A371906 -id:A371906

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A372000

a(n) = product of primes p such that floor(n/p) is odd.

+10
4

1, 2, 6, 3, 15, 10, 70, 35, 105, 42, 462, 77, 1001, 286, 4290, 2145, 36465, 24310, 461890, 46189, 969969, 176358, 4056234, 676039, 3380195, 520030, 1560090, 111435, 3231615, 430882, 13357342, 6678671, 220396143, 25928958, 907513530, 151252255, 5596333435, 589087730, 22974421470, 2297442147

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

The only primes in the sequence are 2 and 3.

We can approach the sequence in a manner akin to A260850, a variant of A008336. Set k = 1. Then for all prime factors p | n, if p | k, divide k by p, otherwise multiply k by p. Then we set a(n) = k. This accounts for the "toggling on or off" of prime factors as n increases.

For n >= 1, A055773(n) | a(n), where A055773(n) = A034386(n) / A034386(floor(n/2)).

LINKS

Michael De Vlieger, Table of n, a(n) for n = 1..3384

Michael De Vlieger, Plot prime(i) | a(n) at (x,y) = (n,i) for n = 1..2048.

FORMULA

a(n) = Product_{k = 1..floor(pi(n)/2)+1} Product_{j = 1+floor(n/(2*k))..floor(n/(2*k-1))} prime(j), where pi(x) = A000720(n).

EXAMPLE

a(1) = 1 since n = 1 is the empty product.

a(2) = 2 since for n = 2, floor(n/p) = floor(2/2) = 1 is odd.

a(3) = 6 since for n = 3 and p = 2, floor(3/2) = 1 is odd, and for p = 3, floor(3/3) = 1 is odd. Hence a(3) = 2*3 = 6.

a(4) = 3 since for n = 4 and p = 2, floor(4/2) = 2 is even, but for p = 3, floor(4/3) = 1 is odd. Therefore, a(n) = 3.

a(5) = 15 since for n = 5, though floor(5/2) = 2 is even, floor(5/3) and floor(5/5) are both odd. Therefore, a(n) = 3*5 = 15, etc.

Table relating a(n) with b(n), diagramming prime factors with "x" that produce a(n), or powers of 2 with "x" that sum to b(n), where b(n) = A371906(n).

Prime factor

1111

n b(n) 23571379 b(n)

----------------------------

1 1 . 0

2 2 x 1

3 6 xx 3

4 3 .x 2

5 15 .xx 6

6 10 x.x 5

7 70 x.xx 13

8 35 ..xx 12

9 105 .xxx 14

10 42 xx.x 11

11 462 xx.xx 27

12 77 ...xx 24

13 1001 ...xxx 56

14 286 x...xx 49

15 4290 xxx.xx 55

16 2145 .xx.xx 54

17 36465 .xx.xxx 118

18 24310 x.x.xxx 117

19 461890 x.x.xxxx 245

20 46189 ....xxxx 240

----------------------------

01234567

Power of 2

MATHEMATICA

Table[Times @@ Select[Prime@ Range@ PrimePi[n], OddQ@ Quotient[n, #] &], {n, 40}] (* or *)

Table[Product[Prime[i], {j, 1 + Floor[PrimePi[n]/2]}, {i, 1 + PrimePi[Floor[n/(2 j)]], PrimePi[Floor[n/(2 j - 1)]]}], {n, 40}]

PROG

(PARI) a(n) = vecprod(select(x->((n\x) % 2), primes([1, n]))); \\ Michel Marcus, Apr 16 2024

(SageMath)

print([prod(p for p in prime_range(n + 1) if is_odd(n//p)) for n in range(1, 41)])

# Peter Luschny, Apr 16 2024

CROSSREFS

Cf. A005117, A008336, A034386, A055773, A260850, A371906.

KEYWORD

nonn,easy

AUTHOR

Michael De Vlieger, Apr 15 2024

STATUS

approved

A371907

a(n) = sum of 2^(k-1) such that floor(n/prime(k)) is even.

