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Least semiprime whose prime indices sum to n.
17

%I #13 Sep 06 2022 10:38:12

%S 4,6,9,14,21,26,34,38,46,58,62,74,82,86,94,106,118,122,134,142,146,

%T 158,166,178,194,202,206,214,218,226,254,262,274,278,298,302,314,326,

%U 334,346,358,362,382,386,394,398,422,446,454,458,466,478,482,502,514,526

%N Least semiprime whose prime indices sum to n.

%C Converges to A100484.

%C After a(4) = 9, also the least squarefree semiprime whose prime indices sum to n.

%C A semiprime is a product of any two prime numbers. A prime index of n is a number m such that the m-th prime number divides n. The multiset of prime indices of n is row n of A112798.

%e The sequence of terms together with their prime indices begins:

%e 4: {1,1} 106: {1,16} 254: {1,31}

%e 6: {1,2} 118: {1,17} 262: {1,32}

%e 9: {2,2} 122: {1,18} 274: {1,33}

%e 14: {1,4} 134: {1,19} 278: {1,34}

%e 21: {2,4} 142: {1,20} 298: {1,35}

%e 26: {1,6} 146: {1,21} 302: {1,36}

%e 34: {1,7} 158: {1,22} 314: {1,37}

%e 38: {1,8} 166: {1,23} 326: {1,38}

%e 46: {1,9} 178: {1,24} 334: {1,39}

%e 58: {1,10} 194: {1,25} 346: {1,40}

%e 62: {1,11} 202: {1,26} 358: {1,41}

%e 74: {1,12} 206: {1,27} 362: {1,42}

%e 82: {1,13} 214: {1,28} 382: {1,43}

%e 86: {1,14} 218: {1,29} 386: {1,44}

%e 94: {1,15} 226: {1,30} 394: {1,45}

%t Table[Min@@Table[Prime[k]*Prime[n-k],{k,n-1}],{n,2,30}]

%t Take[DeleteDuplicates[SortBy[{Times@@#,Total[PrimePi[#]]}&/@Tuples[ Prime[ Range[ 200]],2],{Last,First}],GreaterEqual[#1[[2]],#2[[2]]]&][[All,1]],60] (* _Harvey P. Dale_, Sep 06 2022 *)

%o (PARI) a(n) = vecmin(vector(n-1, k, prime(k)*prime(n-k))); \\ _Michel Marcus_, Dec 03 2020

%Y A024697 is the sum of the same semiprimes.

%Y A098350 has this sequence as antidiagonal minima.

%Y A338904 has this sequence as row minima.

%Y A339114 (this sequence) is the squarefree case for n > 4.

%Y A339115 is the greatest among the same semiprimes.

%Y A001358 lists semiprimes, with odd/even terms A046315/A100484.

%Y A006881 lists squarefree semiprimes, with odd/even terms A046388/A100484.

%Y A037143 lists primes and semiprimes.

%Y A056239 gives the sum of prime indices of n.

%Y A084126 and A084127 give the prime factors of semiprimes.

%Y A087112 groups semiprimes by greater factor.

%Y A320655 counts factorizations into semiprimes.

%Y A332765/A332877 is the greatest squarefree semiprime of weight n.

%Y A338898, A338912, and A338913 give the prime indices of semiprimes, with product A087794, sum A176504, and difference A176506.

%Y A338899, A270650, and A270652 give the prime indices of squarefree semiprimes, with difference A338900.

%Y A338907/A338906 list semiprimes of odd/even weight.

%Y A338907/A338908 list squarefree semiprimes of odd/even weight.

%Y Cf. A000040, A001221, A001222, A014342, A025129, A062198, A112798, A338905, A339116.

%K nonn

%O 2,1

%A _Gus Wiseman_, Nov 28 2020