A078456 -id:A078456 - cp's OEIS frontend

A135212 a(n) = A078456(n)/A120271(n).

1, 1, 2, 4, 8, 576, 1152, 2304, 4608, 18432, 552960, 59719680, 2388787200, 100329062400, 200658124800, 802632499200, 1605264998400, 288947699712000, 6356849393664000, 444979457556480000, 10679506981355520000

Offset: 1

Alexander Adamchuk, Nov 23 2007

nonn

G. C. Greubel, Table of n, a(n) for n = 1..250

Cf. A078456, A120271.

Table[ Det[ DiagonalMatrix[ Table[ Prime[i+1]-1, {i, 1, n-1} ] ] + 1 ], {n, 1, 50} ] / Numerator[ Table[ Sum[ 1/(Prime[i]-1), {i, 1, n} ], {n, 1, 50}] ]

a(n) = matdet(matrix(n-1, n-1, j, k, if (j==k, prime(j+1), 1)))/numerator(sum(k=1, n, 1/(prime(k)-1))); \\ Michel Marcus, Oct 02 2016

a(n) = A078456(n)/A120271(n).

A120271 a(n) = numerator(Sum_{k=1..n} 1/(prime(k)-1)).

Original entry on oeis.org

1, 3, 7, 23, 121, 21, 173, 1597, 17927, 127469, 129317, 43619, 44081, 44521, 1033223, 13538159, 395369371, 132680013, 400467919, 402757063, 1214947859, 1221110939, 50305908619, 50529880549, 101470376303, 509322834499, 8691337402883

Offset: 1

Alexander Adamchuk, Jul 01 2006

a(n) is squarefree except for n = 5, 14, 49, ... where squared prime factors are 11, 211, 479, ...

a(n)/A128646(n) is the asymptotic mean over the positive integers of the number of prime divisors that are not greater than prime(n), counted with multiplicity (cf. A007814, A169611, A356006). - Amiram Eldar, Jul 23 2022

Robert Israel, Table of n, a(n) for n = 1..1495
Eric Weisstein's World of Mathematics, Prime Sums.

Cf. A128646 (denominators), A119686, A006093, A000040.
Cf. A135212, A078456.
Cf. A007814, A169611, A356006.

R:= [seq(1/(ithprime(k)-1),k=1..40)]:
S:= ListTools:-PartialSums(R):
A:= map(numer,S); # Robert Israel, Jan 12 2025

Numerator[Table[Sum[1/(Prime[i]-1),{i,1,n}],{n,1,50}]]
Accumulate[1/(Prime[Range[30]]-1)]//Numerator (* Harvey P. Dale, May 03 2025 *)

a(n) = numerator(sum(k=1, n, 1/(prime(k)-1))); \\ Michel Marcus, Oct 02 2016

a(n) = numerator(Sum_{k=1..n} 1/(prime(k)-1)).

a(n) = A078456(n) * A135212(n). - Alexander Adamchuk, Nov 23 2007

A068499 Numbers m such that m! reduced modulo (m+1) is not zero.

Original entry on oeis.org

1, 2, 3, 4, 6, 10, 12, 16, 18, 22, 28, 30, 36, 40, 42, 46, 52, 58, 60, 66, 70, 72, 78, 82, 88, 96, 100, 102, 106, 108, 112, 126, 130, 136, 138, 148, 150, 156, 162, 166, 172, 178, 180, 190, 192, 196, 198, 210, 222, 226, 228, 232, 238, 240, 250, 256, 262, 268, 270

Offset: 1

Benoit Cloitre, Mar 11 2002

Also n such that tau((n+1)!) = 2* tau(n!)
For n > 2, a(n) is the smallest number such that a(n) !== -1 (mod a(k)+1) for any 1 < k < n. [Franklin T. Adams-Watters, Aug 07 2009]
Also n such that sigma((n+1)!) = (n+2)* sigma(n!), which is the same as A062569(n+1) = (n+2)*A062569(n). - Zak Seidov, Aug 22 2012
This sequence is obtained by the following sieve: keep 1 in the sequence and then, at the k-th step, keep the smallest number, x say, that has not been crossed off before and cross off all the numbers of the form k*(x + 1) - 1 with k > 1. The numbers that are left form the sequence. - Jean-Christophe Hervé, Dec 12 2015
a(n) = A039915(n-1) for 3 < n <= 1000. - Georg Fischer, Oct 19 2018

			Illustration of the sieve: keep 1 = a(1) and then
1st step: take 2 = a(2) and cross off 5, 8, 11, 14, 17, 20, 23, 26, etc.
2nd step: take 3 = a(3) and cross off 7, 11, 15, 19, 23, 27, etc.
3rd step: take 4 = a(4) and cross off 9, 14, 19, 24, etc.
4th step: take 6 = a(5) and cross off 13, 19, 25 etc.
10 is obtained at next step and the smallest crossed off numbers are then 21 and 28. This gives the beginning of the sequence up to 22 = a(10): 1, 2, 3, 4, 6, 10, 12, 16, 18, 22. - _Jean-Christophe Hervé_, Dec 12 2015

Jean-Christophe Hervé, Table of n, a(n) for n = 1..10000
David J. Hemmer and Karlee J. Westrem, Palindrome Partitions and the Calkin-Wilf Tree, arXiv:2402.02250 [math.CO], 2024. See Remark 3.3, p. 6.
Karlee J. Westrem, Schaper numbers, palindrome partitions, and symmetric functions, with applications to characters of the symmetric group, Ph. D. Dissertation, Michigan Tech. Univ. (2025). See p. 49.
Index entries for sequences generated by sieves

Cf. A000040, A039915, A062569, A166460 (almost complement).