+10
2

0, 0, 0, 1, 1, 2, 2, 3, 1, 4, 4, 7, 7, 14, 8, 9, 9, 10, 10, 15, 5, 20, 20, 23, 19, 50, 48, 57, 57, 62, 62, 63, 45, 108, 96, 99, 99, 226, 192, 197, 197, 206, 206, 223, 217, 472, 472, 475, 467, 470, 404, 437, 437, 438, 418, 427, 297, 808, 808, 815, 815, 1838, 1828

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,6

COMMENTS

This is a transform of A372007(n) = s(n). Write the prime indices of k factors prime(k) | s(n) instead as 2^(k-1) and take the sum for all primes p | s(n). Hence, s(14) = 105 = 3*5*7 becomes a(14) = 2^1 + 2^2 + 2^3 = 2 + 4 + 8 = 14.

LINKS

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Michael De Vlieger, "Tiger Stripe" Factors of Primorials, ResearchGate, 2024.

Plot powers 2^(i-1) that sum to a(n) at (x,y) = (n,i) for n = 1..2048, 12X vertical exaggeration.

FORMULA

a(n) = A357215(n) - A371906(n).

EXAMPLE

a(1) = 0 since n = 1 is the empty product.

a(2) = 0 since for n = prime(1) = 2, floor(2/2) = 1 is odd. Therefore a(2) = 0.

a(3) = 0 since for n = 3 and prime(1) = 2, floor(3/2) = 1 is odd, and for prime(2) = 3, floor(3/3) = 1 is odd. Hence a(3) = 0.

a(4) = 1 since for n = 4 and prime(1) = 2, floor(4/2) = 2 is even, but for prime(2) = 3, floor(4/3) = 1 is odd. Therefore, a(4) = 2^(1-1) = 1.

a(8) = 1 since for n = 8, both floor(8/2) and floor(8/3) are even, but both floor(8/5) and floor(8/7) are odd. Therefore, a(8) = 2^(1-1) + 2^(2-1) = 1 + 2 = 3, etc.

Table relating a(n) with b(n), s(n), and t(n), diagramming powers of 2 with "x" that sum to a(n) or b(n), or prime factors with "x" that produce s(n) or t(n). Sequences s(n) = A372007(n), t(n) = A372000(n), c(n) = A034386(n), b(n) = A371906(n), and c(n) = A357215(n) = a(n) + b(n). Column A (at top) shows powers of 2 that sum to a(n), with B same for b(n), while column S represents prime factors of s(n), T same of t(n).

[A] 2^k [B] 2^k

n 0123 a(n) 012345 b(n) c(n) s(n) t(n) v(n)

--------------------------------------------------------

1 . 0 . 0 2^0-1 1 1 P(0)

2 . 0 x 1 2^1-1 1 2 P(1)

3 . 0 xx 3 2^2-1 1 6 P(2)

4 x 1 .x 2 2^2-1 2 3 P(2)

5 x 1 .xx 6 2^3-1 2 15 P(3)

6 .x 2 x.x 5 2^3-1 3 10 P(3)

7 .x 2 x.xx 13 2^4-1 3 70 P(4)

8 xx 3 ..xx 12 2^4-1 6 35 P(4)

9 x 1 .xxx 14 2^4-1 2 105 P(4)

10 ..x 4 xx.x 11 2^4-1 5 42 P(4)

11 ..x 4 xx.xx 27 2^5-1 5 462 P(5)

12 xxx 7 ...xx 24 2^5-1 30 77 P(5)

13 xxx 7 ...xxx 56 2^6-1 30 1001 P(6)

14 .xxx 14 x...xx 49 2^6-1 105 286 P(6)

15 ...x 8 xxx.xx 55 2^6-1 7 4290 P(6)

16 x..x 9 .xx.xx 54 2^6-1 14 2145 P(6)

--------------------------------------------------------

2357 [T] 11

[S] 235713

MATHEMATICA

Table[Total[2^(-1 + Select[Range@ PrimePi[n], EvenQ@ Quotient[n, Prime[#]] &])], {n, 50}]

PROG

(PARI) a(n) = my(vp=primes([1, n])); vecsum(apply(x->2^(x-1), Vec(select(x->(((n\x) % 2)==0), vp, 1)))); \\ Michel Marcus, Apr 30 2024

CROSSREFS

Cf. A357215, A371906, A372007.

KEYWORD

nonn,easy

AUTHOR

Michael De Vlieger, Apr 17 2024

STATUS

approved

A372007

a(n) = product of those prime(k) such that floor(n/prime(k)) is even.

