A007775 -id:A007775 - cp's OEIS frontend

A226120 Decimal expansion of Sum_{n>=1} n^3/(exp(2Pin/7)-1).

1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 9, 0, 1, 6, 1, 7, 6, 7, 8, 8, 8, 6, 6, 2, 6, 7, 5, 5, 8, 4, 3, 5, 9, 3, 0, 5, 8, 5, 5, 4, 4, 5, 3, 3, 3, 4, 8, 0, 2, 5, 4, 8, 9, 7, 8, 4, 3, 4, 0, 6, 1, 0, 9, 9, 4, 3, 8, 7, 3, 7, 8, 5, 0, 6, 7, 1, 4, 8, 0, 1, 7, 9, 1, 6, 2, 7, 1, 3, 6, 6, 2, 1

Offset: 2

Jean-François Alcover, May 27 2013

An almost-integer discovered by Simon Plouffe. The computed sum equals 10 within 15 digits.

			10.00000000000000019016176788866267558435930585544533348025489784340610994387...

Simon Plouffe, Simon Plouffe Home Page
Eric Weisstein's MathWorld, Almost Integer

Cf. A060295 (famous almost-integer: Ramanujan's constant), A226121 (another surprising almost-integer by Simon Plouffe), A007775, A089034.

NSum[n^3/(Exp[2*Pi*n/7] - 1), {n, 1, Infinity}, NSumTerms -> 220,    WorkingPrecision -> 100] // RealDigits[#, 10, 100] & // First

Offset corrected by Rick L. Shepherd, Jan 01 2014

A227863 Numbers congruent to {1,49} mod 120.

Original entry on oeis.org

1, 49, 121, 169, 241, 289, 361, 409, 481, 529, 601, 649, 721, 769, 841, 889, 961, 1009, 1081, 1129, 1201, 1249, 1321, 1369, 1441, 1489, 1561, 1609, 1681, 1729, 1801, 1849, 1921, 1969, 2041, 2089, 2161, 2209, 2281, 2329, 2401, 2449, 2521, 2569, 2641, 2689

Offset: 1

Gary Croft, Nov 01 2013

The squares of natural numbers not divisible by 2, 3 or 5 and therefore the squares of prime numbers >5 are confined to this sequence.

Bruno Berselli, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Cf. A000040, A001248, A007775.

Table[60 n - 6 (-1)^n - 65, {n, 50}] (* Bruno Berselli, Nov 04 2013 *)
LinearRecurrence[{1,1,-1},{1,49,121},50] (* or *) #+{1,49}&/@(120*Range[0,30])//Flatten (* Harvey P. Dale, Jul 13 2025 *)

a(n)=n\2*120+[-71,1][n%2+1] \\ Charles R Greathouse IV, Aug 26 2014

G.f.: x*(1 + 48*x + 71*x^2)/((1 + x)*(1 - x)^2). [Bruno Berselli, Nov 04 2013]
a(n) = 60*n - 6*(-1)^n - 65. [Bruno Berselli, Nov 04 2013]
E.g.f.: 71 + (60*x - 65)*exp(x) - 6*exp(-x). - David Lovler, Sep 10 2022

A274172 Nonsquare composites with all prime factors larger than 5.

Original entry on oeis.org

77, 91, 119, 133, 143, 161, 187, 203, 209, 217, 221, 247, 253, 259, 287, 299, 301, 319, 323, 329, 341, 343, 371, 377, 391, 403, 407, 413, 427, 437, 451, 469, 473, 481, 493, 497, 511, 517, 527, 533, 539, 551, 553, 559, 581, 583, 589

Offset: 1

Dimitris Valianatos, Jun 12 2016

nonn

Nonsquare composites not divisible by 2,3,5.

			377 = 13*29 is a term.

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

Intersection of A007775 and A089229. - Felix Fröhlich, Jun 12 2016

filter:= n -> igcd(n,30)=1 and not issqr(n) and not isprime(n):
select(filter, [seq(i,i=3..1000,2)]); # Robert Israel, May 30 2021

{ for(n=1,600, if(!(isprime(n) || n%2==0 || n%3==0 || n%5==0 || issquare(n)), print1(n", ")))}

is(n) = my(f=factor(n)); if(!issquare(n),f[1,1]>5 && matsize(f)[1]>1,0) \\ David A. Corneth, Jun 12 2016

A343357 7-rough abundant numbers.

Original entry on oeis.org

20169691981106018776756331, 21373852696395930345517903, 21975933054040886129898689, 23476198863254546445077041, 23782174126975753483041047, 23836908704943476736166573, 24137500239684251978741183, 24272002214551310731350839, 24955720586792192723783257, 24986334842265665051802619

Offset: 1

David A. Corneth, Apr 12 2021

nonn

Each term has at least A001276(4) = 15 distinct prime factors and A108227(4) = 18 prime factors counted with multiplicity. - Jianing Song, Apr 13 2021

The smallest term with exactly 15 distinct prime factors is a(830) = 465709156638373299218537971 = 7^3 * 11^2 * 13^2 * 17^2 * 19 * 23 * ... * 61. - Jianing Song, Apr 14 2021

			k = 20169691981106018776756331 is in the sequence as its smallest prime factor is at least 7 and it is abundant as sigma(k) > 2*k.

