David S. Metzler - cp's OEIS frontend

A328946 Product of primorials of consecutive integers (second definition A034386).

1, 1, 2, 12, 72, 2160, 64800, 13608000, 2857680000, 600112800000, 126023688000000, 291114719280000000, 672475001536800000000, 20194424296150104000000000, 606438561613387623120000000000, 18211350005250030322293600000000000, 546886840657658410578476808000000000000

Offset: 0

David S. Metzler, Oct 31 2019

nonn

Similar to superprimorials (A006939), but a term of the sequence is a product of primorials of consecutive integers, not consecutive primes. So after 2# each primorial will repeat at least twice in the product. Also similar to superprimorials in that the exponents of the primes decrease linearly, but here it is linearly in p, not in pi(p).

			a(7) = 1# * 2# * 3# * 4# * 5# * 6# * 7# = 1*2*(2*3)*(2*3)*(2*3*5)*(2*3*5)*(2*3*5*7) = 2^6 * 3^5 * 5^3 * 7^1. Note that in the prime factorization the sum of each prime and its exponent is constant and equal to 7+1 = 8.
a(23) = 2^22 * 3^21 * 5^19 * 7^17 * 11^13 * 13^11 * 17^7 * 19^5 * 23^1. Here each prime and its exponent add to 24.

Product of consecutive elements of A034386.

b:= proc(n) option remember; `if`(n=0, [1$2], (p-> (h->
      [h, h*p[2]])(`if`(isprime(n), n, 1)*p[1]))(b(n-1)))
    end:
a:= n-> b(n)[2]:
seq(a(n), n=0..16);  # Alois P. Heinz, Nov 11 2020

b[n_] := b[n] = If[n==0, {1, 1}, Function[p, Function[h, {h, h p[[2]]}][If[ PrimeQ[n], n, 1] p[[1]]]][b[n - 1]]];
a[n_] := b[n][[2]];
a /@ Range[0, 16] (* Jean-François Alcover, Nov 30 2020, after Alois P. Heinz *)

primo(n) = lcm(primes([2, n])); \\ A034386
a(n) = prod(k=1, n, primo(k)); \\ Michel Marcus, Nov 01 2019

a(n) = Product_{k=1..n} A034386(k) = Product_{p prime, p<=n} p^(n-p+1) = Product_{p prime} p^(max(n-p+1,0)) = Product_{p prime,p+k = n+1 and k >= 0} p^k.

a(n) = lcm(n, a(n-1)^2/a(n-2)). - Jon Maiga, Jul 08 2021

a(0)=1 prepended by Alois P. Heinz, Nov 11 2020

A322966 Denominator of Sum_{d | n} 1/rad(d) where rad = A007947.

Original entry on oeis.org

1, 2, 3, 1, 5, 1, 7, 2, 3, 5, 11, 3, 13, 7, 5, 1, 17, 2, 19, 5, 21, 11, 23, 3, 5, 13, 1, 7, 29, 5, 31, 2, 11, 17, 35, 3, 37, 19, 39, 1, 41, 7, 43, 11, 1, 23, 47, 1, 7, 10, 17, 13, 53, 1, 55, 7, 57, 29, 59, 5, 61, 31, 21, 1, 65, 11, 67, 17, 23, 35, 71, 6, 73, 37, 15, 19, 77, 13, 79, 5

Offset: 1

David S. Metzler, Dec 31 2018

Let rad(n) be the radical of n, which equals the product of all prime factors of n (A007947). Let f(n) = Sum_{d | n} 1/rad(n). The sequence a(n) lists the denominators of the fractions f(n) in lowest terms.

f is a multiplicative function whose value on a prime power is f(p^k) = 1 + k/p. Hence f is a weighted divisor-counting function that weights divisors d higher when they have few and small prime divisors themselves.

			The divisors of 12 are 1,2,3,4,6,12, so f(12) = 1 + (1/2) + (1/3) + (1/2) + (1/6) + (1/6) = 8/3, and a(12) = 3. Alternately, since f is multiplicative, f(12) = f(4)*f(3) = (1+2/2)*(1+1/3) = 8/3. The denominator of f(12) is 3 hence a(12) = 3.

Cf. A007947 (radical), A322965 (numerators).

Numbers n where f(n) increases to a record: A322447.

