A045572 -id:A045572 - cp's OEIS frontend

A084101 Expansion of (1+x)^2/((1-x)*(1+x^2)).

1, 3, 3, 1, 1, 3, 3, 1, 1, 3, 3, 1, 1, 3, 3, 1, 1, 3, 3, 1, 1, 3, 3, 1, 1, 3, 3, 1, 1, 3, 3, 1, 1, 3, 3, 1, 1, 3, 3, 1, 1, 3, 3, 1, 1, 3, 3, 1, 1, 3, 3, 1, 1, 3, 3, 1, 1, 3, 3, 1, 1, 3, 3, 1, 1, 3, 3, 1, 1, 3, 3, 1, 1, 3, 3, 1, 1, 3, 3, 1, 1, 3, 3, 1, 1, 3, 3, 1, 1, 3, 3, 1, 1, 3, 3, 1, 1, 3, 3, 1, 1, 3, 3, 1, 1

Offset: 0

Paul Barry, May 15 2003

Partial sums of A084099. Inverse binomial transform of A000749 (without leading zeros).
From Klaus Brockhaus, May 31 2010: (Start)
Periodic sequence: Repeat 1, 3, 3, 1.
Interleaving of A010684 and A176040.
Continued fraction expansion of (7 + 5*sqrt(29))/26.
Decimal expansion of 121/909.
a(n) = A143432(n+3) + 1 = 2*A021913(n+1) + 1 = 2*A133872(n+3) + 1.
a(n) = A165207(n+1) - 1.
First differences of A047538.
Binomial transform of A084102. (End)
From Wolfdieter Lang, Feb 09 2012: (Start)
a(n) = A045572(n+1) (Modd 5) := A203571(A045572(n+1)), n >= 0.
For general Modd n (not to be confused with mod n) see a comment on A203571. The nonnegative members of the five residue classes Modd 5, called [m] for m=0,1,...,4, are shown in the array A090298 if there the last row is taken as class [0] after inclusion of 0.
(End)

			From _Wolfdieter Lang_, Feb 09 2012: (Start)
Modd 5 of nonnegative odd numbers restricted mod 5:
A045572: 1, 3, 7, 9, 11, 13, 17, 19, 21, 23, ...
Modd 5:  1, 3, 3, 1,  1,  3,  3,  1,  1,  3, ...
(End)

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

Cf. A084102.

Cf. A010684 (repeat 1, 3), A176040 (repeat 3, 1), A178593 (decimal expansion of (7+5*sqrt(29))/26), A143432 (expansion of (1+x^4)/((1-x)*(1+x^2))), A021913 (repeat 0, 0, 1, 1), A133872 (repeat 1, 1, 0, 0), A165207 (repeat 2, 2, 4, 4), A047538 (congruent to 0, 1, 4 or 7 mod 8), A084099 (expansion of (1+x)^2/(1+x^2)), A000749 (expansion of x^3/((1-x)^4-x^4)). - Klaus Brockhaus, May 31 2010

R:=PowerSeriesRing(Integers(), 100); Coefficients(R!( (1+x)^2/((1-x)*(1+x^2)) )); // G. C. Greubel, Feb 28 2019

CoefficientList[Series[(1+x)^2/((1-x)(1+x^2)),{x,0,110}],x] (* or *) PadRight[{},110,{1,3,3,1}] (* Harvey P. Dale, Nov 21 2012 *)

x='x+O('x^100); Vec((1+x)^2/((1-x)*(1+x^2))) \\ Altug Alkan, Dec 24 2015

((1+x)^2/((1-x)*(1+x^2))).series(x, 100).coefficients(x, sparse=False) # G. C. Greubel, Feb 28 2019

a(n) = binomial(3, n mod 4). - Paul Barry, May 25 2003
From Klaus Brockhaus, May 31 2010: (Start)
a(n) = a(n-4) for n > 3; a(0) = a(3) = 1, a(1) = a(2) = 3.
a(n) = (4 - (1+i)*i^n - (1-i)*(-i)^n)/2 where i = sqrt(-1). (End)
E.g.f.: 2*exp(x) + sin(x) - cos(x). - Arkadiusz Wesolowski, Nov 04 2017
a(n) = 2 - (-1)^(n*(n+1)/2). - Guenther Schrack, Feb 26 2019

A105115 Numbers k such that the decimal representation of 1/k is neither terminating nor purely repeating.