David A. Corneth, Table of n, a(n) for n = 1..2187

Cf. A005101, A007775, A047802, A115414.

is(n) = gcd(n, 30) == 1 && sigma(n) > 2*n

A343737 Highly composite 7-rough numbers: numbers that are not divisible by any prime smaller than 7 and whose number of divisors reaches a record.

Original entry on oeis.org

1, 7, 49, 77, 539, 1001, 5929, 7007, 17017, 77077, 119119, 323323, 1310309, 2263261, 7436429, 24895871, 52055003, 215656441, 572605033, 1509595087, 6685349671, 16605545957, 46797447697, 215872097441, 247357937827, 514771924667, 1731505564789, 6692035020671

Offset: 1

Jon E. Schoenfield, Jun 27 2021

nonn

			   n   a(n)   prime factorization     number of divisors
  --  ------  ----------------------  ------------------
   1       1  -                                1
   2       7  7                                2
   3      49  7^2                              3
   4      77  7   * 11                         4
   5     539  7^2 * 11                         6
   6    1001  7   * 11   * 13                  8
   7    5929  7^2 * 11^2                       9
   8    7007  7^2 * 11   * 13                 12
   9   17017  7   * 11   * 13   * 17          16
  10   77077  7^2 * 11^2 * 13                 18
  11  119119  7^2 * 11   * 13   * 17          24

Amiram Eldar, Table of n, a(n) for n = 1..150

Cf. A000005, A002182, A007775, A053624, A343734.

A376318 The number of distinct values of x+y+z (mod n) when xyz = 1 (mod n).

Original entry on oeis.org

1, 1, 2, 1, 4, 2, 7, 2, 4, 4, 11, 2, 13, 7, 8, 3, 17, 4, 19, 4, 14, 11, 23, 4, 20, 13, 11, 7, 29, 8, 31, 6, 22, 17, 28, 4, 37, 19, 26, 8, 41, 14, 43, 11, 16, 23, 47, 6, 49, 20, 34, 13, 53, 11, 44, 14, 38, 29, 59, 8, 61, 31, 28, 11, 52, 22, 67, 17, 46, 28, 71, 8, 73, 37, 40, 19, 77, 26

Offset: 1

W. Edwin Clark, Sep 22 2024

nonn

The values of n for which a(n) = n seem to be A007775, but I have no proof of this.

Charles R Greathouse IV, Table of n, a(n) for n = 1..30000
Charles R Greathouse IV, Program for computing terms (C with PARI)

Cf. A007775, A376296, A376183, A180783, A376427.

a:=proc(n)
local x,y,z,N;
N:=NULL;
for x from 0 to n-1 do
 for y from x to n-1 do
  for z from y to n-1 do
   if (x*y*z) mod n = 1 mod n then N:=N,(x+y+z) mod n; fi;
  od:
 od:
od:
nops({N});
end:

a(n)=my(v=vectorsmall(n)); for(x=1,n, if(gcd(x,n)>1, next); for(y=1,x, if(gcd(y,n)>1, next); my(z=1/Mod(x*y,n)); v[lift(x+y+z)+1]=1)); sum(i=1,n, v[i]) \\ Charles R Greathouse IV, Sep 23 2024

def A376318(n):
    s = set()
    for x in range(n):
        for y in range(x,n):
            try:
                s.add((x+y+pow(x*y%n,-1,n))%n)
            except:
                continue
    return len(s) # Chai Wah Wu, Sep 23 2024

A230113 Digital root of summed Fibonacci and Lucas digital roots indexed by numbers not divisible by 2, 3 or 5.

Original entry on oeis.org

3, 4, 5, 6, 6, 5, 4, 3, 4, 6, 6, 5, 4, 3, 3, 5, 6, 5, 4, 3, 3, 4, 5, 6, 5, 3, 3, 4, 5, 6, 6, 4, 3, 4, 5, 6, 6, 5, 4, 3, 4, 6, 6, 5, 4, 3, 3, 5, 6, 5, 4, 3, 3, 4, 5, 6, 5, 3, 3, 4, 5, 6, 6, 4, 3, 4, 5, 6, 6, 5, 4, 3, 4, 6, 6, 5, 4, 3, 3, 5, 6, 5, 4, 3, 3, 4, 5, 6, 5, 3, 3, 4, 5, 6, 6, 4

Offset: 1

Gary Croft, Dec 20 2013

32-beat repeating sequence is periodically palindromic starting at Length(40), then at Lengths (72)...(104)...(136)...(168)...{+32 terms ... repeat ... n}.

			Referencing A227896 (Fibo) and A233766 (Lucas): 1st Fibo term (1) + 1st Lucas term (2) = 3 = digital root 3. Likewise, 2nd Fibo term (4) + 2nd Lucas term (9) = 13 = digital root 4.

Cf. A007775, A227896, A233766, A000032, A000033.

