A062397 -id:A062397 - cp's OEIS frontend

A366669 a(n) = phi(10^n+1), where phi is Euler's totient function (A000010).

1, 10, 100, 720, 9792, 90900, 990000, 9090900, 94117632, 681410880, 9897840000, 86925373920, 979102080000, 9080325951840, 95255567232000, 712493107200000, 9926748531589120, 90004044661864320, 989999010000000000, 9090909090909090900, 97910150554895155200

Offset: 0

Sean A. Irvine, Oct 15 2023

nonn

Max Alekseyev, Table of n, a(n) for n = 0..331

Cf. A000010, A003021, A057934, A062397, A119704, A295503, A344897, A366668.

Cf. A053285, A366579, A366608, A366618, A366630, A366639, A366658, A366667, A366690, A366716.

EulerPhi[10^Range[0,20] + 1] (* Paul F. Marrero Romero, Nov 10 2023 *)

PARI
```
{a(n) = eulerphi(10^n+1)}
```

a(n) = A000010(A062397(n)). - Paul F. Marrero Romero, Nov 10 2023

A291639 Numbers k such that 0 is the smallest decimal digit of k^3.

Original entry on oeis.org

10, 16, 20, 22, 30, 34, 37, 40, 42, 43, 47, 48, 50, 52, 59, 60, 63, 67, 69, 70, 73, 74, 79, 80, 84, 86, 87, 89, 90, 93, 94, 99, 100, 101, 102, 103, 106, 107, 109, 110, 112, 115, 116, 117, 118, 120, 123, 124, 126, 127, 128, 130, 131, 134, 135, 138, 140, 141

Offset: 1

Colin Barker, Aug 28 2017

The sequence is infinite. For example, A062397(i) is in the sequence for any i > 1, since A168575(i) contains the digit 0 for any i > 1. - Felix Fröhlich, Aug 28 2017

Also contains A008592, and has asymptotic density 1. - Robert Israel, Aug 29 2017

			16 is in the sequence because 16^3 = 4096, the smallest decimal digit of which is 0.

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

Cf. A008592, A062397, A168575, A291640, A291641, A291642, A291643, A291644.

select(n -> min(convert(n^3,base,10))=0, [$1..1000]); # Robert Israel, Aug 29 2017

Select[Range[150],DigitCount[#^3,10,0]>0&] (* Harvey P. Dale, Feb 03 2025 *)

select(k->vecmin(digits(k^3))==0, vector(500, k, k))

A029797 Numbers k such that k^2 and k^3 have the same set of digits.

Original entry on oeis.org

0, 1, 10, 100, 146, 1000, 1203, 1460, 7652, 8077, 8751, 8965, 10000, 10406, 11914, 12030, 12057, 12586, 12768, 12961, 13055, 14202, 14600, 14625, 16221, 19350, 20450, 21539, 22040, 22175, 23682, 24071, 25089, 25201, 25708, 26653, 26981

Offset: 1

Patrick De Geest

Conjecture: there exists some m and N for which a(n) = m + n for all n >= N. [Charles R Greathouse IV, Jun 28 2011]

This conjecture is false. If the conjecture is true then for some N we would have k is in the sequence if k >= n. But 10^e + 1 (A062397) is not in the sequence for any integer e >= 0. - David A. Corneth, Nov 13 2023

			146 is in the sequence as 146^2 = 21316 has digits {1, 2, 3, 6} and 146^3 = 3112136 has digits {1, 2, 3, 6} as well. - _David A. Corneth_, Nov 13 2023

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A029793, A029795, A029800, A062397.

Cf. A011557 (a subsequence).

[ n: n in [0..34*10^4] | Set(Intseq(n^2)) eq Set(Intseq(n^3)) ];  // Bruno Berselli, Jun 28 2011

isA029797(n)=Set(Vec(Str(n^2)))==Set(Vec(Str(n^3))) \\ Charles R Greathouse IV, Jun 28 2011

A087000 Half length of periodic part of decimal expansion of 1/p for those primes having a periodic part of even length.

Original entry on oeis.org

3, 1, 3, 8, 9, 11, 14, 23, 29, 30, 4, 22, 48, 2, 17, 54, 56, 21, 65, 4, 23, 74, 39, 83, 89, 90, 96, 49, 15, 111, 114, 116, 15, 25, 128, 131, 134, 14, 73, 156, 55, 168, 58, 16, 183, 93, 189, 191, 194, 100, 102, 209, 70, 216, 16, 76, 230, 77, 243, 245, 249, 251

Offset: 1

Reinhard Zumkeller, Jul 29 2003

a(n) appears to be the least k such that 10^k+1 is divisible by A028416(n). See A001271. - Michel Marcus, Aug 13 2023

Cf. A028416, A002371, A086999, A087001, A087002, A001271, A062397 (10^n+1).

a(n) = A002371(A049084(A028416(n)))/2.
a(n) = A055642(A086999(n))/2.
a(n) = A055642(A087001(n)) = A055642(A087002(n)).

A215614 Numbers k that are not multiples of 10 and such that sum of digits of k^2 is 7.

