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 1-3 of 3 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

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

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
Showing 1-3 of 3 results.

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

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