cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-7 of 7 results.

A002280 a(n) = 6*(10^n - 1)/9.

Original entry on oeis.org

0, 6, 66, 666, 6666, 66666, 666666, 6666666, 66666666, 666666666, 6666666666, 66666666666, 666666666666, 6666666666666, 66666666666666, 666666666666666, 6666666666666666, 66666666666666666, 666666666666666666, 6666666666666666666, 66666666666666666666, 666666666666666666666
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

a(n-1) = number of Fibonacci numbers F(k), k <= 10^n, which end in 0. a(1)=6 because there are 6 Fibonacci numbers up to 10^2 which end in 0. - Shyam Sunder Gupta and Benoit Cloitre, Aug 15 2002
a(n) is the total number of holes in a certain triangle fractal (start with 10 triangles, 6 holes) after n iterations. See illustration in links. - Kival Ngaokrajang, Feb 21 2015

Links

Crossrefs

Different from A072912. Cf. A073548, etc.
Cf. A002275, A002276, A002277, A002278, A002279, A002281, A002282, A002283, A075415, A178631, A178633.
Cf. A010785, A017233, A073551.

Programs

Formula

a(n) = 6*A002275(n).
From Jaume Oliver Lafont, Feb 03 2009: (Start)
G.f.: 6*x/((1-x)*(1-10*x)).
a(n) = 11*a(n-1) - 10*a(n-2) with a(0)=0, a(1)=6. (End)
a(n) = A178633(n)/A002283(n). - Reinhard Zumkeller, May 31 2010
a(n) = a(n-1) + 6*10^(n-1) with a(0)=0. - Vincenzo Librandi, Jul 22 2010
E.g.f.: 2*exp(x)*(exp(9*x) - 1)/3. - Stefano Spezia, Sep 13 2023
From Elmo R. Oliveira, Jul 21 2025: (Start)
a(n) = A073551(n+1)/2 for n >= 1.
a(n) = A010785(A017233(n-1)) for n >= 1. (End)

A073551 Number of Fibonacci numbers F(k), k <= 10^n, which end in 3.

Original entry on oeis.org

2, 12, 132, 1332, 13332, 133332, 1333332, 13333332, 133333332, 1333333332, 13333333332, 133333333332, 1333333333332, 13333333333332, 133333333333332, 1333333333333332, 13333333333333332, 133333333333333332, 1333333333333333332, 13333333333333333332, 133333333333333333332, 1333333333333333333332, 13333333333333333333332
Offset: 1

Views

Author

Shyam Sunder Gupta, Aug 15 2002

Keywords

Comments

These numbers also have many palindromic divisors. - Jason Earls, Nov 28 2009

Examples

			a(2) = 12 because there are 12 Fibonacci numbers up to 10^2 which end in 3.

References

  • Jason Earls, "Palindions," Mathematical Bliss, Pleroma Publications, 2009, pages 115-120. ASIN: B002ACVZ6O.

Links

Crossrefs

Cf. A000045 (Fibonacci numbers), A073548.

Programs

  • Mathematica
    LinearRecurrence[{11, -10}, {2, 12, 132}, 25] (* Paolo Xausa, Aug 27 2025 *)

Formula

If n>1 then a(n) = (2*10^n - 20)/15. - Robert Gerbicz, Sep 06 2002
From Elmo R. Oliveira, Jul 21 2025: (Start)
G.f.: 2*x*(1 - 5*x + 10*x^2)/((1-x)*(1-10*x)).
E.g.f.: 2*(9 + 15*x - 10*exp(x) + exp(10*x))/15.
a(n) = 2*A073548(n).
a(n) = 11*a(n-1) - 10*a(n-2) for n >= 4. (End)

Extensions

More terms from Robert Gerbicz, Sep 06 2002

A073505 Number of primes == 1 (mod 10) less than 10^n.

Original entry on oeis.org

0, 5, 40, 306, 2387, 19617, 166104, 1440298, 12711386, 113761519, 1029517130, 9401960980, 86516370000
Offset: 1

Views

Author

Shyam Sunder Gupta, Aug 14 2002

Keywords

Comments

Also Pi(n,5,1)
This and the related sequences A073505-A073517 and A006880, A073548-A073565 are included because there is interest in the distribution of primes by their initial or final digits.

Examples

			a(2) = 5 because there are 5 primes == 1 (mod 10) less than 10^2. They are 11, 31, 41, 61 and 71.

Links

Crossrefs

Cf. A006880, A087630, A073506, A073507, A073508, A073509, A073510, A073511, A073512, A073513, A073514, A073515, A073516, A073517.

Programs

  • Mathematica
    c = 0; k = 1; Do[While[k < 10^n, If[PrimeQ[k], c++ ]; k += 10]; Print[c], {n, 1, 10}]

Formula

a(n) + A073506(n) + A073507(n) + A073508(n) + 2 = A006880(n).

Extensions

Edited by Robert G. Wilson v, Oct 03 2002
a(10) from Robert G. Wilson v, Dec 22 2003
a(11)-a(13) from Giovanni Resta, Aug 07 2018

A073506 Number of primes == 3 (mod 10) less than 10^n.

Original entry on oeis.org

1, 7, 42, 310, 2402, 19665, 166230, 1440474, 12712499, 113765625, 1029509448, 9401979904, 86516427946
Offset: 1

Views

Author

Shyam Sunder Gupta, Aug 14 2002

Keywords

Comments

Also Pi(n,5,3)
This and the related sequences A073505-A073517 and A006880, A073548-A073565 are included because there is interest in the distribution of primes by their initial or final digits.

