cp's OEIS Frontend

A357975 Divide all prime indices by 2, round down, and take the number with those prime indices, assuming prime(0) = 1.

%I A357975 #19 Jun 06 2025 14:46:34
%S A357975 1,1,2,1,2,2,3,1,4,2,3,2,5,3,4,1,5,4,7,2,6,3,7,2,4,5,8,3,11,4,11,1,6,
%T A357975 5,6,4,13,7,10,2,13,6,17,3,8,7,17,2,9,4,10,5,19,8,6,3,14,11,19,4,23,
%U A357975 11,12,1,10,6,23,5,14,6,29,4,29,13,8,7,9,10,31
%N A357975 Divide all prime indices by 2, round down, and take the number with those prime indices, assuming prime(0) = 1.
%C A357975 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 A357975 Also the Heinz number of the part-wise half (rounded down) of the partition with Heinz number n, where the Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
%C A357975 Each n appears A000005(n) times at odd positions (infinitely many at even). To see this, note that our transformation does not distinguish between A066207 and A066208.
%H A357975 SiYang Hu, <a href="/A357975/b357975.txt">Table of n, a(n) for n = 1..10000</a>
%F A357975 Completely multiplicative with a(prime(2k)) = prime(k) and a(prime(2k+1)) = prime(k). Cf. A297002.
%F A357975 a(prime(n)) = A109763(n-1).
%e A357975 The prime indices of n = 1501500 are {1,1,2,3,3,3,4,5,6}, so the prime indices of a(n) are {1,1,1,1,2,2,3}; hence we have a(1501500) = 720.
%e A357975 The 6 odd positions of 2124 are: 63, 99, 105, 165, 175, 275, with prime indices:
%e A357975    63: {2,2,4}
%e A357975    99: {2,2,5}
%e A357975   105: {2,3,4}
%e A357975   165: {2,3,5}
%e A357975   175: {3,3,4}
%e A357975   275: {3,3,5}
%t A357975 Table[Times@@(If[#1<=2,1,Prime[Floor[PrimePi[#1]/2]]^#2]&@@@FactorInteger[n]),{n,100}]
%Y A357975 Positions of 1's are A000079.
%Y A357975 Positions of 2's are 3 and A164095.
%Y A357975 Positions of first appearances are A297002, sorted A066207.
%Y A357975 A004526 is floor(n/2), with an extra first zero.
%Y A357975 A056239 adds up prime indices, row-sums of A112798.
%Y A357975 A109763 lists primes of index floor(n/2).
%Y A357975 Cf. A003961, A064988, A064989, A076610, A215366, A248601, A357980.
%K A357975 nonn
%O A357975 1,3
%A A357975 _Gus Wiseman_, Oct 23 2022