Original entry on oeis.org

4, 5, 32, 49, 149, 1049

Offset: 1

Zak Seidov, Aug 17 2012

Except for the number 1, the terms of this sequence and numbers 10^k+1 (A062397) are the only numbers (up to trailing 0's) whose square has sum of digits less than 9. - M. F. Hasler, Sep 23 2014
Is this sequence finite? See also A384095 for a similar problem with digit sum 9. - M. F. Hasler, Jun 20 2025
a(7) > 10^15 if it exists. - David A. Corneth, Jun 21 2025
a(7) > 10^65 if it exists. - Michael S. Branicky, Jun 25 2025
a(7) > 10^700 if it exists. - Max Alekseyev, Jun 27 2025

Cf. A004159 (sum of digits of n^2).
Subsequence of A262711.
Cf. A384094 (similar for digit sum 9), A384095 (subset of "sporadic solutions").

Select[Range[1500],Mod[#,10]!=0&&Total[IntegerDigits[#^2]]==7&] (* Harvey P. Dale, Aug 21 2022 *)

for(n=1,9e9, n%10&&sumdigits(n^2)==7&&print1(n",")) \\ M. F. Hasler, Sep 23 2014

Edited and unproven keywords fini,full removed by Max Alekseyev, Jun 20 2025

A001271 Irregular table read by rows: row n lists prime factors of 10^n +1, with multiplicity.

Original entry on oeis.org

2, 11, 101, 7, 11, 13, 73, 137, 11, 9091, 101, 9901, 11, 909091, 17, 5882353, 7, 11, 13, 19, 52579, 101, 3541, 27961, 11, 11, 23, 4093, 8779, 73, 137, 99990001, 11, 859, 1058313049, 29, 101, 281, 121499449, 7, 11, 13, 211, 241, 2161, 9091, 353, 449, 641, 1409, 69857

Offset: 0

N. J. A. Sloane, revised Jul 13 2009

Except for 2, all these primes are in A028416, and all the primes in A028416 appear here. - Davide Rotondo, Aug 12 2023

			Table begins:
    2;
   11;
  101,
    7,      11, 13;
   73,     137;
   11,    9091;
  101,    9901;
   11,  909091;
   17, 5882353;
  ...

J. Brillhart et al., Factorizations of b^n +- 1. Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 2nd edition, 1985; and later supplements.

Ray Chandler, Rows n=0..310, flattened (from Kamada link)
J. Brillhart et al., Factorizations of b^n +- 1, Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 3rd edition, 2002.
Makoto Kamada, Factorizations of 100...001.
S. S. Wagstaff, Jr., The Cunningham Project

Cf. A028416, A057934, A062397.

Term ordering corrected by Sean A. Irvine, Apr 11 2012

Minor edits to description by Ray Chandler, May 02 2017

A093142 Expansion of g.f. (1-5x)/((1-x)(1-10*x)).

Original entry on oeis.org

1, 6, 56, 556, 5556, 55556, 555556, 5555556, 55555556, 555555556, 5555555556, 55555555556, 555555555556, 5555555555556, 55555555555556, 555555555555556, 5555555555555556, 55555555555555556, 555555555555555556, 5555555555555555556, 55555555555555555556, 555555555555555555556

Offset: 0

Paul Barry, Mar 24 2004

Second binomial transform of 5*A001045(3n)/3+(-1)^n.
Partial sums of A093143.
A convex combination of 10^n and 1.
In general the second binomial transform of k*Jacobsthal(3n)/3+(-1)^n is 1, 1+k, 1+11k, 1+111k, ... This is the case for k=5.

Seiichi Manyama, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (11,-10).

Cf. A001045, A002275, A062397, A093143.

CoefficientList[Series[(1-5x)/((1-x)(1-10x)),{x,0,20}],x] (* or *) LinearRecurrence[{11,-10},{1,6},20] (* Harvey P. Dale, Aug 23 2014 *)

{a(n) = (5*10^n+4)/9} \\ Seiichi Manyama, Sep 14 2019

a(n) = 5*10^n/9 + 4/9.
a(n) = 10*a(n-1) - 4 with a(0)=1. - Vincenzo Librandi, Aug 02 2010
a(n) = 11*a(n-1) - 10*a(n-2), n > 1. - Harvey P. Dale, Aug 23 2014
From Elmo R. Oliveira, Apr 29 2025: (Start)
E.g.f.: exp(x)*(5*exp(9*x) + 4)/9.
a(n) = (A062397(n) + A002275(n))/2. (End)

A098406 a(n) = (10^n + 17)/9.

Original entry on oeis.org

2, 3, 13, 113, 1113, 11113, 111113, 1111113, 11111113, 111111113, 1111111113, 11111111113, 111111111113, 1111111111113, 11111111111113, 111111111111113, 1111111111111113, 11111111111111113, 111111111111111113, 1111111111111111113, 11111111111111111113, 111111111111111111113

Offset: 0

Klaus Brockhaus, Sep 07 2004

A097683 gives numbers k such that a(k) is prime.

			a(5) = (100000 + 17)/9 = 11113.

