A007775 -id:A007775 - cp's OEIS frontend

A145011 First differences of A007775.

6, 4, 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 2, 4, 6, 2

Offset: 1

Ki Punches, Feb 25 2009

Also the first differences of A084968 divided by 7. - Antti Karttunen, May 01 2015

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

Cf. A007775, A084968.

Multiplied by 7: row 4 of A257251.

a145011 n = a145011_list !! (n-1)
a145011_list = zipWith (-) (tail a007775_list) a007775_list
-- Reinhard Zumkeller, Jan 06 2013

Differences[Select[Range[400],GCD[#,30]==1&]] (* Harvey P. Dale, Dec 07 2011 *)

a(n)=[4,6,4,2,4,2][n%8+1] \\ Charles R Greathouse IV, Oct 20 2013

Period 8: repeat 6,4,2,4,2,4,6,2.
a(n) = 2*((abs(abs((n mod 8) - 3) - 1) mod 3) + 1). - Pieter Stadhouders, Mar 09 2010
G.f.: x*(-2*x^7 - 6*x^6 - 4*x^5 - 2*x^4 - 4*x^3 - 2*x^2 - 4*x - 6)/(x^8 - 1). - Chai Wah Wu, Feb 16 2021

Edited by Omar E. Pol, Mar 02 2009

Offset corrected by Reinhard Zumkeller, Jan 06 2013

A140378 Lengths of runs of consecutive primes and nonprimes in A007775.

Original entry on oeis.org

1, 12, 1, 6, 1, 3, 1, 6, 2, 2, 1, 2, 1, 3, 1, 2, 1, 3, 1, 4, 2, 1, 2, 6, 1, 1, 1, 1, 1, 6, 2, 1, 2, 4, 3, 2, 2, 4, 1, 1, 1, 1, 1, 3, 1, 2, 2, 1, 1, 2, 1, 2, 1, 3, 1, 4, 2, 1, 1, 2, 2, 3, 2, 2, 4, 2, 2, 1, 1, 4, 2, 1, 1, 4, 1, 3, 2, 1, 1, 3, 1, 3, 2, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 3, 2, 2, 2

Offset: 1

Juri-Stepan Gerasimov, Jun 13 2008

nonn

Primes can be classified according to their remainder modulo 30: remainder 1 (A136066), 7 (A132231), 11 (A132232), 13 (A132233), 17 (A039949), 19 (A132234), 23 (A132235), or 29 (A132236). In the sequence A007775 of all numbers (prime or nonprime) in any of these remainder classes, we look for runs of numbers that are successively prime or nonprime and place the lengths of these runs in this sequence.

			Groups of runs in A007775 are (1), (7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47), (49), (53, 59, 61, 67, 71, 73), (77), (79, 83,...), which is 1 nonprime followed by 12 primes followed by 1 nonprime followed by 6 primes etc.

Cf. A136066, A132231, A132232, A132233, A039919, A132234, A132235, A132236.

A007775 := proc(n) option remember ; local a; if n = 1 then 1; else for a from A007775(n-1)+1 do if a mod 2 <>0 and a mod 3 <>0 and a mod 5 <> 0 then RETURN(a) ; fi ; od: fi ; end: A := proc() local al,isp,n; al := 0: isp := false ; n := 1: while n< 300 do a := A007775(n) ; if isprime(a) <> isp then printf("%d,",al) ; al := 1; isp := not isp ; else al := al+1 ; fi ; n := n+1: od: end: A() ; # R. J. Mathar, Jun 16 2008

Edited by R. J. Mathar, Jun 16 2008

A227896 32-beat repeating palindromic sequence: digital roots of Fibonacci numbers indexed by the set of natural numbers not divisible by 2, 3 or 5 (A007775).

Original entry on oeis.org

1, 4, 8, 8, 4, 5, 1, 5, 4, 8, 4, 5, 1, 1, 5, 8, 8, 5, 1, 1, 5, 4, 8, 4, 5, 1, 5, 4, 8, 8, 4, 1, 1, 4, 8, 8, 4, 5, 1, 5, 4, 8, 4, 5, 1, 1, 5, 8, 8, 5, 1, 1, 5, 4, 8, 4, 5, 1, 5, 4, 8, 8, 4, 1, 1, 4, 8, 8, 4, 5, 1, 5, 4, 8, 4, 5, 1, 1, 5, 8, 8, 5, 1, 1, 5, 4, 8, 4, 5, 1, 5, 4, 8, 8, 4, 1, 1, 4, 8, 8, 4, 5, 1, 5, 4, 8, 4, 5, 1, 1, 5, 8, 8, 5, 1, 1, 5, 4, 8, 4, 5, 1, 5, 4, 8, 8, 4, 1

Offset: 1

Gary Croft, Oct 14 2013

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1).

