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 1-5 of 5 results.

A095415 Length of repunits of which the prime factor-digit-excess computed by A095414 equals 0.

Original entry on oeis.org

2, 3, 5, 7, 9, 11, 13, 17, 19, 23, 27, 31, 47, 59, 67, 71, 83, 113, 127, 139, 163, 197, 211, 229, 251, 263, 311, 317, 347, 421, 457, 461
Offset: 1

Views

Author

Labos Elemer, Jun 22 2004

Keywords

Comments

541, 701, 857 are also terms. Conjecture: Except for the number 4, A046413 is a subsequence. Conjecture: except for the prime powers 9 and 27, all terms are prime. - Chai Wah Wu, Nov 03 2019
Sequence continues as 467?, 479?, 509?, 541, 557?, 571?, 577?, 593?, 599?, 617?, 643?, 647?, 661?, 673?, 683?, 691?, 701, 727?, 743?, 751?, 757?, 769?, 773?, 821?, 857, 863?, 887?, 911?, 967?, 971?, 977?, 991?, where ? marks uncertain/candidate terms. - Max Alekseyev, Apr 29 2022

Crossrefs

A004023 is a subsequence.
Cf. A002275, A004023, A095370, A095407, A095413-A095418.

Programs

  • Mathematica
    d[1] = -1; d[n_] := Total[ IntegerLength /@ First /@ FactorInteger[(10^n - 1)/9]] - n; Select[ Range[67], d[#] == 0 &] (* Giovanni Resta, Jul 16 2018 *)

Formula

Solutions to A095414(x) = 0.

Extensions

Data corrected and extended by Giovanni Resta, Jul 16 2018
a(29)-a(32) confirmed by Max Alekseyev, Apr 29 2022

A095416 Length of smallest repunit of which the prime factor-digit-excess computed by A095414 equals n.

Original entry on oeis.org

2, 4, 6, 12, 24, 32, 30, 80, 96, 60, 84, 126, 120, 200, 168, 264, 210, 252
Offset: 0

Views

Author

Labos Elemer, Jun 22 2004

Keywords

Comments

a(18), a(19) > 322. a(20) = 300. - Giovanni Resta, Jul 19 2018
a(A095371(n)-1) >= A328899(n). a(18) <= 440, a(19) <= 336, a(21) <= 624, a(22) <= 560, a(23) <= 480, a(24) <= 540, a(25) <= 720, a(26) <= 612, a(27) <= 420, a(28) <= 600, a(30) = 1050, a(31) <= 660, a(32) <= 1400, a(33) <= 900, a(34) <= 1020, a(35) <= 1500, a(36) <= 1380, a(37) <= 840, a(48) <= 1260, a(50) <= 1680. - Chai Wah Wu, Nov 03 2019

Examples

			n=60: concatenated p-set for 60th-repunit is:
371113313741611012112412712161354190919901279612906161418890139526741,
its length=69, so excess=9, 60 is the smallest such repunit

Crossrefs

Cf. A002275, A095370, A095407, A095413-A095418.

Formula

a(n) = Min{x; A095414(x)=n}.

Extensions

Edited by Charles R Greathouse IV, Aug 03 2010
Data corrected and extended by Giovanni Resta, Jul 19 2018
a(0) from Chai Wah Wu, Nov 03 2019

A102146 a(n) = sigma(10^n - 1), where sigma(n) is the sum of positive divisors of n.

Original entry on oeis.org

13, 156, 1520, 15912, 148512, 2042880, 14508000, 162493344, 1534205464, 16203253248, 144451398000, 2063316971520, 14903272088640, 158269280832000, 1614847741624320, 17205180696931968, 144444514193267496
Offset: 1

Views

Author

Jun Mizuki (suzuki32(AT)sanken.osaka-u.ac.jp), Feb 14 2005

Keywords

Links

Crossrefs

Cf. A000203, A001270, A002283, A003020, A005422, A046053, A046107, A046412, A046415, A046416, A046417, A046418, A046419, A046420, A057951, A059892, A061075, A070528, A070529, A081317, A081318, A085035, A095370, A095413, A095414, A095417, A095418, A102347, A102380, A112505, A147556, A295503, A366669.