Ivan Panchenko, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (11,-10).

Cf. A097683, A047855, A002275.

Cf. A002282, A062397.

FromDigits/@Table[PadLeft[{3},n,1],{n,20}] (* Harvey P. Dale, Jun 18 2011 *)

PARI
```
for(n=1,18,print1(((10^n)+17)/9,","))
```

a(1) = 3; a(n) = a(n-1) + 10^(n-1).
a(1) = 3; a(n) = 10*a(n-1) - 17.
a(n) = A047855(n)+1 = A002275(n)+2.
G.f.: (2-19*x)/((10*x-1)*(x-1)). - R. J. Mathar, Jan 27 2017
From Elmo R. Oliveira, Aug 23 2024: (Start)
E.g.f.: exp(x)*(exp(9*x) + 17)/9.
a(n) = A062397(n) - A002282(n).
a(n) = 11*a(n-1) - 10*a(n-2) for n > 1. (End)

a(0) from Ivan Panchenko, Nov 02 2013

A098609 a(n) = 100^n - 1.

Original entry on oeis.org

0, 99, 9999, 999999, 99999999, 9999999999, 999999999999, 99999999999999, 9999999999999999, 999999999999999999, 99999999999999999999, 9999999999999999999999, 999999999999999999999999, 99999999999999999999999999, 9999999999999999999999999999, 999999999999999999999999999999

Offset: 0

Henry Bottomley, Sep 17 2004

Vincenzo Librandi, Table of n, a(n) for n = 0..100
Index entries for linear recurrences with constant coefficients, signature (101,-100).

Cf. A002283, A062397, A098608.

[100^n-1: n in [0..20]]; // Vincenzo Librandi, Sep 23 2016

Table[100^n - 1, {n, 0, 20}] (* Vincenzo Librandi, Sep 23 2016 *)

a(n) = 100*a(n-1) + 99 = A002283(2*n) = A098608(n) - 1.
From Chai Wah Wu, Sep 22 2016: (Start)
a(n) = 101*a(n-1) - 100*a(n-2) for n > 1.
G.f.: 99*x/((x - 1)*(100*x - 1)). (End)
E.g.f.: exp(x)*(exp(99*x) - 1). - Stefano Spezia, Aug 05 2024
a(n) = A002283(n)*A062397(n). - Elmo R. Oliveira, Sep 09 2024

A093135 Expansion of g.f. (1-8x)/((1-x)(1-10*x)).

Original entry on oeis.org

1, 3, 23, 223, 2223, 22223, 222223, 2222223, 22222223, 222222223, 2222222223, 22222222223, 222222222223, 2222222222223, 22222222222223, 222222222222223, 2222222222222223, 22222222222222223, 222222222222222223, 2222222222222222223, 22222222222222222223, 222222222222222222223

Offset: 0

Paul Barry, Mar 24 2004

Second binomial transform of 2*A001045(3*n)/3 + (-1)^n.
Partial sums of A093136.
A convex combination of 10^n and 1.
In general the second binomial transform of k*Jacobsthal(3*n)/3 + (-1)^n is 1, 1+k, 1+11*k, 1+111*k, ... This is the case for k=2.
Essentially the same as A091628 (cf. 2nd formula). - Georg Fischer, Oct 06 2018
a(n) is 3^n represented in bijective base-3 numeration. - Alois P. Heinz, Aug 26 2019

Wikipedia, Bijective numeration.
Index entries for linear recurrences with constant coefficients, signature (11,-10).

Cf. A001045, A002279, A047855, A062397, A091628, A093136.

a(n) = (2*10^n + 7)/9.
a(n) = 10*a(n-1) - 7 (with a(0)=1). - Vincenzo Librandi, Aug 02 2010
From Elmo R. Oliveira, Apr 03 2025: (Start)
E.g.f.: exp(x)*(7 + 2*exp(9*x))/9.
a(n) = 11*a(n-1) - 10*a(n-2).
a(n) = (A062397(n) - A002279(n))/2. (End)

More terms from Elmo R. Oliveira, Apr 03 2025

cp's OEIS Frontend

A366669 a(n) = phi(10^n+1), where phi is Euler's totient function (A000010).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A291639 Numbers k such that 0 is the smallest decimal digit of k^3.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

A029797 Numbers k such that k^2 and k^3 have the same set of digits.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

PARI

A087000 Half length of periodic part of decimal expansion of 1/p for those primes having a periodic part of even length.

Views

Author

Keywords

Comments

Links

Crossrefs

Formula

A215614 Numbers k that are not multiples of 10 and such that sum of digits of k^2 is 7.

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

PARI

Extensions

A001271 Irregular table read by rows: row n lists prime factors of 10^n +1, with multiplicity.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Extensions

A093142 Expansion of g.f. (1-5*x)/((1-x)*(1-10*x)).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A098406 a(n) = (10^n + 17)/9.

Views

Author

Keywords

Comments

Examples

Links

A093142 Expansion of g.f. (1-5x)/((1-x)(1-10*x)).

A093135 Expansion of g.f. (1-8x)/((1-x)(1-10*x)).