Select[Range[300],Mod[#!,#+1]!=0&] (* Harvey P. Dale, Apr 11 2012 *)

{plnt=1 ; nfa=1; mxind=60 ;  for(k=1, 10^7, nfa=nfa*k;
if(nfa % (k+1) != 0 , print1(k, ", "); plnt++ ;
if(mxind <  plnt, break() )))} \\ Douglas Latimer, Apr 25 2012

a(n)=if(n<5,n,prime(n-1)-1) \\ Charles R Greathouse IV, Apr 25 2012

from sympy import prime
def A068499(n): return prime(n-1)-1 if n>3 else n # Chai Wah Wu, Aug 27 2024

For n >= 4, a(n) = prime(n-1) - 1 = A006093(n-1).
For n <> 3, all terms are one less prime. - Zak Seidov, Aug 22 2012
a(n) = Integer part of A078456(n+1)/A078456(n). - Eric Desbiaux, May 07 2013

A070918 Triangle of T(n,k) coefficients of polynomials with first n prime numbers as roots.

Original entry on oeis.org

1, -2, 1, 6, -5, 1, -30, 31, -10, 1, 210, -247, 101, -17, 1, -2310, 2927, -1358, 288, -28, 1, 30030, -40361, 20581, -5102, 652, -41, 1, -510510, 716167, -390238, 107315, -16186, 1349, -58, 1, 9699690, -14117683, 8130689, -2429223, 414849, -41817, 2451, -77, 1

Offset: 0

Rick L. Shepherd, May 20 2002

Analog of the Stirling numbers of the first kind (A008275): The Stirling numbers (beginning with the 2nd row) are the coefficients of the polynomials having exactly the first n natural numbers as roots. This sequence (beginning with first row) consists of the coefficients of the polynomials having exactly the first n prime numbers as roots.

			Row 4 of this sequence is 210, -247, 101, -17, 1 because (x-prime(1))(x-prime(2))(x-prime(3))(x-prime(4)) = (x-2)(x-3)(x-5)(x-7) = x^4 - 17*x^3 + 101*x^2 - 247*x + 210.
Triangle begins:
        1;
       -2,      1;
        6,     -5,       1;
      -30,     31,     -10,      1;
      210,   -247,     101,    -17,      1;
    -2310,   2927,   -1358,    288,    -28,    1;
    30030, -40361,   20581,  -5102,    652,  -41,   1;
  -510510, 716167, -390238, 107315, -16186, 1349, -58, 1;
  ...

Alois P. Heinz, Rows n = 0..140, flattened

Cf. A008275 (Stirling numbers of first kind).
Cf. A005867 (absolute values of row sums).
Cf. A054640 (sum of absolute values of terms in rows).
Cf. A000040, A078456.

T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(mul(x-ithprime(i), i=1..n)):
seq(T(n), n=0..10);  # Alois P. Heinz, Aug 18 2019

Table[CoefficientList[Expand[Times@@(x-Prime[Range[n]])],x],{n,0,10}]// Flatten (* Harvey P. Dale, Feb 12 2020 *)

p=1; for(k=1,10,p=p*(x-prime(k)); for(n=0,k,print1(polcoeff(p,n),",")))

From Alois P. Heinz, Aug 18 2019: (Start)
T(n,k) = [x^k] Product_{i=1..n} (x-prime(i)).
Sum_{k=0..n} |T(n,k)| = A054640(n).
|Sum_{k=0..n} T(n,k)| = A005867(n).
|Sum_{k=0..n} k * T(n,k)| = A078456(n). (End)

First term T(0,0)=1 prepended by Alois P. Heinz, Aug 18 2019

A096294 Triangle T(n,k) read by rows: for n >=0 and n >= k >=0, the fraction of positive integers with exactly k of the first n primes as divisors is T(n,k)/A002110(n).

Original entry on oeis.org

1, 1, 1, 2, 3, 1, 8, 14, 7, 1, 48, 92, 56, 13, 1, 480, 968, 652, 186, 23, 1, 5760, 12096, 8792, 2884, 462, 35, 1, 92160, 199296, 152768, 54936, 10276, 1022, 51, 1, 1658880, 3679488, 2949120, 1141616, 239904, 28672, 1940, 69, 1, 36495360, 82607616

Offset: 0

Matthew Vandermast, Jun 24 2004

Sum of entries in n-th row is A002110(n), the product of the first n primes (primorial numbers, first definition).
From Peter Munn, Apr 10 2017: (Start)
T(n,k) is a count of those integers in any interval of A002110(n) integers that have exactly k of the first n primes as divisors. The count is the same for each such interval because each of the first n primes is a factor of an integer m if and only if it is a factor of m + A002110(n).
A284411(m) is least p=prime(n) such that 2*Sum_{k=0..m-1} T(n,k) < A002110(n).
(End)

			Triangle begins:
1
1 1
2 3 1
8 14 7 1
48 92 56 13 1
480 968 652 186 23 1

First column is A005867; second column is A078456. See also A096180.

Cf. A194156, A284411.

primo(n) = prod(k=1, n, prime(k));
row(n) = {v = vector(n+1); for (k=1, primo(n), f = factor(k)[,1]; v[1+sum(j=1, #f, primepi(f[j])<=n)]++;); v;} \\ Michel Marcus, Apr 29 2017

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A135212 a(n) = A078456(n)/A120271(n).

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A120271 a(n) = numerator(Sum_{k=1..n} 1/(prime(k)-1)).

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A068499 Numbers m such that m! reduced modulo (m+1) is not zero.

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A070918 Triangle of T(n,k) coefficients of polynomials with first n prime numbers as roots.

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A096294 Triangle T(n,k) read by rows: for n >=0 and n >= k >=0, the fraction of positive integers with exactly k of the first n primes as divisors is T(n,k)/A002110(n).

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