A007775 -id:A007775 - cp's OEIS frontend

A141256 An Okazaki-like composition, see A126759.

1, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 4, 2, 5, 3, 2, 2, 6, 2, 7, 2, 3, 4, 8, 2, 2, 5, 2, 3, 9, 2, 10, 2, 4, 6, 3, 2, 11, 7, 5, 2, 12, 3, 13, 4, 2, 8, 14, 2, 15, 2, 6, 5, 16, 2, 4, 3, 7, 9, 17, 2, 18, 10, 3, 2, 5, 4, 19, 6, 8, 3, 20, 2, 21, 11, 2, 7, 22, 5, 23, 2, 2, 12, 24, 3, 6, 13, 9, 4, 25, 2, 26, 8, 10, 14

Offset: 0

Reinhard Zumkeller, Jun 17 2008

nonn

a(5*n) = a(3*n) = a(2*n) = a(n);
for n with GCD(n,30)=1: a(n+30*k)=a(n)+8*k, note: 30=2*3*5,
A000010(30)=8;
for k>1: a(A007775(k-1))=k and a(m)A007775(k-1).

R. Zumkeller, Table of n, a(n) for n = 0..10000

a(n) = if n=0 then 1 else if GCD(n,30)>1 then a(LPD(n)) else 2*floor(n/30) + (if n mod 30 = 1 then 2 else IP(n)-1), with IP=A049084, LPD=A032742 and GCD=A050873.

A206546 Period 8: repeat [1, 7, 11, 13, 13, 11, 7, 1].

Original entry on oeis.org

1, 7, 11, 13, 13, 11, 7, 1, 1, 7, 11, 13, 13, 11, 7, 1, 1, 7, 11, 13, 13, 11, 7, 1, 1, 7, 11, 13, 13, 11, 7, 1, 1, 7, 11, 13, 13, 11, 7, 1, 1, 7, 11, 13, 13, 11, 7, 1, 1, 7, 11, 13, 13, 11, 7, 1, 1, 7, 11, 13, 13, 11, 7, 1, 1, 7, 11, 13, 13, 11, 7, 1, 1, 7, 11, 13, 13, 11, 7, 1, 1, 7, 11, 13, 13, 11, 7, 1, 1, 7, 11, 13, 13, 11, 7, 1

Offset: 1

Wolfdieter Lang, Feb 10 2012

For general Modd n (not to be confused with mod n) see a comment on A203571. The present sequence gives the residues Modd 15 of the positive odd numbers relatively prime to 15 (the positive odd numbers from all reduced residue classes mod 15), shown in A007775. The underlying periodic sequence with period length 30 is [0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,0,14,13,12,11,10,9,8,7,6,5,4,3,2,1], called, with offset 0, P_15 or Modd15.

			Residues Modd 15 of the positive odd numbers relatively prime to 15:
A007775: 1, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 49, ...
Modd 15: 1, 7, 11, 13, 13, 11,  7,  1,  1,  7, 11, 13, 13, 11, ...

Antti Karttunen, Table of n, a(n) for n = 1..8192
Index entries for linear recurrences with constant coefficients, signature (1,-1,1,-1,1,-1,1).

Cf. A007775, A010877.

Cf. A206545 and further crossrefs given there.

PadRight[{},100,{1,7,11,13,13,11,7,1}] (* Harvey P. Dale, Sep 30 2015 *)

a(n)=[1, 7, 11, 13, 13, 11, 7, 1][n%8+1] \\ Charles R Greathouse IV, Jul 17 2016

(define (A206546 n) (list-ref '(1 7 11 13 13 11 7 1) (modulo (- n 1) 8))) ;; Antti Karttunen, Aug 10 2017

a(n) = A007775(n) (Modd 15) := Modd15(A007775(n)), n>=1, with the periodic sequence Modd15 (period length 30) given in the comment section.
O.g.f: x*(1+x^7+7*x*(1+x^5)+11*x^2*(1+x^3)+13*x^3*(1+x))/(1-x^8) = x*(1+x)*(1+6*x+5*x^2+8*x^3+5*x^4+6*x^5+x^6)/(1-x^8).
a(n) = -k^2 + 7k + 1 where k = (n-1) mod 8. - David A. Corneth, Aug 13 2017

A230462 Numbers congruent to {1, 11, 13, 17, 19, or 29} mod 30.