Conjectures from Colin Barker, Sep 22 2019: (Start)
G.f.: x*(3 + x + x^2 + x^3 - x^5 - x^6 - x^7 + x^8 + 2*x^9 - x^11 - x^12 - x^13 + 2*x^15 + 4*x^16) / ((1 - x)*(1 + x^16)).
a(n) = a(n-1) - a(n-16) + a(n-17) for n>17.
(End)

A246508 Digital root of numbers congruent to {1,7,11,13,17,19,23,29} modulo 30.

Original entry on oeis.org

1, 7, 2, 4, 8, 1, 5, 2, 4, 1, 5, 7, 2, 4, 8, 5, 7, 4, 8, 1, 5, 7, 2, 8, 1, 7, 2, 4, 8, 1, 5, 2, 4, 1, 5, 7, 2, 4, 8, 5, 7, 4, 8, 1, 5, 7, 2, 8, 1, 7, 2, 4, 8, 1, 5, 2, 4, 1, 5, 7, 2, 4, 8, 5, 7, 4, 8, 1, 5, 7, 2, 8, 1, 7, 2, 4, 8, 1, 5, 2, 4, 1, 5, 7, 2, 4, 8, 5, 7, 4

Offset: 1

Gary Croft, Nov 14 2014

Period 24 repeating sequence, the digital root squares of which produce period 24 palindromic sequence A240924.

Index entries for linear recurrences with constant coefficients, signature (0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1).

Cf. A007775 (numbers not divisible by 2, 3 or 5), A240924 (digital root of this sequence squared).

a(n) = A010888(A007775(n)). - Michel Marcus, Nov 25 2014

G.f.: ( -x*(1 +7*x +2*x^2 +4*x^3 +8*x^4 +x^5 +5*x^6 +2*x^7 +4*x^8 +x^9 +5*x^10 +7*x^11 +2*x^12 +4*x^13 +8*x^14 +5*x^15 +7*x^16 +4*x^17 +8*x^18 +x^19 +5*x^20 +7*x^21 +2*x^22 +8*x^23) ) / ( (x-1) *(1+x+x^2) *(1+x) *(1-x+x^2) *(1+x^2) *(x^4-x^2+1) *(1+x^4) *(x^8-x^4+1) ). - R. J. Mathar, Sep 22 2016

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

A275591 a(n) = n^2 + 9*n + 1.

Original entry on oeis.org

1, 11, 23, 37, 53, 71, 91, 113, 137, 163, 191, 221, 253, 287, 323, 361, 401, 443, 487, 533, 581, 631, 683, 737, 793, 851, 911, 973, 1037, 1103, 1171, 1241, 1313, 1387, 1463, 1541, 1621, 1703, 1787, 1873, 1961, 2051, 2143, 2237, 2333, 2431, 2531, 2633, 2737

Offset: 0

Miquel Cerda, Aug 02 2016

Also, nonnegative integers m such that 4*m + 77 is a square. The negative values of m are -7, -13, -17, -19.

The product of two consecutive terms belongs to the sequence. In fact: a(k)*a(k+1) = a(k*(k+1)+9*k+1).

Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A028569.

Subsequence of A007775.

Table[n^2 + 9 n + 1, {n, 0, 50}] (* Bruno Berselli, Aug 05 2016 *)

a(n) = n^2 + 9*n + 1 \\ Charles R Greathouse IV, Aug 03 2016

Vec((1+8*x-7*x^2)/(1-x)^3 + O(x^100)) \\ Colin Barker, Aug 04 2016

O.g.f.: (1 + 8*x - 7*x^2)/(1 - x)^3. - Colin Barker, Aug 03 2016
E.g.f.: (1 + 10*x + x^2)*exp(x).
a(n) = a(-n-9) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Colin Barker, Aug 03 2016
a(n) = A048058(n-1) + A008592(n-1) for n>0.
a(n) = 1 + A028569(n). - Omar E. Pol, Aug 02 2016
a(n) + a(-n) = (n-1)^2 + (n+1)^2.
Sum_{i>=0} 1/a(i) = 9736/29393 + tan(sqrt(77)*Pi/2)*Pi/sqrt(77) = 1.301517...

Edited and extended by Bruno Berselli, Aug 05 2016

cp's OEIS Frontend

A226120 Decimal expansion of Sum_{n>=1} n^3/(exp(2*Pi*n/7)-1).

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A227863 Numbers congruent to {1,49} mod 120.

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A274172 Nonsquare composites with all prime factors larger than 5.

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PARI

A343357 7-rough abundant numbers.

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PARI

A343737 Highly composite 7-rough numbers: numbers that are not divisible by any prime smaller than 7 and whose number of divisors reaches a record.

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A376318 The number of distinct values of x+y+z (mod n) when x*y*z = 1 (mod n).

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Maple

PARI

Python

A230113 Digital root of summed Fibonacci and Lucas digital roots indexed by numbers not divisible by 2, 3 or 5.

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A246508 Digital root of numbers congruent to {1,7,11,13,17,19,23,29} modulo 30.

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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.

A226120 Decimal expansion of Sum_{n>=1} n^3/(exp(2Pin/7)-1).

A376318 The number of distinct values of x+y+z (mod n) when xyz = 1 (mod n).