Original entry on oeis.org

6, 12, 14, 15, 18, 22, 24, 26, 28, 30, 34, 35, 36, 38, 42, 44, 45, 46, 48, 52, 54, 55, 56, 58, 60, 62, 65, 66, 68, 70, 72, 74, 75, 76, 78, 82, 84, 85, 86, 88, 90, 92, 94, 95, 96, 98, 102, 104, 105, 106, 108, 110, 112, 114, 115, 116, 118, 120, 122, 124, 126, 130, 132, 134

Offset: 1

David Wasserman, Apr 07 2005

k is in this sequence iff 1) k is divisible by 2 or 5 and 2) k is also divisible by some prime other than 2 and 5. Contains the numbers that are in neither A003592 nor A045572.

The asymptotic density of this sequence is 3/5. - Amiram Eldar, Mar 26 2021

			22 is a member because 1/22 = .045454545..., which has a 0 before the repeating 45.

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

Cf. A003592, A045572.

f[n_, lim_] := Block[{g, a}, g[x_] := First /@ FactorInteger@ x; a = g@ n; Select[Range@ lim, And[1 < GCD[#, n] < #, Length@ Complement[g@ #, a] >= 1] &]]; f[10, 134] (* Michael De Vlieger, Jun 20 2015 *)

A162699 Odd numbers not divisible by 7.

Original entry on oeis.org

1, 3, 5, 9, 11, 13, 15, 17, 19, 23, 25, 27, 29, 31, 33, 37, 39, 41, 43, 45, 47, 51, 53, 55, 57, 59, 61, 65, 67, 69, 71, 73, 75, 79, 81, 83, 85, 87, 89, 93, 95, 97, 99, 101, 103, 107, 109, 111, 113, 115, 117, 121, 123, 125, 127, 129, 131, 135, 137, 139, 141, 143, 145, 149

Offset: 1

Zak Seidov, Jul 11 2009

Numbers coprime to 14. The asymptotic density of this sequence is 3/7. - Amiram Eldar, Oct 18 2020

Amiram Eldar, 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. A045572 (odd numbers not divisible by 5).

s={}; Do[If[Mod[n, 7] > 0, AppendTo[s, n]], {n, 1, 60, 2}]; s
Table[2 n + 1 + 2*Floor[(2 n - 7)/12], {n, 1, 99}]
DeleteCases[Range[1,151,2],?(Divisible[#,7]&)] (* _Harvey P. Dale, Jul 04 2015 *)

a(n)=(14*n-7)\12*2+1 \\ Charles R Greathouse IV, Jun 11 2015

a(n) = 2*n + 1 + 2*floor( (2*n-7)/12 ).

a(n) = a(n-1)+a(n-6)-a(n-7). G.f.: x*(x^2+1)*(x^4+2*x^3+x^2+2*x+1)/ ((1+x)*(1+x+x^2)*(x^2-x+1)*(x-1)^2). - R. J. Mathar, Jul 31 2009

A176553 Numbers m such that concatenations of divisors of m are noncomposites.

Original entry on oeis.org

1, 3, 7, 9, 13, 21, 31, 37, 67, 73, 79, 97, 103, 109, 121, 151, 163, 181, 183, 193, 219, 223, 229, 237, 277, 283, 307, 363, 367, 373, 381, 409, 433, 439, 471, 487, 489, 499, 511, 523, 571, 601, 603, 607, 613, 619, 657, 669, 709, 733, 787, 811, 817, 819, 823, 841, 867

Offset: 1

Jaroslav Krizek, Apr 20 2010

Do all primes p > 5 have a multiple in this sequence? This holds at least for p < 10^4. - Charles R Greathouse IV, Sep 23 2016
Conjecture: this sequence is a subsequence of A003136 (Loeschian numbers). - Davide Rotondo, Jan 02 2022
If m is not in A003136, there is a prime p == 2 (mod 3) such that the exponent of p in the factorization of m is odd, then we have 3 | 1+p | 1+p+p^2+...+p^(2*r-1) | sigma(m), sigma = A000203 is the sum of divisors, so the concatenation of the divisors of m is also divisible by 3. - Jianing Song, Aug 22 2022

			a(6) = 21: the divisors of 21 are 1,3,7,21, and their concatenation 13721 is noncomposite.

