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