A250477 Number of times prime(n) (the n-th prime) occurs as the least prime factor among numbers 1 .. (prime(n)^2 * prime(n+1)): a(n) = A078898(A251720(n)).

6, 8, 12, 21, 33, 45, 63, 80, 116, 148, 182, 232, 265, 296, 356, 433, 490, 548, 625, 674, 740, 829, 919, 1055, 1187, 1252, 1313, 1376, 1446, 1657, 1897, 2029, 2134, 2301, 2484, 2605, 2785, 2946, 3110, 3301, 3439, 3654, 3869, 3978, 4086, 4349, 4811, 5147, 5273, 5395, 5604, 5787, 6049, 6403, 6684, 6954, 7153

Offset

1,1

Comments

a(n) = Position of 6 on row n of array A249821. This is always larger than A250474(n), the position of 4 on row n, as 4 is guaranteed to be the first composite term on each row of A249821.

From Antti Karttunen, Mar 29 2015: (Start)

a(n) = 1 + number of positive integers <= (prime(n)*prime(n+1)) whose smallest prime factor is at least prime(n).

That a(n) > A250474(n) can also be seen by realizing that prime(n) must occur at least as many times as the smallest prime factor for the numbers in range 1 .. (prime(n)^2 * prime(n+1)) than for numbers in (smaller) range 1 .. (prime(n)^3), and also by realizing that a(n) cannot be equal to A250474(n) because each row of A249822 is a permutation of natural numbers.

Or more simply, by considering the comment given in A256447 which follows from the new interpretation given above.

(End)

Links

Antti Karttunen, Table of n, a(n) for n = 1..564

Antti Karttunen, Ratio a(n)/A250474(n+1), plotted up to n=65 with OEIS Plot2-utility

Formula

a(n) = A078898(A251720(n)).

a(1) = 1, a(n) = Sum_{d | A002110(n-1)} moebius(d) * floor(A006094(n) / d). [Follows when A251720, (p_n)^2 * p_{n+1} is substituted to the similar formula given for A078898. Here p_n is the n-th prime (A000040(n)), A006094(n) gives the product p_n * p{n+1} and A002110(n) gives the product of the first n primes. Because the latter is always squarefree, one could use here also Liouville's lambda (A008836) instead of Moebius mu (A008683)].

a(n) = A250474(n) + A256447(n).

Mathematica

f[n_] := Count[Range[Prime[n]^2*Prime[n + 1]], x_ /; Min[First /@ FactorInteger[x]] == Prime@ n]; Array[f, 20] (* Michael De Vlieger, Mar 30 2015 *)

Prog

(PARI)
allocatemem(234567890);
A002110(n) = prod(i=1, n, prime(i));
A250477(n) = { my(m); m = (prime(n) * prime(n+1)); sumdiv(A002110(n-1), d, (moebius(d)*(m\d))); };
for(n=1, 23, print1(A250477(n), ", "));
\\ A more practical program:
(PARI)
allocatemem(234567890);
vecsize = (2^24)-4;
v020639 = vector(vecsize);
v020639[1] = 1; for(n=2, vecsize, v020639[n] = vecmin(factor(n)[, 1]));
A020639(n) = v020639[n];
A250477(n) = { my(p=prime(n), q=prime(n+1), u=p*q, k=1, s=1); while(k <= u, if(A020639(k) >= p, s++); k++); s; };
for(n=1, 564, write("b250477.txt", n, " ", A250477(n)));
\\ Antti Karttunen, Mar 29 2015

Crossrefs

Column 6 of A249822. Cf. also A250474 (column 4), A250478 (column 8).

First differences: A256446. Cf. also A256447, A256448.

Cf. A002110, A006094, A020639, A078898, A249821, A251720, A281890.

Sequence in context: A315870 A212350 A077670 * A315871 A315872 A315873

Adjacent sequences: A250474 A250475 A250476 * A250478 A250479 A250480

Keyword

nonn

Author

Antti Karttunen, Dec 14 2014

Status

approved