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 1-6 of 6 results.

A308968 Table, read by rows: row n contains the prime factors of A001008(n) (numerator of n-th harmonic number), with multiplicity.

Original entry on oeis.org

1, 3, 11, 5, 5, 137, 7, 7, 3, 11, 11, 761, 7129, 11, 11, 61, 97, 863, 13, 13, 509, 29, 43, 919, 1049, 1117, 29, 41233, 17, 17, 8431, 37, 1138979, 19, 19, 39541, 37, 7440427, 5, 11167027, 18858053, 3, 23, 23, 53, 227, 761, 583859, 5, 577, 467183, 109, 312408463
Offset: 1

Views

Author

M. F. Hasler, Jul 03 2019

Keywords

Comments

Row 1 is taken to be {1} instead of being empty, by convention.
Length, first = smallest and last = largest term of the rows are given in A308967, A308970 and A308971, respectively. See A308969 for prime divisors without repetition.

Examples

			   n | A001008(n) written as product of primes
-----+---------------------------------------------
   1 | 1 (empty product)
   2 | 3
   3 | 11
   4 | 5 * 5
   5 | 137
   6 | 7 * 7
   7 | 3 * 11 * 11
   8 | 761
   9 | 7129
  10 | 11 * 11 * 61
  11 | 97 * 863
  12 | 13 * 13 * 509
  13 | 29 * 43 * 919
  14 | 1049 * 1117
  15 | 29 * 41233
  16 | 17 * 17 * 8431
  17 | 37 * 1138979
  18 | 19 * 19 * 39541
  19 | 37 * 7440427
  20 | 5 * 11167027
etc.

Links

Crossrefs

Cf. A001008, A308967 (row lengths, for n > 1).
Cf. A308969 (prime divisors without repetition), A308970 (column 1 = first / smallest term of each row), A308971 (last / greatest term in each row).

Programs

  • PARI
    A308968_row(n)={if(n>1, concat(apply(f->vector(f[2],i,f[1]), Col(factor(A001008(n)))~)),[1])}

A308970 Smallest prime factor of A001008(n), numerator of n-th harmonic number; a(1) = 1.

Original entry on oeis.org

1, 3, 11, 5, 137, 7, 3, 761, 7129, 11, 97, 13, 29, 1049, 29, 17, 37, 19, 37, 5, 18858053, 3, 761, 5, 109, 34395742267, 521, 29, 43, 31, 109, 2917, 269, 3583, 397, 37, 10839223, 199, 737281, 41, 85691034670497533, 7, 140473, 109, 1553, 47, 911, 7, 23982193, 61, 227, 53, 941, 5953
Offset: 1

Views

Author

M. F. Hasler, Jul 03 2019

Keywords

Comments

Row 1 is taken to be {1} instead of being empty, by convention.
This is the first column of A308968 and A308969, which list the prime factors of A001008.

Examples

			   n | A001008(n) written as product of primes
-----+------------------------------------------
   1 | 1 (empty product)
   2 | 3
   3 | 11
   4 | 5 * 5
   5 | 137
   6 | 7 * 7
   7 | 3 * 11 * 11
   8 | 761
   9 | 7129
  10 | 11 * 11 * 61
  11 | 97 * 863
  12 | 13 * 13 * 509
  13 | 29 * 43 * 919
  14 | 1049 * 1117
  15 | 29 * 41233
  16 | 17 * 17 * 8431
  17 | 37 * 1138979
  18 | 19 * 19 * 39541
  19 | 37 * 7440427
  20 | 5 * 11167027
etc., therefore this sequence = 1, 3, 11, 5, 137, 7, 3, 761, ...

Links

Crossrefs

Cf. A001008.
Cf. A308967 (number of prime factors), A308968 (table of factorization), A308969 (table of prime divisors), A308971 (greatest prime factor) of A001008(n).

Programs

  • Mathematica
    FactorInteger[#][[1,1]]&/@Numerator[HarmonicNumber[Range[60]]] (* Harvey P. Dale, Jul 02 2025 *)
  • PARI
    row(n)={if(n>1, factor(A001008(n))[1,1], 1)}

A308971 Largest prime factor of A001008(n), numerator of n-th harmonic number; a(1) = 1.

Original entry on oeis.org

1, 3, 11, 5, 137, 7, 11, 761, 7129, 61, 863, 509, 919, 1117, 41233, 8431, 1138979, 39541, 7440427, 11167027, 18858053, 227, 583859, 467183, 312408463, 34395742267, 215087, 375035183, 4990290163, 17783, 2667653736673, 535919, 199539368321, 15088528003, 137121586897
Offset: 1

Views

Author

M. F. Hasler, Jul 03 2019

Keywords

Comments

Initial terms coincide with A120299 = greatest prime factor of Stirling numbers of first kind A000254. They differ when the unreduced denominator of H(n), equal to n!, is divisible by this factor, i.e., A120299(n) <= n. Can this ever happen?

Examples

			   n | A001008(n) written as product of primes
-----+------------------------------------------
   1 | 1 (empty product)
   2 | 3
   3 | 11
   4 | 5 * 5
   5 | 137
   6 | 7 * 7
   7 | 3 * 11 * 11
   8 | 761
   9 | 7129
  10 | 11 * 11 * 61
  11 | 97 * 863
  12 | 13 * 13 * 509
  13 | 29 * 43 * 919
  14 | 1049 * 1117
  15 | 29 * 41233
  16 | 17 * 17 * 8431
  17 | 37 * 1138979
  18 | 19 * 19 * 39541
  19 | 37 * 7440427
  20 | 5 * 11167027
etc., therefore this sequence = 1, 3, 11, 5, 137, 7, 11, 761, 7129, 61, ...