Cf. A007775, A030132.

a(n)=fibonacci(n\8*6 + 9 + 3*(n+1)\2*2 - max(5, (n-2)%8)*2)%9 \\ Charles R Greathouse IV, Aug 26 2014

a(n)=[1,4,8,8,4,5,1,5,4,8,4,5,1,1,5,8,8,5,1,1,5,4,8,4,5,1,5,4,8,8,4,1][n%32+1] \\ Charles R Greathouse IV, Aug 26 2014

Vec(x*(1 + 3*x + 4*x^2 - 4*x^4 + x^5 - 4*x^6 + 4*x^7 - x^8 + 4*x^9 - 4*x^10 + x^11 - 4*x^12 + 4*x^14 + 3*x^15 + x^16) / ((1 - x)*(1 + x^16)) + O(x^100)) \\ Colin Barker, Sep 21 2019

From Colin Barker, Sep 21 2019: (Start)
G.f.: x*(1 + 3*x + 4*x^2 - 4*x^4 + x^5 - 4*x^6 + 4*x^7 - x^8 + 4*x^9 - 4*x^10 + x^11 - 4*x^12 + 4*x^14 + 3*x^15 + x^16) / ((1 - x)*(1 + x^16)).
a(n) = a(n-1) - a(n-16) + a(n-17) for n>17.
(End)

A373469 Least odd k such that C(2k, k) == 1 (mod A007775(n)), or 0 if no such k exists.

Original entry on oeis.org

1, 17, 13, 2383, 37, 3, 3391, 185, 129, 419, 95, 139, 7, 7373, 497, 21, 89, 27, 319, 7, 23, 191, 277, 25, 33635, 137, 1957, 347, 879, 889, 47, 57, 411, 263, 63, 57, 63, 143, 62561, 363, 1679, 861, 285735, 1017, 545, 2605, 913, 1873, 735, 206349, 817, 407, 485, 49, 7605, 179817

Offset: 1

M. F. Hasler, Jul 12 2024

A007775 lists the odd numbers not divisible by 3 or 5. It seemed that these are exactly the odd numbers not in A086748 (= odd m such that C(2k,k) == 1 (mod m) has no odd solution k), i.e., the numbers in A086748 would exactly be the odd multiples of 3 and 5, but so far there was no proof or disproof for that. The present sequence gives an explicit proof, if it exists, for each x in A007775, that x is not in A086748.
It is highly possible that a(n) = 0 for n with m = A007775(n) divisible by three or more distinct primes, in which case values of k such that C(2k,k) coprime to m, let alone C(2k,k) == 1 (mod m), are very sparse and possibly finite. See A030979 for a similar problem. - Max Alekseyev, Jul 14 2024
Examples for moduli that have 3 distinct prime factors >5: a(603) = 57 associated with modulus A007775(603) = 2261 = 7*17*19.  a(4333) = 23 associated with modulus A007775(4333) = 16247 = 7*11*211. a(6621) = 1709 associated with 11*37*61. a(6797)=19999 assocated with 7*11*331. - R. J. Mathar, Aug 09 2024

M. F. Hasler and Max Alekseyev, Table of n, a(n) for n = 1..266

Cf. A007775 (odd numbers not divisible by 3 or 5), A086748 (odd m such that C(2k,k)==1 (mod m) has no odd solution k).

Cf. A030979.

/* helper function: compute C(n,k) mod prime p */
LucasT(n,k,p)={if(n>=k, my(kp = digits(k,p), np = digits(n,p)[-#kp..-1]); prod(i=1, #kp, binomial(np[i], kp[i]), Mod(1,p)))}
is1(k,f)={for(i=1,matsize(f)[1], LucasT(2*k, k, f[i,1])==1||return); vecmax(f[,2])==1 || binomial(2*k,k)%factorback(f)==1}
apply( {A373469(n, m=A007775(n), f=factor(m))=!f || forstep(k=3, oo, 2, is1(k,f) && return(k))}, [1..50])

a(43)-a(56) from Max Alekseyev, Jul 12 2024

A375668 The maximum exponent in the prime factorization of the 7-rough numbers (A007775).

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Amiram Eldar, Aug 23 2024

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A007775, A051903, A375039, A375667.

If[# == 1, 0, Max[FactorInteger[#][[;; , 2]]]] & /@ Select[Range[300], CoprimeQ[#, 30] &]

lista(nmax) = print1(0, ", "); for(n = 2, nmax, if(gcd(n, 30) == 1, print1(vecmax(factor(n)[,2]), ", ")));

a(n) = A051903(A007775(n)).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 1 + Sum_{k>=2} (1 - 1/((1-1/2^k) * (1-1/3^k) * (1-1/5^k) * zeta(k))) = 1.05546104674564363968... .
In general, the asymptotic mean of the maximum exponent in the prime factorization of the p-rough numbers (numbers that are not divisible by any prime smaller than p) is 1 + Sum_{k>=2} (1 - 1/(zeta(k) * Product_{primes q < p} (1-1/q^k))).