Examples

			a(2)=7 because there are 7 primes == 3 (mod 10) less than 10^2. They are 3, 13, 23, 43, 53, 73 and 83.

Links

Crossrefs

Cf. A006880, A087631, A073505, A073507, A073508, A073509, A073510, A073511, A073512, A073513, A073514, A073515, A073516, A073517.

Programs

  • Mathematica
    c = 0; k = 3; Do[While[k < 10^n, If[PrimeQ[k], c++ ]; k += 10]; Print[c], {n, 1, 10}]

Formula

A073505(n) + a(n) + A073507(n) + A073508(n) + 2 = A006880(n).

Extensions

Edited by Robert G. Wilson v, Oct 03 2002
a(10) from Robert G. Wilson v, Dec 22 2003
a(11)-a(13) from Giovanni Resta, Aug 07 2018

A073507 Number of primes == 7 (mod 10) less than 10^n.

Original entry on oeis.org

1, 6, 46, 308, 2411, 19621, 166211, 1440495, 12712314, 113764039, 1029518337, 9401997000, 86516367790
Offset: 1

Views

Author

Shyam Sunder Gupta, Aug 14 2002

Keywords

Comments

This and the related sequences A073505-A073517 and A006880, A073548-A073565 are included because there is interest in the distribution of primes by their initial or final digits.

Examples

			a(2)=6 because there are 6 primes == 7 (mod 10) less than 10^2. They are 7, 17, 37, 47, 67 and 97.

Crossrefs

Cf. A006880, A087632, A073505, A073506, A073508, A073509, A073510, A073511, A073512, A073513, A073514, A073515, A073516, A073517.

Programs

  • Mathematica
    c = 0; k = 7; Do[While[k < 10^n, If[PrimeQ[k], c++ ]; k += 10]; Print[c], {n, 1, 10}]

Formula

A073505(n) + A073506(n) + a(n) + A073508(n) + 2 = A006880(n).

Extensions

Edited by Robert G. Wilson v, Oct 03 2002
a(10) from Robert G. Wilson v, Dec 22 2003
a(11)-a(13) from Giovanni Resta, Aug 07 2018

A073508 Number of primes == 9 (mod 10) less than 10^n.

Original entry on oeis.org

0, 5, 38, 303, 2390, 19593, 166032, 1440186, 12711333, 113761326, 1029509896, 9401974132, 86516371101
Offset: 1

Views

Author

Shyam Sunder Gupta, Aug 14 2002

Keywords

Comments

This and the related sequences A073505-A073517 and A006880, A073548-A073565 are included because there is interest in the distribution of primes by their initial or final digits.

Examples

			a(2) = 5 because there are 5 primes == 9 (mod 10) less than 10^2. They are 19, 29, 59, 79 and 89.

Links

Crossrefs

Cf. A006880, A087633, A073505, A073506, A073507, A073509, A073510, A073511, A073512, A073513, A073514, A073515, A073516, A073517.

Programs

  • Mathematica
    c = 0; k = 9; Do[While[k < 10^n, If[PrimeQ[k], c++ ]; k += 10]; Print[c], {n, 1, 10}]

Formula

A073505(n) + A073506(n) + A073507(n) + a(n) + 2 = A006880(n).

Extensions

Edited by Robert G. Wilson v, Oct 03 2002
a(10) from Robert G. Wilson v, Dec 22 2003
a(11)-a(13) from Giovanni Resta, Aug 07 2018

A073554 Number of Fibonacci numbers F(k), k <= 10^n, which end in 7.

Original entry on oeis.org

0, 14, 134, 1334, 13334, 133334, 1333334, 13333334, 133333334, 1333333334, 13333333334, 133333333334, 1333333333334, 13333333333334, 133333333333334, 1333333333333334, 13333333333333334, 133333333333333334, 1333333333333333334, 13333333333333333334, 133333333333333333334, 1333333333333333333334, 13333333333333333333334
Offset: 1

Views

Author

Shyam Sunder Gupta, Aug 15 2002

Keywords

Examples

			a(2) = 14 because there are 14 Fibonacci numbers up to 10^2 which end in 7.

Links

Crossrefs

Cf. A073548 (end in 2), A073549 (6), A073550 (1), A073551 (3), (A073552 (4)), A073553 (5), this sequence (7), A073555 (8), A073556 (9).
Cf. A000045, A067275.

Programs

  • Mathematica
    Join[{0},Table[10 FromDigits[PadRight[{1},n,3]]+4,{n,30}]] (* Harvey P. Dale, Mar 29 2023 *)

Formula

If n>1 then a(n) = (2*10^n + 10)/15. - Robert Gerbicz, Sep 06 2002
a(n) = A073550(n) for n >= 3. - Georg Fischer, Oct 13 2022
From Elmo R. Oliveira, Jul 22 2025: (Start)
G.f.: 2*x^2*(7 - 10*x)/((1-x)*(1-10*x)).
E.g.f.: 2*(-6 - 15*x + 5*exp(x) + exp(10*x))/15.
a(n) = 2*A067275(n) for n >= 2.
a(n) = 11*a(n-1) - 10*a(n-2) for n > 3. (End)

Extensions

More terms from Robert Gerbicz, Sep 06 2002
Showing 1-7 of 7 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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