+10
2

1, 1, 1, 2, 2, 3, 3, 6, 2, 5, 5, 30, 30, 105, 7, 14, 14, 21, 21, 210, 10, 55, 55, 330, 66, 429, 143, 2002, 2002, 15015, 15015, 30030, 910, 7735, 221, 1326, 1326, 12597, 323, 3230, 3230, 33915, 33915, 746130, 49742, 572033, 572033, 3432198, 490314, 1225785, 24035

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,4

COMMENTS

The only primes in the sequence are 2, 3, 5, and 7.

LINKS

Michael De Vlieger, Table of n, a(n) for n = 1..7591

Michael De Vlieger, Plot prime(i) | a(n) at (x,y) = (n,i) for n = 1..2048, 12X vertical exaggeration.

Michael De Vlieger, "Tiger Stripe" Factors of Primorials, ResearchGate, 2024.

FORMULA

a(n) = A034386(n) / A372000(n).

a(n) = Product_{k = 1..pi(n)} Product_{j = 1+floor(n/(2*k+1))..floor(n/(2*k))} prime(j), where pi(x) = A000720(n).

EXAMPLE

a(1) = 1 since n = 1 is the empty product.

a(2) = 1 since for n = 2, floor(n/p) = floor(2/2) = 1 is odd.

a(3) = 1 since for n = 3 and p = 2, floor(3/2) = 1 is odd, and for p = 3, floor(3/3) = 1 is odd.

a(4) = 2 since for n = 4 and p = 2, floor(4/2) = 2 is even, but for p = 3, floor(4/3) = 1 is odd. Therefore, a(4) = 2.

a(5) = 2 since for n = 5, though floor(5/2) = 2 is even, floor(5/3) and floor(5/5) are both odd. Therefore, a(5) = 2.

a(8) = 6 since for n = 8, both floor(8/2) and floor(8/3) are even, but both floor(8/5) and floor(8/7) are odd. Therefore, a(8) = 2*3 = 6, etc.

Table relating a(n) with b(n), s(n), and t(n), diagramming prime factors with "x" that produce a(n) or b(n), or powers of 2 with "x" that sum to s(n) or t(n). Sequences b(n) = A372000(n), c(n) = A034386(n), s(n) = A371907(n), t(n) = A371906(n), and v(n) = A357215(n) = s(n) + t(n). Column A represents prime factors of a(n), B same of b(n), while column S (at bottom) shows powers of 2 that sum to s(n), with T same for t(n). P(n) = A002110(n).

[A] [B] 11

n 2357 a(n) 235713 b(n) c(n) s(n) t(n) v(n)

--------------------------------------------------------

1 . 1 . 1 P(0) 0 0 2^0-1

2 . 1 x 2 P(1) 0 1 2^1-1

3 . 1 xx 6 P(2) 0 3 2^2-1

4 x 2 .x 3 P(2) 1 2 2^2-1

5 x 2 .xx 15 P(3) 1 6 2^3-1

6 .x 3 x.x 10 P(3) 2 5 2^3-1

7 .x 3 x.xx 70 P(4) 2 13 2^4-1

8 xx 6 ..xx 35 P(4) 3 12 2^4-1

9 x 2 .xxx 105 P(4) 1 14 2^4-1

10 ..x 5 xx.x 42 P(4) 4 11 2^4-1

11 ..x 5 xx.xx 462 P(5) 4 27 2^5-1

12 xxx 30 ...xx 77 P(5) 7 24 2^5-1

13 xxx 30 ...xxx 1001 P(6) 7 56 2^6-1

14 .xxx 105 x...xx 286 P(6) 14 49 2^6-1

15 ...x 7 xxx.xx 4290 P(6) 8 55 2^6-1

16 x..x 14 .xx.xx 2145 P(6) 9 54 2^6-1

--------------------------------------------------------

0123 012345

[S] 2^k [T] 2^k

MATHEMATICA

Table[Times @@ Select[Prime@ Range@ PrimePi[n], EvenQ@ Quotient[n, #] &], {n, 51}] (* or *)

Table[Product[Prime[i], {j, PrimePi[n]}, {i, 1 + PrimePi[Floor[n/(2 j + 1)]], PrimePi[Floor[n/(2 j)]]}], {n, 51}]

PROG

(PARI) a(n) = my(vp=primes([1, n])); vecprod(select(x->(((n\x) % 2)==0), vp)); \\ Michel Marcus, Apr 30 2024

CROSSREFS

Cf. A005117, A034386, A371907, A372000.

KEYWORD

nonn,easy

AUTHOR

Michael De Vlieger, Apr 17 2024

STATUS

approved

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