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 1-8 of 8 results.

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

A095414 Excess of total number of distinct prime factor digits of n-th repunit over n, the number of digits of n-th repunit itself.

Original entry on oeis.org

-1, 0, 0, 1, 0, 2, 0, 2, 0, 1, 0, 3, 0, 1, 2, 3, 0, 3, 0, 3, 3, 1, 0, 4, 2, 2, 0, 4, 1, 6, 0, 5, 2, 3, 3, 4, 1, 1, 1, 5, 1, 6, 2, 4, 3, 3, 0, 5, 1, 4, 3, 4, 2, 4, 3, 6, 3, 3, 0, 9, 2, 1, 6, 6, 2, 5, 0, 6, 3, 5, 0, 6, 1, 3, 6, 3, 3, 5, 2, 7, 2, 3, 0, 10, 2, 4
Offset: 1

Views

Author

Labos Elemer, Jun 22 2004

Keywords

Comments

a(n) <= A095370(n) - 1 since the product of a k digit number and an m digit number has at least k+m-1 digits. - Chai Wah Wu, Nov 03 2019

Examples

			n=9: r9 = 111111111 = 3*3*37*333667, with a total of 9 digits among the distinct prime factors; the excess is a(9) = 9 - 9 = 0;
n=30: r30 = 111....1111 = 3*7*11*13*31*37*41*211*241*271*2161*9091*2906161, with a total of 36 digits among the distinct prime factors, so the excess a(30) = 36 - 30 = 6.

Links

Crossrefs

Cf. A002275, A095370, A095407, A095413, A095415, A095416, A095417, A095418.

Programs

  • Mathematica
    a[1] = -1; a[n_] := Total[IntegerLength /@ First /@ FactorInteger[(10^n - 1)/9]] - n; Array[a, 60] (* Giovanni Resta, Jul 16 2018 *)

Formula

a(n) = A095407(A002275(n)) - n = A095413(n) - n.

Extensions

Data corrected by Giovanni Resta, Jul 16 2018

A095408 Total number of decimal digits in all distinct prime factors of n minus number of digits in n.

Original entry on oeis.org

-1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 1, 0, 0, -1, 1, -1, 0, 0, 1, 0, -1, 1, 1, 0, 0, 0, 1, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, -1, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, -1, 1, 2, 0, 1, 1, 1, 0, 0, 0, 1, 0, 1, 1, 2, 0, 0, -1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 0, 0, 1, -1, 0, 1, 0, 0, 0, 0, 0, -1, 0, 1, 0, -1, 0, 1, 0, 0, 0, 0, 0, 0, -1
Offset: 1

Views

Author

Labos Elemer, Jun 21 2004

Keywords

Examples

			n=22: prime divisors are {2,11}, a(22) = 3-2 = 1.
n=63: prime divisors are {3,7}, a(63) = 2-2 = 0.
n=100: prime divisors are {2,5}, a(100) = 2-3 = -1.

Links

Crossrefs

Cf. A055642, A095407.

Programs

  • Mathematica
    ffi[x_] :=Flatten[FactorInteger[x]] lf[x_] :=Length[FactorInteger[x]] ba[x_] :=Table[Part[ffi[x], 2*j-1], {j, 1, lf[x]}] tdp[x_] :=Flatten[Table[IntegerDigits[Part[ba[x], j]], {j, 1, lf[x]}], 1] pl[x_] :=Length[tdp[x]] nl[x_] :=Length[IntegerDigits[x]] t1=Table[nl[w], {w, 1, 1000}];t2=Table[pl[w], {w, 1, 1000}];t2-t1
(* Second program: *)
Array[Total@ IntegerLength[FactorInteger[#][[All, 1]]] - IntegerLength@ # - Boole[# == 1] &, 108] (* Michael De Vlieger, Dec 16 2017 *)
  • PARI
    A095407(n) = vecsum(apply(p->#digits(p), factor(n)[, 1]));
A095408(n) = (A095407(n) - #digits(n)); \\ Antti Karttunen, Dec 16 2017

Formula

a(n) = A095407(n) - A055642(n).

A095411 Numbers k such that total number of decimal digits of all distinct prime factors of k is larger than the number of digits of k.

