A095372 -id:A095372 - cp's OEIS frontend

A095374 One less than the number of divisors of 2*n + 1.

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

Offset: 1

Labos Elemer, Jun 07 2004

Number of special divisors of A095372(n) with A095372(k) form.

			A095372(22) is divisible by {91, 9091, 90909091, 90909090909091, A095372(22)}, thus a(22)=5.
G.f.= x + x^2 + x^3 + 2*x^4 + x^5 + x^6 + 3*x^7 + x^8 + x^9 + 3*x^10 + x^11 + 2*x^12 + ...

Michael De Vlieger, Table of n, a(n) for n = 1..5000
Gerzson Kéri, The factorization of compressed Chebyshev polynomials and other polynomials related to multiple-angle formulas, Annales Univ. Sci. Budapest (Hungary, 2022) Sect. Comp., Vol. 53, 93-108.

Cf. A000005, A001620, A095372, A095373, A023645, A099774.

g[x_]:=1+90*(100^x-1)/99 t=Table[1+90*(100^n-1)/99, {n, 1, 35}]; Do[Print[{w, is=Intersection[Divisors[g[w]], t], Length[is]}], {w, 1, 35}]
Table[DivisorSigma[0,2n+1],{n,90}]-1 (* Harvey P. Dale, Oct 31 2015 *)
Table[Sum[1 - Ceiling@ # + Floor@ # &[(n + i)/(n - i + 1)], {i, n}], {n, 87}] (* Michael De Vlieger, Feb 27 2017 *)

{a(n) = if( n<0, 0, numdiv(2*n + 1) - 1)} /* Michael Somos, Aug 30 2012 */

a(n) = A023645(2*n + 1) = A000005(2*n + 1) - 1.
a(n) = Sum_{i=1..n} 1-ceiling((n+i)/(n-i+1))+floor((n+i)/(n-i+1)). - Wesley Ivan Hurt, Feb 26 2017
O.g.f.: Sum_{n >= 1} x^n/(1 - x^(2*n+1)) = x/(1 - x) + Sum_{n >= 1} x^(2*n*(n+1))*(1 + x^(2*n+1))/(1 - x^(2*n+1)). - Peter Bala, Mar 04 2019
a(n) = A099774(n+1) - 1. - Bernard Schott, Mar 04 2019
Sum_{k=1..n} a(k) ~ n * (log(n) + 2*gamma - 3 + 3*log(2)) / 2, where gamma is Euler's constant (A001620). - Amiram Eldar, Mar 15 2025

A274743 Repunits with odd indices multiplied by 99, i.e., 99*(1, 111, 11111, 1111111, ...).

Original entry on oeis.org

99, 10989, 1099989, 109999989, 10999999989, 1099999999989, 109999999999989, 10999999999999989, 1099999999999999989, 109999999999999999989, 10999999999999999999989, 1099999999999999999999989, 109999999999999999999999989, 10999999999999999999999999989

Offset: 1

Rodolfo A. Fiorini, Jul 04 2016

It is apparent that the reciprocals of the terms in the sequence give an increasing sequence of periodic terms similar to A095372, but with the initial term equal to "01". The leading zero is important (see links). Furthermore, the reciprocals of the terms give a sequence of even growing periods, starting from 2, with delta = 4 (i.e., 2, 6, 10, 14, 18, ...).

Adding "11" to each term gives the binary representation of the n-th iteration of "Rule 14" elementary cellular automaton starting with a single ON (black cell) as in A266299.

			a(2) = 101*10989 - 100*99 = 1099989.

Rodolfo A. Fiorini, How Random is Your Tomographic Noise? A Number Theoretic Transform (NTT) Approach, Fundamenta Informaticae, 135(1-2), 2014, 135-170.
Rodolfo A. Fiorini, Computerized tomography noise reduction by CICT optimized exponential cyclic sequences (OECS) co-domain, Fundamenta Informaticae, vol.141 (2015), 115-134.
Index entries for linear recurrences with constant coefficients, signature (101,-100).

Cf. A002275, A095372, A100706, A266299, A274755.

[11*(10^(2*n-1)-1) : n in [1..20]]; // Wesley Ivan Hurt, Jul 04 2016

A274743:=n->11*(10^(2*n-1)-1): seq(A274743(n), n=1..20); # Wesley Ivan Hurt, Jul 04 2016

Array[99(10^(2 # - 1) - 1)/9 &, 15] (* Michael De Vlieger, Jul 04 2016 *)
99*Table[FromDigits[PadRight[{},2n+1,1]],{n,0,15}] (* Harvey P. Dale, Jul 22 2019 *)

Vec(99*x*(1+10*x)/((1-x)*(1-100*x)) + O(x^99)) \\ Altug Alkan, Jul 05 2016

a002275(n) = (10^n-1)/9
a(n) = 99*a002275(2*n-1) \\ Felix Fröhlich, Jul 05 2016

a(n) = 101*a(n-1) - 100*a(n-2) for n>2, with a(0)= 99 and a(1)= 10989.
a(n) = 99*A100706(n-1).
G.f.: 99*x*(1 + 10*x)/((1 - x)*(1 - 100*x)). - Ilya Gutkovskiy, Jul 04 2016
a(n) = 11*(10^(2*n-1)-1). - Wesley Ivan Hurt, Jul 04 2016
E.g.f.: 11*(9 - 10*exp(x) + exp(100*x))/10. - Stefano Spezia, Aug 05 2024

A152577 a(n) = 10^(2*n - 1) + 1.