A135766 Multiply sequence A007775 (1 7 11 13 ...) by sequence A000351 (1 5 25 125 ...).

Original entry on oeis.org

1, 7, 5, 11, 35, 25, 13, 55, 175, 125, 17, 65, 275, 875, 625, 19, 85, 325, 1375, 4375, 3125, 23, 95, 425, 1625, 6875, 21875, 15625, 29, 115, 475, 2125, 8125, 34375, 109375, 78125, 31, 145, 575, 2375, 10625, 40625, 171875, 546875, 390625, 37, 155, 725, 2875

Offset: 1

Alford Arnold, Nov 29 2007

			A(13) = 275 since A007775(3) * A000351(3) = 11 times 25.

Cf. A135764, A135765, A007775 (not divisible by 2, 3, or 5), A000351 (powers of five).

T(n,k) = A007775(n-k)*A000351(k), n>0, 0<=kR. J. Mathar, Jan 07 2008

More terms from R. J. Mathar, Jan 07 2008

A233766 Digital root of Lucas numbers (A030133) indexed by natural numbers not divisible by 2, 3 or 5 (A007775).

Original entry on oeis.org

2, 9, 6, 7, 2, 9, 3, 7, 9, 7, 2, 9, 3, 2, 7, 6, 7, 9, 3, 2, 7, 9, 6, 2, 2, 9, 6, 7, 2, 9, 3, 7, 9, 7, 2, 9, 3, 2, 7, 6, 7, 9, 3, 2, 7, 9, 6, 2, 2, 9, 6, 7, 2, 9, 3, 7, 9, 7, 2, 9, 3, 2, 7, 6, 7, 9, 3, 2, 7, 9, 6, 2, 2, 9, 6, 7, 2, 9, 3, 7, 9, 7, 2, 9, 3, 2, 7, 6, 7, 9, 3, 2, 7, 9, 6, 2

Offset: 1

Gary Croft, Dec 15 2013

32-beat repeating sequence.

Cf. A000032, A007775, A030133.

Conjectures from Colin Barker, Sep 22 2019: (Start)
G.f.: x*(2 + 9*x + 6*x^2 + 7*x^3 + 2*x^4 + 9*x^5 + 3*x^6 + 7*x^7 + 9*x^8 + 7*x^9 + 2*x^10 + 9*x^11 + 3*x^12 + 2*x^13 + 7*x^14 + 6*x^15 + 7*x^16 + 9*x^17 + 3*x^18 + 2*x^19 + 7*x^20 + 9*x^21 + 6*x^22 + 2*x^23) / ((1 - x)*(1 + x)*(1 - x + x^2)*(1 + x^2)*(1 + x + x^2)*(1 - x^2 + x^4)*(1 + x^4)*(1 - x^4 + x^8)).
a(n) = a(n-24) for n>24.
(End)

A236866 Positions of primes in A007775 (numbers not divisible by 2, 3 or 5).

Original entry on oeis.org

2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 15, 16, 17, 18, 19, 20, 22, 23, 24, 26, 27, 28, 29, 30, 31, 34, 35, 37, 38, 40, 41, 42, 44, 45, 47, 48, 49, 51, 52, 53, 54, 57, 60, 61, 62, 63, 64, 65, 67, 69, 71, 72, 73, 74, 75, 76, 79, 82, 83, 84, 85, 89, 90, 93, 94, 95

Offset: 1

Alex Ratushnyak, Jan 31 2014

From Antti Karttunen, Feb 01 2014: (Start)
Positions of primes among natural numbers coprime to 30.
Term 1 is missing from the sequence, because A007775(1)=1 is not considered a prime, terms 2 - 13 are all present, and 14 is the next term missing from here, as A007775(14)=49 is the first composite in that sequence.
(End)

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A007775, A181709.

from sympy import isprime
i=0
for n in range(1000):
    if n%2 and n%3 and n%5:
        i+=1   # A007775(i)=n
        if isprime(n):  print(i, end=', ')

;; With Antti Karttunen's IntSeq-library.
(define A236866 (MATCHING-POS 1 1 (lambda (n) (prime? (A007775 n)))))
;; Where a slow version of A007775 can be defined for example like this,
(define A007775 (MATCHING-POS 1 1 (lambda (n) (= 1 (gcd n 30)))))
;; from Antti Karttunen, Feb 01 2014

A295869 Numbers not divisible by 2, 3 or 5 (A007775) with digital root 8.