Bill McEachen, Table of n, a(n) for n = 1..10000

Cf. A037278, A176554, A176555.

Subsequence of A045572.

Select[Range[10^3], ! CompositeQ@ FromDigits@ Flatten@ IntegerDigits@ Divisors@ # &] (* Michael De Vlieger, Sep 23 2016 *)

is(n)=my(d=divisors(n)); d[1]="1"; isprime(eval(concat(d))) || n==1 \\ Charles R Greathouse IV, Sep 23 2016

from sympy import divisors, isprime
def ok(m): return m==1 or isprime(int("".join(str(d) for d in divisors(m))))
print([m for m in range(1, 900) if ok(m)]) # Michael S. Branicky, Feb 05 2022

Edited and extended by Charles R Greathouse IV, Apr 30 2010

Data corrected by Bill McEachen, Nov 03 2021

A082768 Numbers that begin with 1, 3, 7 or 9.

Original entry on oeis.org

1, 3, 7, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119

Offset: 1

Amarnath Murthy, Apr 18 2003

Cf. A082769, A082770, A045572 (Numbers that end with 1, 3, 7 or 9).

Cf. A082769, A082770, A089743.

isA082768 := proc(n)
    convert(n,base,10) ;
    if op(-1,%) in {1,3,7,9} then
        true;
    else
        false;
    end if;
end proc:
A082768 := proc(n)
    if n = 1 then
        1;
    else
        for a from procname(n-1)+ 1 do
            if isA082768(a) then
                return a;
            end if;
        end do:
    end if;
end proc:
seq(A082768(n),n=1..120) ; # R. J. Mathar, Aug 27 2025

Select[Range[119], MemberQ[{1, 3, 7, 9}, First[IntegerDigits[#]]] &] (* Jayanta Basu, Jun 24 2013 *)

More terms from David Wasserman, Jul 28 2005

Offset changed by Andrew Howroyd, Sep 29 2024

A178505 Decimal form of the period of 1/n for n such that gcd(n,10)=1. Leading zeros are suppressed.

Original entry on oeis.org

3, 142857, 1, 9, 76923, 588235294117647, 52631578947368421, 47619, 434782608695652173913, 37, 344827586206896551724137931, 32258064516129, 3, 27, 25641, 2439, 23255813953488372093

Offset: 1

Michel Lagneau, May 29 2010

The numbers n are A045572, and the corresponding periods are A002329.

			3 is in the sequence because 1/3 = 0.3333...
142857 is in the sequence because 1/7 = 0.142857 142857 ...
1 is in the sequence because 1/9 = 0.1111....

Ray Chandler, Table of n, a(n) for n = 1..406 (terms up to 1000 digits)

Cf. A002329, A036275, A045572, A060284.

with(numtheory): nn:= 100: T:=array(1..nn):k:=1: U:=array(1..nn):k:=1: for n from 2 to 200 do:x:=1/n:for p from 1 to 200 while(irem(10^p,n)<>1 or gcd(n,10)<> 1) do:od: if irem(10^p,n) = 1 and gcd(n,10) = 1 then y:=floor(x*10^p): T[k]:=y: U[k]:=n : k:=k+1:else fi:od:print(T):

a(n) = A060284(A045572(n+1)). [R. J. Mathar, Jun 26 2010]

Name corrected by T. D. Noe, Jul 07 2010

A072182 A Wallis pair (x,y) satisfies sigma(x^2) = sigma(y^2); sequence gives x's for Wallis pairs with x < y (ordered by values of x, then y).

Original entry on oeis.org

4, 12, 28, 36, 44, 52, 68, 76, 84, 92, 108, 116, 124, 132, 148, 156, 164, 172, 188, 196, 204, 212, 228, 236, 244, 252, 268, 276, 284, 292, 308, 316, 324, 326, 332, 348, 356, 364, 372, 388, 396, 404, 406, 412, 428, 436, 444, 452, 468, 476, 484, 492, 508, 516

Offset: 1

N. J. A. Sloane, Oct 19 2002

4*A045572 is included in this sequence. - Benoit Cloitre, Oct 22 2002

D. Johnson remarks that some terms are repeated, e.g., a(139)=a(140)=1284 forms a Wallis pair with A072186(139)=1528 and also with A072186(140)=1605. - M. F. Hasler, Sep 15 2013

			The first few pairs are all multiples of the first pair (4,5): (4, 5), (12, 15), (28, 35), (36, 45), (44, 55), (52, 65), ...

