A038111 -id:A038111 - cp's OEIS frontend

A356094 a(n) = denominator((prime(n)-1)/prime(n)#), where prime(n)# = A002110(n) is the n-th primorial.

2, 3, 15, 35, 231, 5005, 255255, 1616615, 10140585, 462120945, 6685349671, 1236789689135, 30425026352721, 311494317420715, 13367169186706335, 1253429172199617105, 33151040519900217915, 3909612711980232366109, 119065478046670712967865, 7970583287524270870963077

Offset: 1

Amiram Eldar, Jul 26 2022

See A356093 for details.

Amiram Eldar, Table of n, a(n) for n = 1..351

Cf. A002110, A356093 (numerators).

Similar sequences: A038111, A338560, A340819, A341432, A342451, A342480.

primorial[n_] := Product[Prime[i], {i, 1, n}]; Denominator[Table[(Prime[i] - 1)/primorial[i], {i, 1, 20}]]

a(n) = denominator((prime(n)-1)/factorback(primes(n))); \\ Michel Marcus, Jul 26 2022

from math import gcd
from sympy import primorial, prime
def A356094(n): return (p:=primorial(n))//gcd(p,prime(n)-1) # Chai Wah Wu, Jul 26 2022

A377734 Number of integers less than n that have the same smallest prime factor as n.

Original entry on oeis.org

0, 0, 0, 1, 0, 2, 0, 3, 1, 4, 0, 5, 0, 6, 2, 7, 0, 8, 0, 9, 3, 10, 0, 11, 1, 12, 4, 13, 0, 14, 0, 15, 5, 16, 2, 17, 0, 18, 6, 19, 0, 20, 0, 21, 7, 22, 0, 23, 1, 24, 8, 25, 0, 26, 3, 27, 9, 28, 0, 29, 0, 30, 10, 31, 4, 32, 0, 33, 11, 34, 0, 35, 0, 36, 12, 37, 2, 38, 0, 39

Offset: 1

Ilya Gutkovskiy, Nov 05 2024

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Least Prime Factor.

Cf. A020639, A038110, A038111, A047983, A078898, A334655, A335097, A377730.

Table[Length[Select[Range[n - 1], If[# == 1, 1, FactorInteger[#][[1, 1]]] == If[n == 1, 1, FactorInteger[n][[1, 1]]] &]], {n, 80}]
seq[len_] := Module[{t = Table[FactorInteger[n][[1,1]], {n, 1, len}], s = Table[0, {len}]}, Do[s[[i]] = Count[t[[1;;i-1]], t[[i]]], {i, 1, len}]; s]; seq[80] (* Amiram Eldar, Nov 21 2024 *)

a(n) = if (n>1, my(p=vecmin(factor(n)[,1])); sum(k=2, n-1, p == vecmin(factor(k)[,1])), 0); \\ Michel Marcus, Nov 16 2024

a(n) = |{j < n : lpf(j) = lpf(n)}|.
a(n) = A078898(n) - 1.
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = Sum_{k>=1} (A038110(k)/A038111(k))^2 = 0.2847976823663... . - Amiram Eldar, Nov 21 2024

A306353 Number of composites among the first n composite numbers whose least prime factor p is that of the n-th composite number.

Original entry on oeis.org

1, 2, 3, 1, 4, 5, 6, 2, 7, 8, 9, 3, 10, 11, 1, 12, 4, 13, 14, 15, 5, 16, 2, 17, 18, 6, 19, 20, 21, 7, 22, 23, 1, 24, 8, 25, 26, 3, 27, 9, 28, 29, 30, 10, 31, 4, 32, 33, 11, 34, 35, 36, 12, 37, 2, 38, 39, 13, 40, 41, 5, 42, 14, 43, 44, 3, 45, 15, 46, 6, 47, 48, 16, 49, 50, 51, 17, 52, 53, 54, 18, 55, 56, 7

Offset: 1

Jamie Morken and Vincenzo Librandi, Feb 09 2019

Composites with least prime factor p are on that row of A083140 which begins with p
Sequence with similar values: A122005.
Sequence written as a jagged array A with new row when a(n) > a(n+1):
  1,  2,  3,
  1,  4,  5,  6,
  2,  7,  8,  9,
  3, 10, 11,
  1, 12,
  4, 13, 14, 15,
  5, 16,
  2, 17, 18,
  6, 19, 20, 21,
  7, 22, 23,
  1, 24,
  8, 25, 26,
  3, 27,
  9, 28, 29, 30.
A153196 is the list B of the first values in successive rows with length 4.
  B is given by the formula for A002808(x)=A256388(n+3), an(x)=A153196(n+2)
For example: A002808(26)=A256388(3+3), an(26)=A153196(3+2).
A243811 is the list of the second values in successive rows with length 4.
A047845 is the list of values in the second column and A104279 is the list of values in the third column of the jagged array starting on the second row.
Sequence written as an irregular triangle C with new row when a(n)=1:
  1,2,3,
  1,4,5,6,2,7,8,9,3,10,11,
  1,12,4,13,14,15,5,16,2,17,18,6,19,20,21,7,22,23,
  1,24,8,25,26,3,27,9,28,29,30,10,31,4,32,33,11,34,35,36,12,37,2,38,39,13,40,41,5,42,14,43,44,3,45,15,46,6,47,48,16,49,50,51,17,52,53,54,18,55,56,7,57,19,58,4,59.
A243887 is the last value in each row of C.
The second value D on the row n > 1 of the irregular triangle C is a(A053683(n)) or equivalently A084921(n). For example for row 3 of the irregular triangle:
  D = a(A053683(3)) = a(16) = 12 or D = A084921(3) = 12. This is the number of composites < A066872(3) with the same least prime factor p as the A053683(3) = 16th composite, A066872(3) = 26.
The number of values in each row of the irregular triangle C begins: 3,11,18,57,39,98,61,141,265,104,351,268,...
The second row of the irregular triangle C is A117385(b) for 3 < b < 15.
The third row of the irregular triangle C has similar values as A117385 in different order.

