Jeff Burch - cp's OEIS frontend

A181431 For n not divisible by 10 and using as many leading zeros as needed, smallest number whose inverse has only digits of n in its period.

9, 45, 3, 225, 18, 15, 142857, 1125, 11, 909, 825, 76923, 714285, 66, 61875, 588235294117647, 55, 52631578947368421, 47619, 4545, 434782608695652173913, 4125, 396, 384615, 37, 3571425, 344827586206896551724137931, 32258064516129

Offset: 1

Jeff Burch, Oct 19 2010

			a(13)=76923 because 1/76923=.000013 000013 000013... and is the smallest number whose inverse contains only 13 in its period.

A181098 Primefree centuries (i.e., numbers k such that no prime exists between 100k and 100k+99).

Original entry on oeis.org

16718, 26378, 31173, 39336, 46406, 46524, 51782, 55187, 58374, 58452, 60129, 60850, 63338, 63762, 67898, 69587, 71299, 75652, 78035, 78269, 80277, 83674, 84213, 89052, 95490, 97080, 100881, 101527, 103438, 105916, 111772, 112967

Offset: 1

Jeff Burch, Oct 02 2010

nonn

The first consecutive terms are 473267, 473268; see A190639. - M. F. Hasler, May 15 2011

			16718 is a term because there is no prime between 1671800 and 1671899.

T. D. Noe, Table of n, a(n) for n = 1..1000

Cf. A038822 (number of primes between 100n and 100n+99), A186311 (first occurrences).

Cf. A186393-A186408 (1 to 16 primes), A186509 (17 primes), A361723 (18 primes).

Flatten[Position[Differences[PrimePi[100*Range[0,113000]]],0]]-1 (* Harvey P. Dale, Dec 18 2021 *)

is(n)=nextprime(100*n)>100*n+99 \\ Charles R Greathouse IV, Apr 28 2015

a(n) = n + 100n/log n - O(n/log^2 n). - Charles R Greathouse IV, Sep 08 2017

A181086 Sorted version of A061075.

Original entry on oeis.org

3, 11, 13, 37, 101, 137, 271, 2161, 4649, 8779, 9091, 9901, 27961, 52579, 69857, 333667, 459691, 513239, 909091, 2906161, 5882353, 10838689, 39526741, 99990001, 121499449, 265371653, 1056689261, 1058313049, 5363222357, 5964848081

Offset: 1

Jeff Burch, Oct 01 2010

nonn

Warning: There is some doubt as to whether this sequence is correct. It would be good to have confirmation that the terms shown are correct and that there are no missing terms. - Editors of OEIS, May 01 2017

Cf. A061075.

A180351 Smallest m such that tau(m!) > 10^(10^n).

Original entry on oeis.org

5, 53, 928, 13999, 187661, 2344658, 28039129, 325695551

Offset: 0

Jeff Burch, Aug 29 2010

			a(0)=5 with tau(5!) = A000005(120) = 16 > 10^(10^0) = 10.
a(1)=53 with tau(53!) = A000005(427...0000) = 16174080000 > 10^(10^1) = 10^10.

min{m: A027423(m) > 10^(10^n)}.

a(n) = A098109(10^n).

A179952 Add 1 to all the divisors of n. a(n) = number of perfect squares in the set.

Original entry on oeis.org

0, 0, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 2, 1, 0, 1, 0, 0, 1, 0, 0, 3, 0, 0, 1, 0, 0, 2, 0, 1, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 2, 0, 0, 4, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0, 0, 2, 0, 0, 2, 1, 0, 1, 0, 0, 1, 1, 0, 3, 0, 0, 2, 0, 0, 1, 0, 2, 1, 0, 0, 1, 0, 0, 1, 1, 0, 2, 0, 0, 1, 0, 0, 4, 0, 0, 2, 0, 0, 1, 0, 1, 3

Offset: 1

Jeff Burch, Aug 03 2010

nonn

Number of k>=2 such that both k-1 and k+1 divide n. - Joerg Arndt, Jan 06 2015

			a(24)=3 because the divisors of 24 are 1,2,3,4,6,8,12,24. Adding one to each gives 2,3,4,5,7,9,13,25 and of those 4,9 and 25 are perfect squares.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

N:= 1000: # to get a(1) to a(N)
A:= Vector(N):
for n from 2 to floor(sqrt(N+1)) do
  for k from 1 to floor(N/(n^2-1)) do
      A[k*(n^2-1)]:= A[k*(n^2-1)]+1
   od
od;
convert(A,list); # Robert Israel, Jan 06 2015

a179952[n_] := Count[Sqrt[Divisors[#] + 1], Integer] & /@ Range@n; a179952[105] (* _Michael De Vlieger, Jan 06 2015 *)

a(n) = sumdiv(n, d, issquare(d+1)); \\ Michel Marcus, Jan 06 2015

G.f.: Sum_{n>=2} x^(n^2-1) / (1 - x^(n^2-1)). - Joerg Arndt, Jan 06 2015

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 3/4. - Amiram Eldar, Jan 19 2024

A174907 Write the prime factorization of the factorial numbers (p1^q1p2^q2...*pn^qn) where p1,p2... are the primes and q1,q2... are the exponents. The exponents that change with each successive factorial change to the numbers in the sequence.