Original entry on oeis.org

1, 11, 13, 17, 19, 29, 31, 41, 43, 47, 49, 59, 61, 71, 73, 77, 79, 89, 91, 101, 103, 107, 109, 119, 121, 131, 133, 137, 139, 149, 151, 161, 163, 167, 169, 179, 181, 191, 193, 197, 199, 209, 211, 221, 223, 227, 229, 239, 241, 251, 253, 257, 259, 269

Offset: 1

Gary Croft, Oct 20 2013

Reduces sieving for all twin primes (A001097) except (3,5) and (5,7) to 6/30 or 20% of natural numbers.
This is subset of natural numbers not divisible by 2, 3 or 5 (A007775).
Also A128464(n) and A128464(n)+2 interleaved, with a(n) = 1. - Peter Bala, Oct 28 2013
a(2)..a(10) form a block of 9 primes {11, 13, 17, 19, 29, 31, 41, 43, 47}. Up to 3*10^10 there is only one such block which includes 11 primes: {18873497, 18873499, 18873509, 18873511, 18873521, 18873523, 18873527, 18873529, 18873539, 18873541, 18873551}. Do larger such blocks exist? (None found up to 10^11.) - Mikk Heidemaa, Dec 22 2017

Iain Fox, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,1,-1).

Cf. A001097, A007775, A128464, A132247.

[n : n in [0..400] | n mod 30 in [1, 11, 13, 17, 19, 29]]; // Wesley Ivan Hurt, Jul 22 2016

A230462:=n->30*floor(n/6)+[1, 11, 13, 17, 19, 29][(n mod 6)+1]: seq(A230462(n), n=0..100); # Wesley Ivan Hurt, Jul 22 2016

LinearRecurrence[{1,0,0,0,0,1,-1},{1,11,13,17,19,29,31},60] (* Harvey P. Dale, Dec 01 2015 *)
ParallelCombine[Select[#, MemberQ[{1, 11, 13, 17, 19, 29}, Mod[#, 30]] &] &, Range[10^4]] (* Mikk Heidemaa, Dec 12 2017 *)
CoefficientList[ Series[(1 + 10x + 2x^2 + 4x^3 + 2x^4 + 10x^5 + x^6)/((-1 + x)^2 (1 + x + x^2 + x^3 + x^4 + x^5)), {x, 0, 60}], x] (* Robert G. Wilson v, Jan 09 2018 *)

a(n)=n\6*30+[-1,1,11,13,17,19][n%6+1] \\ Charles R Greathouse IV, Oct 29 2013

first(n) = Vec(x*(1 + 10*x + 2*x^2 + 4*x^3 + 2*x^4 + 10*x^5 + x^6)/((1 + x)*(1 + x + x^2)*(x^2 - x + 1)*(x - 1)^2) + O(x^(n+1))) \\ Iain Fox, Dec 29 2017

G.f.: x*(1+10*x+2*x^2+4*x^3+2*x^4+10*x^5+x^6) / ( (1+x)*(1+x+x^2)*(x^2-x+1)*(x-1)^2 ). - R. J. Mathar, Jul 07 2015
From Wesley Ivan Hurt, Jul 22 2016: (Start)
a(n) = a(n-1) + a(n-6) - a(n-7) for n>7; a(n) = a(n-6) + 30 for n>6.
a(n) = (30*n - 15 - 6*cos(n*Pi/3) + 6*cos(2*n*Pi/3) + 9*cos(n*Pi) + 6*sqrt(3)*sin(n*Pi/3) - 2*sqrt(3)*sin(2*n*Pi/3))/6.
a(6k) = 30k-1, a(6k-1) = 30k-11, a(6k-2) = 30k-13, a(6k-3) = 30k-17, a(6k-4) = 30k-19, a(6k-5) = 30k-29. (End)
a(n) = 5*n + ceiling(7/79 - ((((14654/4883)^n mod 6) mod 5) + n mod 3 + 1) mod 7). - Mikk Heidemaa, Dec 13 2017
a(n + 6) = a(n) + 30. - David A. Corneth, Jan 15 2018
Sum_{n>=1} (-1)^(n+1)/a(n) = (cot(Pi/30) - sec(2*Pi/15) * tan(Pi/15)) * Pi/30. - Amiram Eldar, Jul 29 2024

New name and initial term from Omar E. Pol, Oct 27 2013

A236208 Numbers not divisible by 2, 5 or 11.