Links

Crossrefs

Cf. A001008, A006530.
Cf. A308967 (number of prime factors), A308968 (table of factorization), A308969 (table of prime divisors), A308970 (smallest prime factor) of A001008(n).

Programs

  • Mathematica
    Array[FactorInteger[Numerator@HarmonicNumber[#]][[-1, 1]] &, 35] (* Michael De Vlieger, Jul 04 2019 *)
  • PARI
    a(n)={if(n>1, factor(A001008(n))[1,1], 1)}

Formula

a(n) = A006530(A001008(n)). - Amiram Eldar, Feb 24 2020

A308969 Table, read by rows: row n contains the prime divisors of A001008 (numerator of n-th harmonic number), without repetitions.

Original entry on oeis.org

1, 3, 11, 5, 137, 7, 3, 11, 761, 7129, 11, 61, 97, 863, 13, 509, 29, 43, 919, 1049, 1117, 29, 41233, 17, 8431, 37, 1138979, 19, 39541, 37, 7440427, 5, 11167027, 18858053, 3, 23, 53, 227, 761, 583859, 5, 577, 467183, 109, 312408463
Offset: 1

Views

Author

M. F. Hasler, Jul 03 2019

Keywords

Comments

Row 1 is taken to be {1} instead of being empty, by convention.

Examples

			   n | A001008(n) written as product of primes
-----+---------------------------------------------
   1 | 1 (empty product)
   2 | 3
   3 | 11
   4 | 5 * 5   (So 5 is the only prime divisor, and row(4) = {5}.)
   5 | 137
   6 | 7 * 7
   7 | 3 * 11 * 11       whence row(7) = {3, 11}.
   8 | 761
   9 | 7129
  10 | 11 * 11 * 61      whence row(10) = {11, 61}.
  11 | 97 * 863
  12 | 13 * 13 * 509     whence row(16) = {13, 509}.
  13 | 29 * 43 * 919     whence row(13) = {29, 43, 919}.
  14 | 1049 * 1117
  15 | 29 * 41233
  16 | 17 * 17 * 8431    whence row(16) = {17, 8431}.
  17 | 37 * 1138979
  18 | 19 * 19 * 39541   whence row(18) = {19, 39541}.
  19 | 37 * 7440427
  20 | 5 * 11167027
etc.

Crossrefs

Cf. A001008.
Cf. A308967 (number of prime factors), A308968 (table of factorization), A308970 & A308971 (smallest & greatest prime factor) of A001008(n).

Programs

  • Mathematica
    Table[FactorInteger[Numerator[HarmonicNumber[n]]][[All,1]],{n,30}]// Flatten (* Harvey P. Dale, Sep 14 2020 *)
  • PARI
    row(n)={if(n>1, factor(A001008(n))[,1]~, [1])}

A326227 Indices of nonsquarefree numerators (A001008) of harmonic numbers H(n) = Sum_{k=1..n} 1/k.

Original entry on oeis.org

4, 6, 7, 10, 12, 16, 18, 22, 28, 29, 30, 36, 40, 42, 46, 52, 58, 60, 66, 70, 72, 78, 82, 88, 96, 100, 102, 106, 108, 112, 126, 130, 136, 138, 148, 150, 156, 162, 166, 172, 178, 180, 190, 192, 196, 198, 210, 222, 226, 228, 232, 238, 240, 250, 256, 262, 268, 270, 276
Offset: 1

Views

Author

M. F. Hasler, Jul 03 2019

Keywords

Comments

It appears that the first term of A001008 having a cubic factor is A001008(848) = 11^3 * 1871 * C359.
By Wolstenholme's Theorem, p^2 divides A001008(p-1) whenever p >= 5 is prime (cf. A076637); see A308968 for illustration. Therefore, A006093 \ {1, 2} (primes - 1) is a subsequence. (Thanks to Bernard Schott.)

Crossrefs

Cf. A001008, A076637, A322434.
Cf. A308967 (number of prime factors), A308968 (table of factorization), A308969 (table of prime divisors), A308970 & A308971 (smallest & largest prime factor) of A001008(n).

Programs

  • PARI
    is_A326227(n)={n>3&&vecmax(factor(A001008(n))[,2])>1} \\ Add ,0 in factor() for much faster but possibly incorrect results [false negative].
for(n=1,oo, is_A326227(n) && print1(n","))

A349049 Number of prime factors (with multiplicity) of the denominator of the harmonic number H(n) = Sum_{k=1..n} 1/k.

Original entry on oeis.org

0, 1, 2, 3, 4, 3, 4, 5, 7, 7, 8, 8, 9, 9, 9, 10, 11, 10, 11, 10, 9, 9, 10, 11, 13, 13, 15, 15, 16, 16, 17, 18, 17, 17, 17, 17, 18, 18, 18, 18, 19, 18, 19, 20, 20, 20, 21, 21, 23, 23, 23, 23, 24, 23, 23, 23, 23, 23, 24, 24, 25, 25, 24, 25, 25, 24, 25, 25, 26, 26, 27, 28, 29, 29, 29, 29, 28
Offset: 1

Views

Author

Kam Kong, Nov 07 2021

Keywords

Crossrefs

Cf. A001222, A002805, A308967.

Programs

  • PARI
    my(h=0); for(n=1,77,h+=1/n;print1(bigomega(denominator(h)),", ")); \\ Joerg Arndt, Nov 07 2021
  • Python
    from sympy import harmonic, factorint
def a(n): return sum(factorint(harmonic(n).denominator).values())
print([a(n) for n in range(1, 78)]) # Michael S. Branicky, Nov 07 2021
  • SageMath
    [sloane.A001222(A002805(n)) for n in range(1, 78)]

Formula

a(n) = A001222(A002805(n)).
Showing 1-6 of 6 results.

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