Original entry on oeis.org

17, 53, 71, 89, 107, 143, 161, 179, 197, 233, 251, 269, 287, 323, 341, 359, 377, 413, 431, 449, 467, 503, 521, 539, 557, 593, 611, 629, 647, 683, 701, 719, 737, 773, 791, 809, 827, 863, 881, 899, 917, 953, 971, 989, 1007, 1043, 1061, 1079, 1097, 1133

Offset: 1

Gary Croft, Mar 24 2018

Numbers == {17, 53, 71, 89} mod 90 with additive sum sequence 17{+36+18+18+18} {repeat ...}. Includes all prime numbers >5 with digital root 8.

			17+36=53; 53+18=71; 71+18=89; 89+18=107; 107+36=143.

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).

Intersection of A007775 and A017257.

Filtered([1..1200],n->n mod 2<>0 and n mod 3 <>0 and n mod 5<>0 and n-9*Int((n-1)/9)=8); # Muniru A Asiru, May 30 2018

select(n->modp(n,2)<>0 and modp(n,3)<>0 and modp(n,5)<>0 and n-9*floor((n-1)/9)=8,[$1..1200]); # Muniru A Asiru, May 30 2018

Vec(x*(17 + 36*x + 18*x^2 + 18*x^3 + x^4) / ((1 - x)^2*(1 + x)*(1 + x^2)) + O(x^60)) \\ Colin Barker, Mar 26 2018

Numbers == {17, 53, 71, 89} mod 90.
From Colin Barker, Mar 26 2018: (Start)
G.f.: x*(17 + 36*x + 18*x^2 + 18*x^3 + x^4) / ((1 - x)^2*(1 + x)*(1 + x^2)).
a(n) = (5 + 9*(-1)^n - (9+9*i)*(-i)^n - (9-9*i)*i^n + 90*n) / 4, where i=sqrt(-1).
a(n) = a(n-1) + a(n-4) - a(n-5) for n>5.
(End)

A301617 Numbers not divisible by 2, 3 or 5 (A007775) with digital root 1.

Original entry on oeis.org

1, 19, 37, 73, 91, 109, 127, 163, 181, 199, 217, 253, 271, 289, 307, 343, 361, 379, 397, 433, 451, 469, 487, 523, 541, 559, 577, 613, 631, 649, 667, 703, 721, 739, 757, 793, 811, 829, 847, 883, 901, 919, 937, 973, 991, 1009, 1027, 1063, 1081, 1099

Offset: 1

Gary Croft, Mar 24 2018

Numbers == {1, 19, 37, 73} mod 90 with additive sum sequence 1{+18+18+36+18} {repeat ...}. Includes all prime numbers > 7 with digital root 1.

			1+18=19; 19+18=37; 37+36=73; 73+18=91; 91+18=109.

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).

Cf. A177704, A045572.

Intersection of A007775 and A017173.

seq(seq(i+90*j,i=[1,19,37,73]),j=0..30); # Robert Israel, Mar 25 2018

LinearRecurrence[{1,0,0,1,-1},{1,19,37,73,91},50] (* Harvey P. Dale, Dec 14 2019 *)

a(n) = 1 + 18 * (n - 1 + n\4) \\ David A. Corneth, Mar 24 2018

Vec(x*(1 + 18*x + 18*x^2 + 36*x^3 + 17*x^4) / ((1 - x)^2*(1 + x)*(1 + x^2)) + O(x^60)) \\ Colin Barker, Mar 24 2018

n == {1, 19, 37, 73} mod 90.
a(n + 1) = a(n) + 18 * A177704(n + 1). - David A. Corneth, Mar 24 2018
From Colin Barker, Mar 24 2018: (Start)
G.f.: x*(1 + 18*x + 18*x^2 + 36*x^3 + 17*x^4) / ((1 - x)^2*(1 + x)*(1 + x^2)).
a(n) = a(n-1) + a(n-4) - a(n-5) for n>5.
(End)

The missing term 1081 added to the sequence by Colin Barker, Mar 24 2018

cp's OEIS Frontend

A145011 First differences of A007775.

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A140378 Lengths of runs of consecutive primes and nonprimes in A007775.

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A227896 32-beat repeating palindromic sequence: digital roots of Fibonacci numbers indexed by the set of natural numbers not divisible by 2, 3 or 5 (A007775).

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A373469 Least odd k such that C(2k, k) == 1 (mod A007775(n)), or 0 if no such k exists.

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A375668 The maximum exponent in the prime factorization of the 7-rough numbers (A007775).

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A135766 Multiply sequence A007775 (1 7 11 13 ...) by sequence A000351 (1 5 25 125 ...).

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A233766 Digital root of Lucas numbers (A030133) indexed by natural numbers not divisible by 2, 3 or 5 (A007775).

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A236866 Positions of primes in A007775 (numbers not divisible by 2, 3 or 5).

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