A008365 -id:A008365 - cp's OEIS frontend

A038512 Nonprime numbers with smallest prime factor >= 13.

169, 221, 247, 289, 299, 323, 361, 377, 391, 403, 437, 481, 493, 527, 529, 533, 551, 559, 589, 611, 629, 667, 689, 697, 703, 713, 731, 767, 779, 793, 799, 817, 841, 851, 871, 893, 899, 901, 923, 943, 949, 961, 989, 1003, 1007, 1027, 1037, 1073, 1079

Offset: 1

Jeff Burch

nonn

Composite numbers k such that k^120 mod 2310 = 1. - Gary Detlefs, May 02 2012

The asymptotic density of this sequence is 16/77. - Amiram Eldar, Mar 22 2021

T. D. Noe, Table of n, a(n) for n = 1..10000

Intersection of A002808 and A008365.

for n from 1 to 1079 do if n^120 mod 2310=1 then print(n) fi od; # Gary Detlefs, May 02 2012

Select[Range[1000], ! PrimeQ[#] && FactorInteger[#][[1, 1]] >= 13 &] (* T. D. Noe, Mar 16 2013 *)

913=11*83 deleted by Jean-Marc Rebert, Jul 08 2009
923=13*71 added by Howard Berman (howard_berman(AT)hotmail.com), Jan 22 2010
1 removed by T. D. Noe, Mar 16 2013

A246541 Take the squares of all P_(n+2)-rough numbers less than the (n+1)-th primorial and mod each by the (n+1)-th primorial. There will be a(n) different results.

Original entry on oeis.org

1, 2, 6, 30, 180, 1440, 12960, 142560, 1995840, 29937600, 538876800, 10777536000, 226328256000, 5205549888000, 135344297088000, 3924984615552000, 117749538466560000, 3885734769396480000, 136000716928876800000, 4896025809439564800000, 190945006568143027200000

Offset: 1

John B. Yin, Aug 29 2014

nonn

The P_(n+2)-rough numbers less than the (n+1)-th primorial also comprise the reduced residue system of the (n+1)-th primorial.

The conjectured formula from Jon E. Schoenfield is true. This can be seen by considering that each odd prime p has exactly (p+1)/2 quadratic residues (mod p), of which (p-1)/2 are nonzero. The P_(n+2)-rough numbers less than the (n+1)-th primorial comprise all combinations of nonzero residues modulo the first n+1 primes. So for each odd prime p, the p-1 nonzero residues map to (p-1)/2 (nonzero) residues after squaring. - Bert Dobbelaere, Aug 09 2023

			For n=2, P_(n+2) = 7.
The 7-rough numbers less than 2*3*5 are 1,7,11,13,17,19,23,29.
The squares of those numbers mod 2*3*5 are 1,19,1,19,19,1,19,1.
There are 2 different results: 1 and 19; so a(2) = 2.

Eric Weisstein's World of Mathematics, Primorial
Eric Weisstein's World of Mathematics, Rough Number

Cf. A002110 (primorial).
Cf. k-rough numbers A007310 (k=5), A007775 (k=7), A008364 (k=11), A008365 (k=13), A008366 (k=17), A166061 (k=19), A166063 (k=23).
Cf. A323739.

import java.util.TreeSet;
for(int z = 1; z < 10 ; z++) {
int n = z;
int numNumPerLine = 210;
int[] primes = {2,3,5,7,11,13,17,19,23,29,31,37,41,43};
int numRepeats = 1;
int numSpaces = 1;
for(int i = 0; i < n + 1; i++) {
numSpaces *= (primes[i] - 1);
}
int counter = 0;
long integerLength = 1;
for(int i = 0; i < n + 1; i++) {
integerLength *= primes[i];
}
TreeSet numResults = new TreeSet();
numSpaces/=2;
for(int i = 1; i < integerLength / 2; i+=2) {
boolean isInList = true;
for(int j = 1; j < n + 1; j++) {
if(i % primes[j] == 0) {
isInList = false;
}
}
if(isInList) {
long k = i % integerLength;
if(k != 0) {
long l = (k * k) % integerLength;
if(!numResults.contains(l)) {
numResults.add(l);
}
}
}
}
System.out.println(numResults.size());
}

a(n) = {hp = prod(k=1, n+1, prime(k)); rp = prod(k=1, n+2, prime(k)); v = []; for (i=1, hp, if (gcd(i, rp) == 1, nv = i^2 % hp; if (! vecsearch(v, nv), v = vecsort(concat(v, nv))););); #v;} \\ Michel Marcus, Sep 06 2014

Conjecture: a(n) = (1/2^n)*Product_{j=1..n} (prime(j+1)-1) = A005867(n+1)/2^n. - Jon E. Schoenfield, Feb 20 2019

a(n) = A323739(n+1). - Bert Dobbelaere, Aug 09 2023

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A038512 Nonprime numbers with smallest prime factor >= 13.

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A246541 Take the squares of all P_(n+2)-rough numbers less than the (n+1)-th primorial and mod each by the (n+1)-th primorial. There will be a(n) different results.

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