Original entry on oeis.org

1, 3, 7, 9, 13, 17, 19, 21, 23, 27, 29, 31, 37, 39, 41, 43, 47, 49, 51, 53, 57, 59, 61, 63, 67, 69, 71, 73, 79, 81, 83, 87, 89, 91, 93, 97, 101, 103, 107, 109, 111, 113, 117, 119, 123, 127, 129, 131, 133, 137, 139, 141, 147, 149, 151, 153, 157, 159, 161, 163

Offset: 1

Oleg P. Kirillov, Jan 20 2014

Numbers coprime to 110. The asymptotic density of this sequence is 4/11. - Amiram Eldar, Oct 23 2020

Alois P. Heinz, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (2, -1, 0, 0, -1, 2, -1, 0, 0, -1, 2, -1, 0, 0, -1, 2, -1, 0, 0, -1, 2, -1, 0, 0, -1, 2, -1, 0, 0, -1, 2, -1, 0, 0, -1, 2, -1).

Cf. A007775, A235583, A236204.

Select[Range[200], CoprimeQ[110, #] &] (* Amiram Eldar, Oct 23 2020 *)

G.f.: (x^36 +x^35 +2*x^34 -2*x^33 +2*x^32 +x^31 -x^30 +2*x^29 -2*x^28 +4*x^27 -x^26 -x^25 +6*x^24 -6*x^23 +4*x^22 -x^21 +x^20 +4*x^19 -6*x^18 +4*x^17 +x^16 -x^15 +4*x^14 -6*x^13 +6*x^12 -x^11 -x^10 +4*x^9 -2*x^8 +2*x^7 -x^6 +x^5 +2*x^4 -2*x^3 +2*x^2 +x +1)*x / ((x+1) *(x^2+1) *(x^4-x^3+x^2-x+1) *(x^4+1) *(x^8-x^6+x^4-x^2+1) *(x^16-x^12+x^8-x^4+1) *(x-1)^2). - Alois P. Heinz, Feb 13 2014

A240924 Digital root of squares of numbers not divisible by 2, 3 or 5.

Original entry on oeis.org

1, 4, 4, 7, 1, 1, 7, 4, 7, 1, 7, 4, 4, 7, 1, 7, 4, 7, 1, 1, 7, 4, 4, 1, 1, 4, 4, 7, 1, 1, 7, 4, 7, 1, 7, 4, 4, 7, 1, 7, 4, 7, 1, 1, 7, 4, 4, 1, 1, 4, 4, 7, 1, 1, 7, 4, 7, 1, 7, 4, 4, 7, 1, 7, 4, 7, 1, 1, 7, 4, 4, 1

Offset: 1

Gary Croft, Aug 15 2014

This period 24 repeating sequence is palindromic.

			The first 8 numbers not divisible by 2, 3 or 5 are 1,7,11,13,17,19,23,29; with squares 1,49,121,169,289,361,529,841 and digital root sequence of 1,4,4,7,1,1,7,4.

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (2,-1,-1,2,-1,-1,2,-2,1,1,-2,1,1,-2,1).

Cf. A007775, A166923.

Vec(x*(1 + x)^2*(1 - 4*x^2 + 12*x^3 - 27*x^4 + 45*x^5 - 53*x^6 + 45*x^7 - 27*x^8 + 12*x^9 - 4*x^10 + x^12) / ((1 - x)*(1 - x + x^2)*(1 - x^2 + x^4)*(1 - x^4 + x^8)) + O(x^100)) \\ Colin Barker, Sep 21 2019

A240924 = [1 + (n*n-1) % 9 for n in range(1,10**3,2) if n % 3 and n % 5 ]
# Chai Wah Wu, Sep 03 2014

From Colin Barker, Sep 21 2019: (Start)
G.f.: x*(1 + x)^2*(1 - 4*x^2 + 12*x^3 - 27*x^4 + 45*x^5 - 53*x^6 + 45*x^7 - 27*x^8 + 12*x^9 - 4*x^10 + x^12) / ((1 - x)*(1 - x + x^2)*(1 - x^2 + x^4)*(1 - x^4 + x^8)).
a(n) = 2*a(n-1) - a(n-2) - a(n-3) + 2*a(n-4) - a(n-5) - a(n-6) + 2*a(n-7) - 2*a(n-8) + a(n-9) + a(n-10) - 2*a(n-11) + a(n-12) + a(n-13) - 2*a(n-14) + a(n-15) for n>15.
(End)

A330137 Numbers m such that 1 < gcd(m, 30) < m and m does not divide 30^e for e >= 0.