Original entry on oeis.org

6, 22, 26, 30, 33, 34, 38, 39, 42, 44, 46, 51, 52, 55, 57, 58, 60, 62, 65, 66, 68, 69, 70, 74, 76, 77, 78, 82, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95, 99, 102, 110, 114, 130, 132, 138, 143, 154, 156, 165, 170, 174, 182, 186, 187, 190, 195, 198, 202, 204, 206, 209
Offset: 1

Views

Author

Labos Elemer, Jun 21 2004

Keywords

Examples

			For k=55: 2 digits, prime set={5,11} with {5,1,1} digits, 3>2, so 55 is a term.

Links

Crossrefs

Cf. A055642, A095407, A095408, A095409, A095410.

Programs

  • Mathematica
    q[n_] := Total[IntegerLength /@ FactorInteger[n][[;; , 1]]] > IntegerLength[n]; q[1] = False; Select[Range[210], q] (* Amiram Eldar, Mar 25 2025 *)

Formula

Solutions to A095407(x) > A055642(x).

Extensions

Name corrected by Amiram Eldar, Mar 25 2025

A095409 Numbers k such that total number of decimal digits of all distinct prime factors of k is smaller than number of digits of k.

Original entry on oeis.org

1, 16, 25, 27, 32, 49, 64, 81, 100, 108, 112, 121, 125, 128, 135, 144, 147, 160, 162, 169, 175, 189, 192, 196, 200, 216, 224, 225, 243, 245, 250, 256, 288, 289, 320, 324, 343, 361, 375, 384, 392, 400, 405, 432, 441, 448, 486, 500, 512, 529, 567, 576, 625, 640
Offset: 1

Views

Author

Labos Elemer, Jun 21 2004

Keywords

Examples

			k = 100: prime set = {2,5}, 3 digits and 2 digits of prime factors, so 100 is a term.
k = 147: prime set = {3,7}, 3 digits and 2 digits of prime factors, so 147 is a term.

Links

Crossrefs

Cf. A055642, A095407, A095408, A095410, A095411.

Programs

  • Mathematica
    q[n_] := Total[IntegerLength /@ FactorInteger[n][[;; , 1]]] < IntegerLength[n]; q[1] = True; Select[Range[640], q] (* Amiram Eldar, Mar 25 2025 *)

Formula

Solutions to A095407(x) < A055642(x).

A095410 Numbers n such that total number of decimal digits of all distinct prime factors of n equals the number of digits of n.

Original entry on oeis.org

2, 3, 4, 5, 7, 8, 9, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 23, 24, 28, 29, 31, 35, 36, 37, 40, 41, 43, 45, 47, 48, 50, 53, 54, 56, 59, 61, 63, 67, 71, 72, 73, 75, 79, 80, 83, 89, 96, 97, 98, 101, 103, 104, 105, 106, 107, 109, 111, 113, 115, 116, 117, 118, 119, 120, 122
Offset: 1

Views

Author

Labos Elemer, Jun 21 2004

Keywords

Examples

			n=184, 3 digits,prime set={2,23} also with 3 digits {2,2,3}.

Links

Crossrefs

Cf. A055642, A095407, A095408, A095409, A095411.

Programs

  • Mathematica
    ffi[x_] :=Flatten[FactorInteger[x]] lf[x_] :=Length[FactorInteger[x]] ba[x_] :=Table[Part[ffi[x], 2*j-1], {j, 1, lf[x]}] tdp[x_] :=Flatten[Table[IntegerDigits[Part[ba[x], j]], {j, 1, lf[x]}], 1] pl[x_] :=Length[tdp[x]] nl[x_] :=Length[IntegerDigits[x]] t1=Table[nl[w], {w, 1, 1000}];t2=Table[pl[w], {w, 1, 1000}];t2-t1 Flatten[Position[t2-t1, 0]]
Rest[Select[Range[200],Length[Flatten[IntegerDigits/@Transpose[ FactorInteger[ #]][[1]]]]==IntegerLength[#]&]] (* Harvey P. Dale, Oct 22 2011 *)

Formula

Solutions to A095407[x]=A055642[x].

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
Showing 1-8 of 8 results.

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