Original entry on oeis.org

1, 1, 3, 1, 4, 2, 1, 7, 4, 8, 2, 1, 10, 5, 1, 11, 2, 6, 3, 15, 1, 16, 8, 1, 18, 4, 9, 3, 19, 2, 1, 22, 10, 6, 23, 2, 13, 25, 4, 1, 26, 14, 7, 1, 31, 15, 3, 32, 2, 8, 5, 34, 17, 1, 35, 2, 18, 3, 38, 9, 1, 39, 19, 6, 1, 41, 4, 21, 10, 42, 2, 1, 46, 22, 8, 47, 12, 23, 3, 49, 4, 1, 50, 26, 13, 5, 53

Offset: 2

Jeff Burch, Apr 01 2010

nonn

A177442 Smallest a(n) such that the prime factorization of a(n)! contains at least one factor to each exponent between 1 and n.

Original entry on oeis.org

2, 6, 15, 22, 55, 68, 85, 145, 261, 296, 333, 370, 492, 533, 590, 885, 944, 1005, 1143, 1207, 1278, 2831, 2980, 3048, 3175, 3302, 3429, 3576, 3725, 3874, 4023, 4775, 4966, 7368, 7582, 7805, 8289, 8606, 9036, 9036, 9287, 9599, 9599, 9824, 13902, 14233

Offset: 1

Jeff Burch, May 08 2010

nonn

			15! = 2^11 * 3^6 * 5^3 * 7^2 * 11 * 13 and is the smallest number whose prime factorization has at least one factor with exponent 1, one factor with exponent 2, and one factor with exponent 3, so a(3)=15.

Amiram Eldar, Table of n, a(n) for n = 1..276

Cf. A000142.

f[n_] := Module[{e = Union[FactorInteger[n!][[;; , 2]]]}, FirstPosition[Differences[e], ?(# > 1 &)][[1]]]; f[2] = f[3] = 1; seq[len] := Module[{s = Table[0, {len}], n = 1, c = 0, i}, While[c < len, n++; i = f[n]; Do[If[s[[j]] == 0, c++; s[[j]] = n], {j, 1, Min[i, len]}]]; s]; seq[40] (* Amiram Eldar, Apr 20 2023 *)

Corrected and extended by D. S. McNeil, May 22 2010

A139543 First n digits of 10^n!.

Original entry on oeis.org

3, 93, 402, 2846, 28242, 826393

Offset: 1

Jeff Burch, Jun 10 2008

A098120 Number of times n-th term of A025487 occurs in A073039.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 1, 4, 1, 3, 1, 2, 1, 1, 1, 2, 1, 1, 3, 1, 8, 4, 2, 1, 1, 1, 1, 4, 1, 2, 1, 1, 1, 1, 4, 1, 9, 2, 1, 4, 1, 6, 5, 1, 1, 4, 1, 3, 2, 1, 1, 1, 16, 2, 1, 1, 1, 4, 1, 3, 2, 1, 1, 1, 8, 2, 1, 4, 1, 1, 9, 5, 4, 1, 3, 8, 2, 6, 1, 1, 1, 8, 2, 1, 1, 1, 1, 3, 1, 4, 1, 12, 3, 2, 2, 1, 1, 1, 1, 8, 2, 1, 1, 1

Offset: 1

Jeff Burch, Sep 24 2004

nonn

Cf. A025487, A073039.

A098718 Position of n! in A025487.

Original entry on oeis.org

1, 2, 4, 8, 17, 34, 67, 125, 224, 391, 666, 1108, 1797, 2887, 4552, 7088, 10875, 16495, 24756, 36766, 54084, 78858, 114018, 163558, 232965, 329478, 462996, 646551, 897699, 1239395, 1702142, 2325845, 3162865, 4281304, 5769761, 7742941, 10348857, 13778106, 18275141

Offset: 1

Jeff Burch, Sep 29 2004

nonn

			A025487(34) = 720 = 6! so a(6) = 34. - _David A. Corneth_, Sep 19 2019

Cf. A000142, A025487, A098719.

A025487(a(n)) = n!. - Amiram Eldar, Jun 20 2019

More terms from Amiram Eldar, Jun 20 2019
a(35) corrected by Amiram Eldar, Jul 26 2019
a(36)-a(39) from David A. Corneth, Sep 19 2019

cp's OEIS Frontend

User: Jeff Burch

A181431 For n not divisible by 10 and using as many leading zeros as needed, smallest number whose inverse has only digits of n in its period.

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A181098 Primefree centuries (i.e., numbers k such that no prime exists between 100*k and 100*k+99).

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A181086 Sorted version of A061075.

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A180351 Smallest m such that tau(m!) > 10^(10^n).

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A179952 Add 1 to all the divisors of n. a(n) = number of perfect squares in the set.

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A174907 Write the prime factorization of the factorial numbers (p1^q1*p2^q2*...*pn^qn) where p1,p2... are the primes and q1,q2... are the exponents. The exponents that change with each successive factorial change to the numbers in the sequence.

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A177442 Smallest a(n) such that the prime factorization of a(n)! contains at least one factor to each exponent between 1 and n.

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Mathematica

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A139543 First n digits of 10^n!.

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A098120 Number of times n-th term of A025487 occurs in A073039.

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A098718 Position of n! in A025487.

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A181098 Primefree centuries (i.e., numbers k such that no prime exists between 100k and 100k+99).

A174907 Write the prime factorization of the factorial numbers (p1^q1p2^q2...*pn^qn) where p1,p2... are the primes and q1,q2... are the exponents. The exponents that change with each successive factorial change to the numbers in the sequence.