I. Kaplansky, The challenges of Fermat, Wallis and Ozanam (and several related challenges): II. Fermat's second challenge, Preprint, 2002.

Donovan Johnson, Table of n, a(n) for n = 1..10000

Cf. A072186, A075768, A075769.

Cf. A000203, A000290.

a072182 n = a072182_list !! (n-1)
(a072182_list, a072186_list) = unzip wallisPairs
  wallisPairs = [(x, y) | (y, sy) <- tail ws,
                          (x, sx) <- takeWhile ((< y) . fst) ws, sx == sy]
                where ws = zip [1..] $ map a000203 $ tail a000290_list
-- Reinhard Zumkeller, Sep 17 2013

w = {}; m = 550;
Do[q = DivisorSigma[1, x^2]; sq = Sqrt[q] // Floor; Do[If[q == DivisorSigma[1, y^2], AppendTo[w, {x, y}]], {y, x+1, sq}], {x, 1, m}];
w[[All, 1]] (* Jean-François Alcover, Oct 01 2019 *)

{w=[]; m=550; for(x=1,m,q=sigma(x^2); sq=sqrtint(q); for(y=x+1,sq,if(q==sigma(y^2), w=concat(w,[[x,y]])))); for(j=1,matsize(w)[2],print1(w[j][1],","))}

Extended by Klaus Brockhaus and Benoit Cloitre, Oct 22 2002

A072186 A Wallis pair (x,y) satisfies sigma(x^2) = sigma(y^2); sequence gives y's for Wallis pairs with x < y (ordered by values of x).

Original entry on oeis.org

5, 15, 35, 45, 55, 65, 85, 95, 105, 115, 135, 145, 155, 165, 185, 195, 205, 215, 235, 245, 255, 265, 285, 295, 305, 315, 335, 345, 355, 365, 385, 395, 405, 407, 415, 435, 445, 455, 465, 485, 495, 505, 489, 515, 535, 545, 555, 565, 585, 595, 605, 615, 635

Offset: 1

N. J. A. Sloane, Oct 19 2002

5*A045572 is included in this sequence. - Benoit Cloitre, Oct 22 2002

			The first few pairs are all multiples of the first pair (4,5): (4, 5), (12, 15), (28, 35), (36, 45), (44, 55), (52, 65), ...

I. Kaplansky, The challenges of Fermat, Wallis and Ozanam (and several related challenges): II. Fermat's second challenge, Preprint, 2002.

Donovan Johnson, Table of n, a(n) for n = 1..10000

Cf. A072182, A075768, A075769.

a072186 n = a072186_list !! (n-1)
-- a072186_list defined in A072182.  -- Reinhard Zumkeller, Sep 18 2013

w = {}; m = 550;
Do[q = DivisorSigma[1, x^2]; sq = Sqrt[q] // Floor; Do[If[q == DivisorSigma[1, y^2], AppendTo[w, {x, y}]], {y, x + 1, sq}], {x, 1, m}];
w[[All, 2]] (* Jean-François Alcover, Oct 01 2019 *)

{w=[]; m=550; for(x=1,m,q=sigma(x^2); sq=sqrtint(q); for(y=x+1,sq,if(q==sigma(y^2), w=concat(w,[[x,y]])))); for(j=1,matsize(w)[2],print1(w[j][2],","))}

Extended by Klaus Brockhaus and Benoit Cloitre, Oct 22 2002

A075607 a(1) = 1, a(n) = smallest number not occurring earlier such that the concatenation a(n-1) and a(n) is a prime.

Original entry on oeis.org

1, 3, 7, 9, 11, 17, 21, 13, 19, 31, 37, 27, 29, 39, 23, 33, 43, 49, 51, 47, 59, 53, 81, 61, 63, 67, 79, 93, 41, 57, 83, 69, 71, 77, 89, 99, 73, 121, 97, 87, 103, 91, 127, 123, 113, 111, 109, 133, 117, 101, 107, 119, 129, 169, 151, 141, 131, 143, 137, 147, 139

Offset: 1

Amarnath Murthy, Sep 28 2002

Almost certainly a permutation of A045572. - David W. Wilson, Jan 15 2005

Paolo P. Lava, Table of n, a(n) for n = 1..10000

Cf. A075608, A075609, A075610.