Original entry on oeis.org

11, 1001, 100001, 10000001, 1000000001, 100000000001, 10000000000001, 1000000000000001, 100000000000000001, 10000000000000000001, 1000000000000000000001, 100000000000000000000001

Offset: 1

Cino Hilliard, Dec 08 2008

These numbers are all divisible by 11. This follows from the identity a^n - b^n = (a+b)*(a^(n-1) - a^(n-2)*b + ... + b^(n-1)) for odd values of n. In this example a=10 and b=1 so a+b = 11. The sum of digits rule for divisibility by 11 also applies.

Bisection of A000533. Also, bisection of A062397. a(n) is also A084508(n+1) written in base 2. a(n) is also A087289(n-1) written in base 2. a(n) is also the concatenation of "1", 2(n-1) digits "0" and "1". - Omar E. Pol, Dec 13 2008

			From _Omar E. Pol_, Dec 14 2008: (Start)
n ....... a(n)
1 ....... 11
2 ...... 1001
3 ..... 100001
4 .... 10000001
5 ... 1000000001
(End)

Index entries for linear recurrences with constant coefficients, signature (101,-100).

Cf. A000533, A062397, A084508, A087289. - Omar E. Pol, Dec 13 2008

LinearRecurrence[{101,-100},{11,1001},20] (* Harvey P. Dale, Nov 05 2015 *)

g(n)=forstep(x=1,n,2,y=(10^x+1);print1(y","))

a(n) = 100*a(n-1) - 99 (with a(1)=11). - Vincenzo Librandi, Dec 14 2010
G.f.: -11*x*(-1+10*x) / ( (100*x-1)*(x-1) ). - R. J. Mathar, Sep 01 2011
a(n) = 11*A095372(n-1). - R. J. Mathar, Sep 01 2011
a(n) = 101*a(n-1)-100*a(n-2). - Wesley Ivan Hurt, Apr 24 2021
E.g.f.: (exp(100*x) + 10*exp(x) - 11)/10. - Stefano Spezia, Mar 13 2025

A097209 Primes of the form (10^p + 1)/11 (corresponding p are in A001562).

Original entry on oeis.org

9091, 909091, 909090909090909091, 909090909090909090909090909091, 9090909090909090909090909090909090909090909090909091, 909090909090909090909090909090909090909090909090909090909090909091

Offset: 1

Rick L. Shepherd, Jul 30 2004

nonn

Equivalently, primes of the form 9090...9091 (A054416(n)-1 copies of 90 followed by 91), the subsequence of all primes in A095372.

These primes appear in A187614 because the decimal representation of their reciprocal contains only the digits 0, 1, 8, and 9.

Cf. A001562, A054416, A095372.

A351320 a(n) is the unique integer k such that k * A116436(n) = 1.A116436(n).1 where "." stands for concatenation.

Original entry on oeis.org

111, 101, 87, 23, 21, 83, 21, 21, 27, 101, 87, 29, 23, 21, 33, 21, 83, 21, 39, 101, 87, 23, 21, 21, 21, 83, 101, 87, 59, 23, 21, 99, 57, 21, 27, 21, 101, 87, 29, 23, 21, 83, 69, 21, 71, 21, 101, 87, 33, 23, 21, 21, 83, 21, 101, 87, 23, 21, 27, 21, 39, 21, 83, 101, 87, 29, 23, 21, 21, 107, 21, 101

Offset: 1

Bernard Schott, Feb 07 2022

Except for a(1) = 111, which is unique, all terms appear infinitely many times and belong to this set of fifteen integers: {21, 23, 27, 29, 33, 39, 57, 59, 69, 71, 83, 87, 99, 101, 107}; see A329914.

The corresponding indices where these integers appear the first time are respectively: 5, 4, 9, 12, 15, 19, 33, 29, 43, 45, 6, 3, 32, 2, 70.

			A116436(1) = 1 and 111 * 1 = 1.1.1, hence a(1) = 111.
A116436(2) = 11 and 101 * 11 = 1.11.1, hence a(2) = 101.
A116436(32) = 112359550561797752809 and 99 * 112359550561797752809 = 1.112359550561797752809.1 hence a(32) = 99 (see Penguin reference).

David Wells, 112359550561797732809 entry, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1997, p. 196.

Cf. A116436, A329914, A329915.

M such that k*M=1M1 for: A095372 \ {1} (k=21), A331630 (k=23), A351237 (k=83), A351238 (k=87), A351239 (k=101).

A116436(k) = {local(l, d, lb, ub); d=divisors(10^(k+1)+1); l=[]; lb=10^(k-1); ub=10*lb; for(i=1, #d, if(d[i]>=lb&&d[i]A116436
a(n) = {my(v6=[], i=1); while (#v6 < n, v6 = concat(v6, A116436(i)); i++); my(x= v6[n]); my(k=1); while (eval(Str(1, x, 1)) % x, k++); eval(Str(1, x, 1))/x;} \\ Michel Marcus, Feb 10 2022

a(n) * A116436(n) = 1.A116436(n).1 where "." stands for concatenation.

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A095374 One less than the number of divisors of 2*n + 1.

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A274743 Repunits with odd indices multiplied by 99, i.e., 99*(1, 111, 11111, 1111111, ...).

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A152577 a(n) = 10^(2*n - 1) + 1.

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A097209 Primes of the form (10^p + 1)/11 (corresponding p are in A001562).

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A351320 a(n) is the unique integer k such that k * A116436(n) = 1.A116436(n).1 where "." stands for concatenation.

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