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Least positive integer with n more semiprime divisors than semi-sums of prime indices.
9

%I #16 Nov 18 2023 13:12:45

%S 1,90,630,2310,6930,34650,30030,90090,450450,570570,510510,1531530,

%T 7657650,14804790,11741730,9699690,29099070,145495350

%N Least positive integer with n more semiprime divisors than semi-sums of prime indices.

%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 We define a semi-sum of a multiset to be any sum of a 2-element submultiset. This is different from sums of pairs of elements. For example, 2 is the sum of a pair of elements of {1}, but there are no semi-sums.

%C Are all primorials after 210 included?

%F a(n) is the least positive integer such that A086971(a(n)) - A366739(a(n)) = n.

%e The terms together with their prime indices begin:

%e 1: {}

%e 90: {1,2,2,3}

%e 630: {1,2,2,3,4}

%e 2310: {1,2,3,4,5}

%e 6930: {1,2,2,3,4,5}

%e 34650: {1,2,2,3,3,4,5}

%e 30030: {1,2,3,4,5,6}

%e 90090: {1,2,2,3,4,5,6}

%e 450450: {1,2,2,3,3,4,5,6}

%e 570570: {1,2,3,4,5,6,8}

%e 510510: {1,2,3,4,5,6,7}

%t nn=10000;

%t w=Table[Length[Union[Subsets[prix[n],{2}]]]-Length[Union[Total/@Subsets[prix[n],{2}]]],{n,nn}];

%t spnm[y_]:=Max@@NestWhile[Most,y,Union[#]!=Range[0,Max@@#]&];

%t Table[Position[w,k][[1,1]],{k,0,spnm[w]}]

%o (Python)

%o from itertools import count

%o from sympy import factorint, primepi

%o from sympy.utilities.iterables import multiset_combinations

%o def A367093(n):

%o for k in count(1):

%o c, a = 0, set()

%o for s in (sum(p) for p in multiset_combinations({primepi(i):j for i,j in factorint(k).items()},2)):

%o if s not in a:

%o a.add(s)

%o else:

%o c += 1

%o if c > n:

%o break

%o if c == n:

%o return k # _Chai Wah Wu_, Nov 13 2023

%Y The first part (semiprime divisors) is A086971, firsts A220264.

%Y The second part (semi-sums of prime indices) is A366739, firsts A367097.

%Y All sums of pairs of prime indices are counted by A367095.

%Y The non-binary version is A367105.

%Y A001222 counts prime factors (or prime indices), distinct A001221.

%Y A001358 lists semiprimes, squarefree A006881, conjugate A065119.

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

%Y A299701 counts subset-sums of prime indices, positive A304793.

%Y Semiprime divisors are listed by A367096 and have:

%Y - square count: A056170

%Y - sum: A076290

%Y - squarefree count: A079275

%Y - count: A086971

%Y - firsts: A220264

%Y Cf. A000720, A001248, A008967, A117855, A304792, A365541, A365920, A366738, A366740, A366753.

%K nonn,more

%O 0,2

%A _Gus Wiseman_, Nov 05 2023

%E a(12)-a(16) from _Chai Wah Wu_, Nov 13 2023

%E a(17) from _Chai Wah Wu_, Nov 18 2023