			First composite 4, least prime factor is 2, first case for 2 so a(1)=1.
Next composite 6, least prime factor is 2, second case for 2 so a(2)=2.
Next composite 8, least prime factor is 2, third case for 2 so a(3)=3.
Next composite 9, least prime factor is 3, first case for 3 so a(4)=1.
Next composite 10, least prime factor is 2, fourth case for 2 so a(5)=4.

Jamie Morken, Table of n, a(n) for n = 1..10000

Cf. A002808, A256388, A056608, A083140, A122005, A153196, A243811, A047845, A104279, A243887, A117385, A216244, A084921, A066872, A053683, A038110, A038111, A020639.

counts = {}
values = {}
For[i = 2, i < 130, i = i + 1,
If[PrimeQ[i], ,
x = PrimePi[FactorInteger[i][[1, 1]]];
  If[Length[counts] >= x,
   counts[[x]] = counts[[x]] + 1;
   AppendTo[values, counts[[x]]], AppendTo[counts, 1];
   AppendTo[values, 1]]]]
   (* Print[counts] *)
   Print[values]

c(n) = for(k=0, primepi(n), isprime(n++)&&k--); n; \\ A002808
a(n) = my(c=c(n), lpf = vecmin(factor(c)[,1]), nb=0); for(k=2, c, if (!isprime(k) && vecmin(factor(k)[,1])==lpf, nb++)); nb; \\ Michel Marcus, Feb 10 2019

a(n) is approximately equal to A002808(n)*(A038110(x)/A038111(x)), with A000040(x)=A020639(A002808(n)).
For example if n=325, a(325)~= A002808(325)*(A038110(2)/A038111(2)) with A000040(2)=A020639(A002808(325)).
This gives an estimate of 67.499... and the actual value of a(n)=67.

A252941 Irregular triangle T(n,k) read by rows: T(1,1) = 1, otherwise row n lists the prime factors of A098550(n), with duplicates omitted.

Original entry on oeis.org

1, 2, 3, 2, 3, 2, 3, 5, 2, 7, 5, 2, 3, 5, 2, 3, 5, 7, 2, 7, 2, 5, 3, 7, 2, 5, 3, 2, 11, 3, 13, 11, 13, 3, 11, 2, 13, 3, 5, 2, 7, 3, 13, 2, 17, 2, 3, 5, 17, 2, 3, 5, 11, 2, 17, 5, 13, 2, 3, 7, 13, 2, 3, 5, 7, 2, 19, 3, 7, 19, 2, 3, 7, 5, 19, 2, 11

Offset: 1

Bob Selcoe, Mar 22 2015

Row n contains the distinct prime factors of A098550(n), in increasing order. For example, when n=13, A098550(13) = 35 and T(13,k) = [5,7].
Because A098550 is a permutation of the natural numbers, this sequence is infinite and contains every prime infinitely often.
Primes appear in order; that is, first appearance of prime(j) occurs prior to first appearance of prime(j+1).
T(n,1) = A251101(n), which are the smallest prime factors of A098550(n), n>1.
For n>1, let each coefficient in T(n,1) be prime(i). The ratio that each coefficient appears in T(j,1) {j=1..n} approaches A038110(i)/A038111(i) as j increases.  For example, prime(4) = 7: as j increases, the ratio that 7 appears in T(j,1) approaches 4/105, because A038110(4)/A038111(4) = 4/105.

			Triangle begins T(1,1):
1
2
3
2
3
2
3 5
2 7
5
2 3
5
2 3
5 7
2
7
2 5
3 7
2 5
3
2 11
e.g., n=13: A098550(13) = 35; T(13,k) = 5,7.

David L. Applegate, Hans Havermann, Bob Selcoe, Vladimir Shevelev, N. J. A. Sloane, and Reinhard Zumkeller, The Yellowstone Permutation, arXiv preprint arXiv:1501.01669, 2015.

Cf. A098550.

cp's OEIS Frontend

A356094 a(n) = denominator((prime(n)-1)/prime(n)#), where prime(n)# = A002110(n) is the n-th primorial.

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A377734 Number of integers less than n that have the same smallest prime factor as n.

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A306353 Number of composites among the first n composite numbers whose least prime factor p is that of the n-th composite number.

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A252941 Irregular triangle T(n,k) read by rows: T(1,1) = 1, otherwise row n lists the prime factors of A098550(n), with duplicates omitted.

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