     	42	 78
 70	 1141140	 5953	21111101    	  	 79
 71	 1381380	 6162	211111001   	  	 80
 72	 1492260	 6176	21111011    	  	 81
 73	 1531530	 6794	1211111     	43	 82
 74	 1711710	 6915	12111101    	  	 83
 75	 2042040	 7416	3111111     	  	 84
 76	 2282280	 7543	31111101    	  	 85
 77	 2552550	 8169	1121111     	44	 86
 78	 2852850	 8301	11211101    	  	 87
 79	 3063060	 8523	2211111     	  	 88
 80	 3423420	 8660	22111101    	  	 89
 81	 3573570	 9196	1112111     	45	 90
 82	 3993990	 9335	11121101    	  	 91
 83	 4084080	 9370	4111111     	  	 92
 84	 4564560	 9515	41111101    	  	 93
 85	 4594590	 9902	1311111     	  	 94
 86	 5105100	10201	2121111     	  	 95
 87	 5615610	10736	1111211     	46	 96
 88	 6276270	10885	11112101    	  	   
 89	 6636630	11356	1111121     	47	 98
 90	 7147140	11451	2112111     	  	 99
 91	 7417410	11508	11111201    	  	   
 92	 7657650	11717	1221111     	  	100
 93	 8168160	11781	5111111     	  	101
 94	 8558550	11874	12211101    	  	   
 95	 8678670	12415	1111112     	48	102
 96	 9699690	19473	11111111    	49	104
 97	11741730	20093	111111101   	  	105
 98	13123110	20417	111111011   	  	106
 99	14804790	20941	1111111001  	  	107
100	15825810	21201	11111110001 	  	108
101	16546530	21257	1111110101  	  	109
102	17687670	21516	11111101001 	  	110
103	18888870	21931	111111100001	  	111
104	19399380	24521	21111111    	50	112
105	23483460	25237	211111101   	  	113
106	26246220	25602	211111011   	  	114
107	29099070	28156	12111111    	51	115
108	35225190	28933	121111101   	  	116

  +---------------------------------+
  | Observations regarding Table 1. |
  +---------------------------------+

1. Of the numbers m = A300156(n) that do not appear in A244052, all m < 870 appear in A294492.
2. The only term of A002182 = indices of records in A000005, present in A300156 is 1.
3. Suppose we take records in -1*A299990(n). The records are A000027(n) at indices A000079(n) 
   in A299990, since A243822(p^e) = 0 for e >= 0. Hence the construction of A299990(n) as the difference
   A243822(n) - A000005(n) = A010846(n) - 2*A000005(n), as A010846(n) = A000005(n) + A243822(n).

eof