a = {{1}}; Do[k = 2; While[Nand[! MemberQ[a, #], PrimeQ@ FromDigits@ Join[a[[n - 1]], #]] &@ Set[d, IntegerDigits@ k], k++]; AppendTo[a, d], {n, 2, 61}]; FromDigits /@ a (* Michael De Vlieger, May 08 2017 *)

A075607(n,show=0,a=1,u=[])={for(n=2,n,show&&print1(a","); u=setunion(u,[a]); while(#u>1&&u[2]==u[1]+2,u=u[2..-1]); forstep(k=u[1]+2, 9e9, 2, setsearch(u,k)&&next; isprime(eval(Str(a,k))) && (a=k) && break)); a} \\ Use 2nd, 3rd or 4th optional arg to print intermediate terms, to use another starting value or to exclude some terms. - M. F. Hasler, Nov 25 2015

More terms from Michel ten Voorde Jun 23 2003

A083754 a(1) = 1 and then smallest odd number not occurring earlier such that the concatenation a(1)a(2)a(3)... is a prime.

Original entry on oeis.org

1, 3, 7, 11, 9, 27, 63, 31, 53, 21, 13, 83, 33, 39, 49, 51, 77, 87, 307, 29, 229, 281, 151, 173, 481, 41, 99, 157, 177, 17, 357, 213, 231, 171, 271, 557, 67, 113, 463, 159, 119, 57, 247, 147, 563, 409, 353, 391, 179, 1051, 209, 19, 153, 621, 287, 567, 313, 117, 363

Offset: 1

Amarnath Murthy and Meenakshi Srikanth (menakan_s(AT)yahoo.com), May 06 2003

Conjecture: all odd numbers not of the type 10k+5 are members.
Some of the larger entries may only correspond to probable primes.
Values corresponding to a(6)=27 (A083755(5)) through a(59)=363 (A083755(58), a 149-digit value) have been certified prime with Primo. - Rick L. Shepherd, May 10 2003
Since we begin with 1 and thereafter have more than a single decimal digit, all terms must be in A045572, the sequence that contains all positive integers relatively prime to 10. - Michael De Vlieger, Oct 30 2020.

			13,137,13711, etc. are primes.(1379 is not a prime) hence 11 is the next member after 7.

Michael De Vlieger, Table of n, a(n) for n = 1..501
Michael De Vlieger, Plot of a(n) for 1 <= n <= 500, showing congruency of a(n) with 1 (orange), 3 (green), 7 (blue), or 9 (magenta) (mod 10).

Cf. A045572, A083755.

Block[{c = 1, a = {1}, f, g}, f[m_, n_] := m*10^(1 + Floor[Log10[n]]) + n; g[n_] := (5 n + Mod[3 n + 2, 4] - 4)/2; Do[Block[{j = 2, k, d, t}, While[Nand[FreeQ[a, Set[k, g[j] ]], PrimeQ[Set[d, f[c, k]]]], j++]; c = d; AppendTo[a, k]], {i, 59}]; a] (* Michael De Vlieger, Oct 30 2020 *)

{used_before(v, n) = for (l=1,matsize(v)[2], if(v[l]==n, return(1))); return(0)} {A083754=[1]; p=A083754[1]; A083755=[]; print1(A083754[1],","); for (m=2,151, k=1; while (used_before(A083754,k)||!isprime(tmp_p=p*(10^length(Str(k)))+k), k=k+2); p=tmp_p; A083755=concat(A083755,p); A083754=concat(A083754,k); print1(A083754[m],",")); A083755}

More terms from Rick L. Shepherd, May 08 2003

cp's OEIS Frontend

A084101 Expansion of (1+x)^2/((1-x)*(1+x^2)).

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A105115 Numbers k such that the decimal representation of 1/k is neither terminating nor purely repeating.

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A162699 Odd numbers not divisible by 7.

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A176553 Numbers m such that concatenations of divisors of m are noncomposites.

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A082768 Numbers that begin with 1, 3, 7 or 9.

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A178505 Decimal form of the period of 1/n for n such that gcd(n,10)=1. Leading zeros are suppressed.

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A072182 A Wallis pair (x,y) satisfies sigma(x^2) = sigma(y^2); sequence gives x's for Wallis pairs with x < y (ordered by values of x, then y).

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