Original entry on oeis.org

14, 21, 22, 26, 28, 33, 34, 35, 38, 39, 42, 44, 46, 51, 52, 55, 56, 57, 58, 62, 63, 65, 66, 68, 69, 70, 74, 76, 78, 82, 84, 85, 86, 87, 88, 92, 93, 94, 95, 98, 99, 102, 104, 105, 106, 110, 111, 112, 114, 115, 116, 117, 118, 122, 123, 124, 126, 129, 130, 132, 134

Offset: 1

Michael De Vlieger, Dec 02 2019

Numbers m that are neither 5-smooth nor reduced residues mod 30. Such numbers m have at least 1 prime factor p <= 5 and at least 1 prime factor q > 5.
Complement of the union of A007775 and A051037.
Analogous to A105115 for A120944(2) = 10. This sequence applies to the second primorial in A120944, i.e., 30 = A002110(2).

			14 is in the sequence since gcd(14, 30) = 2 and 14 does not divide 30^e with integer e >= 0.
15 is not in the sequence since 15 | 30.
16 is not in the sequence since 16 | 30^4.
17 is not in the sequence since 17 is coprime to 30.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A002110, A007775, A051037, A105115, A120944, A306999, A307589, A316991, A316992, A330136.

With[{nn = 135, k = 30}, Select[Range@ nn, And[1 < GCD[#, k] < #, PowerMod[k, Floor@ Log2@ nn, #] != 0] &]]

A374290 7-rough powerful numbers: numbers k coprime to 30 such that if a prime p divides k then p^2 also divides k.

Original entry on oeis.org

1, 49, 121, 169, 289, 343, 361, 529, 841, 961, 1331, 1369, 1681, 1849, 2197, 2209, 2401, 2809, 3481, 3721, 4489, 4913, 5041, 5329, 5929, 6241, 6859, 6889, 7921, 8281, 9409, 10201, 10609, 11449, 11881, 12167, 12769, 14161, 14641, 16129, 16807, 17161, 17689, 18769

Offset: 1

Amiram Eldar, Jul 02 2024

This sequence is closed under multiplication.

The least term that is not a power of a prime (A000961) is a(25) = 7^2*11^2 = 5929.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Rough Number.
Index entries for sequences related to powerful numbers.

Intersection of A007775 and A001694.
Intersection of A229829 and A062739.
Intersection of A047201 and A374289.
Cf. A000961, A082695.

powQ[n_] := n == 1 || AllTrue[FactorInteger[n][[;; , 2]], # > 1 &]; Select[Range[20000], CoprimeQ[#, 30] && powQ[#] &]

is(k) = gcd(k, 30) == 1 && ispowerful(k);

Sum_{n>=1} 1/a(n) = 80*zeta(2)*zeta(3)/(147*zeta(6)) = (80/147) * A082695 = 1.05773955745... .

In general, the sum of reciprocals of the p-rough powerful numbers is (zeta(2)*zeta(3)/zeta(6)) * Product_{prime q < p} ((q-1)*q/(q^2-q+1)).

A070884 7 + x where x is congruent to {0, 4, 6, 10, 12, 16, 22, 24} mod 30.

Original entry on oeis.org

7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 49, 53, 59, 61, 67, 71, 73, 77, 79, 83, 89, 91, 97, 101, 103, 107, 109, 113, 119, 121, 127, 131, 133, 137, 139, 143, 149, 151, 157, 161, 163, 167, 169, 173, 179, 181, 187, 191, 193, 197, 199, 203, 209, 211, 217, 221

Offset: 0

Timothy McAlee Sr., May 24 2002

Sequence contains many primes.

A007775 without the first term. Strictly speaking, the sequence should include the 1, because 1=7-6 and -6 = 24 mod 30. [From R. J. Mathar, Sep 25 2008]

			7+0=7, 7+4=11, 7+6=13, 7+10=17, 7+12=19, 7+16=23, ...