Array[Denominator@ DivisorSum[#, 1/Apply[Times, FactorInteger[#][[All, 1]] ] &] &, 80] (* Michael De Vlieger, Jan 19 2019 *)

a(n) = denominator(sumdiv(n, d, 1/factorback(factor(d)[, 1]))) \\ David A. Corneth, Jan 01 2019

A322965 Numerator of Sum_{d | n} 1/rad(d).

Original entry on oeis.org

1, 3, 4, 2, 6, 2, 8, 5, 5, 9, 12, 8, 14, 12, 8, 3, 18, 5, 20, 12, 32, 18, 24, 10, 7, 21, 2, 16, 30, 12, 32, 7, 16, 27, 48, 10, 38, 30, 56, 3, 42, 16, 44, 24, 2, 36, 48, 4, 9, 21, 24, 28, 54, 3, 72, 20, 80, 45, 60, 16, 62, 48, 40, 4, 84, 24, 68, 36, 32, 72, 72, 25, 74, 57, 28

Offset: 1

David S. Metzler, Dec 31 2018

Let rad(n) be the radical of n, which equals the product of all prime factors of n (A007947). Let g(n) = 1/rad(n) and let f(n) = Sum_{d | n} g(d). This is a multiplicative function whose value on a prime power is f(p^k) = 1 + k/p. Hence f is a weighted divisor-counting function that weights divisors d higher when they have few and small prime divisors themselves. The sequence a(n) lists the numerators of the fractions f(n) in lowest terms.

If p is prime, then a(p^k) = p+k if p does not divide k, 1 + k/p if it does. In particular, a(p^p) = 2. - Robert Israel, Jan 25 2019

			The divisors of 12 are 1,2,3,4,6,12, so f(12) = 1 + (1/2) + (1/3) + (1/2) + (1/6) + (1/6) = 8/3 and a(12) = 8. Alternately, since f is multiplicative, f(12) = f(4)*f(3) = (1+2/2)*(1+1/3) = 8/3.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A007947 (radical), A322966 (denominators), A008473 (unreduced numerators, i.e., f(n)*rad(n)), A082695.

Numbers n where f(n) increases to a record: A322447.

rad:= n -> convert(numtheory:-factorset(n),`*`):
f:= proc(n) numer(add(1/rad(d),d=numtheory:-divisors(n))) end proc:
map(f, [$1..100]); # Robert Israel, Jan 25 2019

Array[Numerator@ DivisorSum[#, 1/Apply[Times, FactorInteger[#][[All, 1]]] &] &, 71] (* Michael De Vlieger, Jan 19 2019 *)

rad(n) = factorback(factor(n)[, 1]); \\ A007947
a(n) = numerator(sumdiv(n, d, 1/rad(d))); \\ Michel Marcus, Jan 10 2019

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)/A322966(k) = zeta(2)*zeta(3)/zeta(6) (A082695). - Amiram Eldar, Dec 09 2023

More terms from Michel Marcus, Jan 19 2019

A322447 Numbers k where Sum_{d | k} 1/rad(d) increases to a record.

Original entry on oeis.org

1, 2, 4, 8, 12, 16, 24, 32, 48, 72, 96, 144, 192, 288, 384, 576, 864, 1152, 1728, 2304, 3456, 4608, 5184, 6912, 10368, 13824, 20736, 27648, 41472, 55296, 62208, 82944, 124416, 165888, 207360, 248832, 331776, 373248, 414720, 497664, 622080, 746496, 829440, 995328

Offset: 1

David S. Metzler, Dec 08 2018

nonn

Let rad(n) be the radical of n, which equals the product of all prime factors of n (A007947). Let g(n) = 1/rad(n) and let f(n) = Sum_{d | n} g(d). This is a multiplicative function whose value on a prime power is f(p^k) = 1 + k/p. Hence f is a weighted divisor-counting function that weights divisors d higher when they have few and small prime divisors themselves. This sequence lists the values where f(n) increases to a record, analogously to highly composite numbers (A002182) or superabundant numbers (A004394). The numbers in this sequence are much smoother than those in the other two sequences, since the definition of f(n) strongly disfavors a lack of smoothness in n.