Programs

  • Mathematica
    DivisorSigma[1,10^Range[20]-1] (* Harvey P. Dale, Jan 05 2012 *)
  • PARI
    a(n) = sigma(10^n-1); \\ Michel Marcus, Apr 22 2017

Formula

a(n) = A000203(A002283(n)). - Ray Chandler, Apr 22 2017

A095413 Total number of decimal digits of all distinct prime factors of the n-th repunit.

Original entry on oeis.org

0, 2, 3, 5, 5, 8, 7, 10, 9, 11, 11, 15, 13, 15, 17, 19, 17, 21, 19, 23, 24, 23, 23, 28, 27, 28, 27, 32, 30, 36, 31, 37, 35, 37, 38, 40, 38, 39, 40, 45, 42, 48, 45, 48, 48, 49, 47, 53, 50, 54, 54, 56, 55, 58, 58, 62, 60, 61, 59, 69, 63, 63, 69, 70, 67, 71, 67
Offset: 1

Views

Author

Labos Elemer, Jun 22 2004

Keywords

Examples

			n=10: 10th repunit = 1111111111 = 11*41*271*9091; distinct prime factors have a total of 11 decimal digits, so a(10)=11.
n=27: 27th repunit = 111111111111111111111111111 = 3^3*37*757*333667*440334654777631, with 28 prime factor digits, a(27)=28.

Links

Crossrefs

Cf. A095414, A095415, A095416, A095417, A095418, A002275, A095407, A095370.

Programs

  • Mathematica
    a[1] = 0; a[n_] := Total[IntegerLength /@ First /@ FactorInteger[(10^n - 1) /9]]; Array[a, 70] (* Giovanni Resta, Jul 09 2018 *)
  • PARI
    a(n) = vecsum(apply(x->#Str(x), factor((10^n-1)/9)[,1])); \\ Michel Marcus, Jul 09 2018

Formula

a(n) = A095407(A002275(n)).
a(n) < A095370(n) + n. - Chai Wah Wu, Nov 04 2019

A095418 Excess of sum of all decimal digits of distinct prime factors for n-th repunit over corresponding digit-sum for repunit itself (which is n).

Original entry on oeis.org

-1, 0, 10, 0, 10, 20, 30, 17, 32, 26, 34, 35, 49, 53, 42, 51, 43, 74, 0, 56, 95, 77, 0, 81, 38, 94, 97, 106, 104, 80, 109, 123, 108, 96, 97, 132, 100, 65, 145, 136, 141, 184, 145, 173, 123, 99, 139, 172, 196, 120, 170, 176, 179, 213, 161, 169, 122, 201, 217, 184, 211, 216
Offset: 1

Views

Author

Labos Elemer, Jun 22 2004

Keywords

Examples

			n=60: concatenated distinct-prime factor-set for 60th-repunit is:
371113313741611012112412712161354190919901279612906161418890139526741,
its digit sum is 244, so a(60) = 244 - 60 = 184.
The value of this excess-sum is zero if n=2,4,19,23.

Links

Crossrefs

Cf. A095402, A002275, A095370, A095413, A095414, A095415, A095416, A095417.

Programs

  • Mathematica
    a[1] = -1; a[n_] := Total@ Flatten[IntegerDigits /@ First /@ FactorInteger[(10^n - 1)/9]] - n; Array[a, 62] (* Giovanni Resta, Jul 19 2018 *)

Formula

a(n) = A095402(A002275(n)) - n = A095417(n) - n.

Extensions

Data corrected by Giovanni Resta, Jul 19 2018
Showing 1-5 of 5 results.

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

Content is available under The OEIS End-User License Agreement.