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

$a = 0; while ((($a % 30 == 0 or $a % 30 == 4 or $a % 30 == 6 or $a % 30 == 10 or $a % 30 == 12 or $a % 30 == 16 or $a % 30 == 22 or $a % 30 == 24) and eval("print \"\".(7+\$a).\" \"; return 0;")) or ++$a) { }

G.f.: ( 7+4*x+2*x^2+4*x^3+2*x^4+4*x^5+6*x^6+2*x^7-x^8 ) / ( (1+x)*(x^2+1)*(x^4+1)*(x-1)^2 ). - R. J. Mathar, Sep 22 2016

More terms from Jim McCann (jmccann(AT)umich.edu), Jul 17 2002

A138045 Triangle read by rows: largest proper divisor of n as a table, ones excluded.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 7, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Mats Granvik, Mar 02 2008

The numbers in the triangle form lines that begin at T(A001248,A000040). The first line of numbers from the right, is T(A005843,A000027). The second line is T(A016945,A005408). The third line is T(A084967,A007310).

			The first few terms of the table are:
  0
  0,0
  0,0,0
  0,2,0,0
  0,0,0,0,0
  0,0,3,0,0,0
  0,0,0,0,0,0,0
  0,0,0,4,0,0,0,0
  0,0,3,0,0,0,0,0,0

Antti Karttunen, Table of n, a(n) for n = 1..23220 (the first 215 rows of the triangle).
Eric Weisstein's World of Mathematics, Proper Divisor.

Cf. A001248, A016945, A084967, A007310, A127093, A032742, A007775.

up_to = 23220; \\ binomial(215+1,2)
A032742(n) = if(1==n,n,n/vecmin(factor(n)[,1]));
A138045tr(n, k) = if((k>1) && (A032742(n)==k), k, 0);
A138045list(up_to) = { my(v = vector(up_to), i=0); for(n=1,oo, for(k=1,n, i++; if(i > up_to, return(v)); v[i] = A138045tr(n,k))); (v); };
v138045 = A138045list(up_to);
A138045(n) = v138045[n]; \\ Antti Karttunen, Dec 24 2018

T(n,k) = if k==A032742(n) and n(T(n,k))==n(A032742(n)) and k>1 then k else 0 (1<=k<=n), T(1,1)=0.

A140887 Number of nonprimes in [30n - 30, 30n] coprime to 30.

Original entry on oeis.org

1, 1, 1, 2, 3, 2, 3, 2, 3, 3, 4, 2, 3, 4, 2, 3, 3, 6, 3, 3, 3, 2, 4, 4, 4, 3, 5, 2, 4, 4, 4, 4, 4, 3, 3, 4, 2, 5, 5, 4, 3, 4, 4, 2, 6, 5, 5, 3, 4, 1, 6, 3, 4, 2, 5, 4, 4, 5, 4, 4, 5, 6, 1, 5, 5, 5, 3, 3, 5, 3, 5, 3, 6, 5, 4, 4, 3, 5, 4, 1, 5, 4, 5, 7, 3, 5, 5, 5, 4, 1, 3, 4, 5, 3, 5, 4, 4, 5, 4, 6, 4, 5, 3, 6, 6

Offset: 1

Juri-Stepan Gerasimov, Jul 06 2008

For almost all n, a(n) = phi(30) = 8. It first attains this value at a(361).

			{1,7,11,13,17,19,23,29} are coprime to 30 in [0, 30]; 1 is nonprime, so a(1) = 1.
{31,37,41,43,47,49,53,59} are coprime to 30 in [30, 60]; 49 is nonprime, so a(2) = 1.

Cf. A007775.

Table[Count[Range[30n-30,30n],?(CoprimeQ[#,30]&&!PrimeQ[#]&)],{n,110}] (* _Harvey P. Dale, Oct 04 2012 *)

Rewritten and extended by Charles R Greathouse IV, Mar 26 2010

cp's OEIS Frontend

A141256 An Okazaki-like composition, see A126759.

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A206546 Period 8: repeat [1, 7, 11, 13, 13, 11, 7, 1].

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A230462 Numbers congruent to {1, 11, 13, 17, 19, or 29} mod 30.

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A236208 Numbers not divisible by 2, 5 or 11.

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A240924 Digital root of squares of numbers not divisible by 2, 3 or 5.

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A330137 Numbers m such that 1 < gcd(m, 30) < m and m does not divide 30^e for e >= 0.

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A374290 7-rough powerful numbers: numbers k coprime to 30 such that if a prime p divides k then p^2 also divides k.

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A070884 7 + x where x is congruent to {0, 4, 6, 10, 12, 16, 22, 24} mod 30.