			The divisors of 12 are 1,2,3,4,6,12, so f(12) = 1 + (1/2) + (1/3) + (1/2) + (1/6) + (1/6) = 8/3, which exceeds f(n) for n = 1,...,11. Alternately, since f is multiplicative, f(12) = f(4)*f(3) = (1+2/2)*(1+1/3).
f(207360) = f(2^9)*f(3^4)*f(5) = (11/2)*(7/3)*(6/5) = 15.4, which exceeds f(n) for n < 207360. (Note that this is the first value of the sequence that is divisible by 5; earlier values are all 3-smooth.)

Charlie Neder, Table of n, a(n) for n = 1..200

Cf. A007947 (radical), A002182, A004394.

Also smooth numbers: A003586, A051037, A002473.

rad[n_] := Times @@ (First@# & /@ FactorInteger@n); f[n_] := DivisorSum[n, 1/rad[#] &]; fm = 0; s = {}; Do[f1 = f[n]; If[f1 > fm, fm = f1; AppendTo[s, n]], {n, 1, 10^6}]; s (* Amiram Eldar, Dec 08 2018 *)

rad(n) = factorback(factorint(n)[, 1]); \\ A007947
lista(nn) = {my(m=0, newm); for (n=1, nn, newm = sumdiv(n, d, 1/rad(d)); if (newm > m, m = newm; print1(n, ", ")););} \\ Michel Marcus, Dec 09 2018

A256108 Positions of nonzero digits in binary expansion of Pi.

Original entry on oeis.org

-1, 0, 3, 6, 11, 12, 13, 14, 15, 16, 18, 19, 21, 23, 25, 29, 33, 38, 40, 41, 43, 47, 48, 53, 57, 58, 60, 63, 64, 68, 71, 72, 76, 77, 80, 81, 85, 87, 91, 93, 94, 95, 103, 104, 106, 107, 108, 114, 115, 116, 119, 120, 122, 126, 129, 131, 134, 141, 144, 147, 148, 149, 155, 159

Offset: 1

David S. Metzler, Mar 14 2015

Nonzero entries in A004601 (re-indexed to start at -1 and ascend).

The binary positions (exponents) are negated for convenience (as is standard practice). By the results of the PiHex project, the number 1,000,000,000,000,060 (for example) eventually appears in this sequence. Submitted on 3/14/15, (decimal) Pi Day.

			The most significant nonzero binary digit of pi occurs in the 2^1 position. Then there is a digit in the 2^0 position, then the 2^(-3) position, etc. Negate the exponents appearing to get this sequence.

Paolo Xausa, Table of n, a(n) for n = 1..10000
Steve Pagliarulo, Stu's pi page: base 2
Colin Percival, PiHex Home Page
Wikipedia, PiHex

Cf. A004601 (Pi in base 2), A051480.

PositionIndex[First[RealDigits[Pi, 2, 200]]][1] - 2 (* Paolo Xausa, Aug 04 2024 *)

A256108_upto(N)={localbitprec(N+20); [i-2|i<-[1..-20+#N=concat(binary(Pi))], N[i]]} \\ M. F. Hasler, Jul 27 2024

Pi = Sum_{n>=0} 2^(-a(n)).

This sequence A256108 = { i | A004601(1-i) = 1 }. - M. F. Hasler, Jul 27 2024

A224078 Numbers that are superior highly composite and colossally abundant.

Original entry on oeis.org

2, 6, 12, 60, 120, 360, 2520, 5040, 55440, 720720, 1441440, 4324320, 21621600, 367567200, 6983776800, 321253732800, 6064949221531200, 791245405339403414400, 37188534050951960476800, 581442729886633902054768000

Offset: 1

David S. Metzler, Mar 30 2013

This should be all of the terms. Intersection of {highly composite} and {superabundant} has 449 terms, cf. A002182. The shapes (pattern of exponents of prime factors) of {superior highly composite} and {colossally abundant} seem to diverge for good after the last term listed here.

Finite intersection of A002201 and A004490.

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User: David S. Metzler

A328946 Product of primorials of consecutive integers (second definition A034386).

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A322966 Denominator of Sum_{d | n} 1/rad(d) where rad = A007947.

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A322965 Numerator of Sum_{d | n} 1/rad(d).

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A322447 Numbers k where Sum_{d | k} 1/rad(d) increases to a record.

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A256108 Positions of nonzero digits in binary expansion of Pi.

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A224078 Numbers that are superior highly composite and colossally abundant.

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