A095418 -id:A095418 - cp's OEIS frontend

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

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

Offset: 1

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

nonn

Max Alekseyev, Table of n, a(n) for n = 1..352 (terms 1..322 from Ray Chandler)
C. Caldwell, Sigma function.

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.

DivisorSigma[1,10^Range[20]-1] (* Harvey P. Dale, Jan 05 2012 *)

a(n) = sigma(10^n-1); \\ Michel Marcus, Apr 22 2017

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

Labos Elemer, Jun 22 2004

			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.

Max Alekseyev, Table of n, a(n) for n = 1..352 (first 322 terms from Giovanni Resta)

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

a[1] = 0; a[n_] := Total[IntegerLength /@ First /@ FactorInteger[(10^n - 1) /9]]; Array[a, 70] (* Giovanni Resta, Jul 09 2018 *)

a(n) = vecsum(apply(x->#Str(x), factor((10^n-1)/9)[,1])); \\ Michel Marcus, Jul 09 2018

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

Labos Elemer, Jun 22 2004

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

			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.

Max Alekseyev, Table of n, a(n) for n = 1..352 (first 322 terms from Giovanni Resta)

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

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

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

Data corrected by Giovanni Resta, Jul 16 2018

A095417 Sum of all decimal digits of distinct prime factors for n-th repunit.

Original entry on oeis.org

0, 2, 13, 4, 15, 26, 37, 25, 41, 36, 45, 47, 62, 67, 57, 67, 60, 92, 19, 76, 116, 99, 23, 105, 63, 120, 124, 134, 133, 110, 140, 155, 141, 130, 132, 168, 137, 103, 184, 176, 182, 226, 188, 217, 168, 145, 186, 220, 245, 170, 221, 228, 232, 267, 216, 225, 179, 259

Offset: 1

Labos Elemer, Jun 22 2004

			n=60: concatenated distinct-prime factor-set for 60th-repunit is:
371113313741611012112412712161354190919901279612906161418890139526741,
its digit sum=a(60)=244.

Max Alekseyev, Table of n, a(n) for n = 1..352 (first 322 terms from Giovanni Resta)

Cf. A095402, A002275, A095370, A095413-A095418.

a[1]=0; a[n_] := Total[ Flatten[ IntegerDigits /@ First /@ FactorInteger[(10^n - 1)/9]]]; Array[a, 60] (* Giovanni Resta, Jul 19 2018 *)

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

Data corrected by Giovanni Resta, Jul 19 2018

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

Labos Elemer, Jun 22 2004

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

A004023 is a subsequence.

Cf. A002275, A004023, A095370, A095407, A095413-A095418.

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

Solutions to A095414(x) = 0.

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

Labos Elemer, Jun 22 2004

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

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

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

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

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

cp's OEIS Frontend

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

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

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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.

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A095417 Sum of all decimal digits of distinct prime factors for n-th repunit.

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

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

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