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 1-4 of 4 results.

A340158 Numbers m such that m, m + 1, m + 2, m + 3 and m + 4 have k, 2k, 3k, 4k and 5k divisors respectively.

Original entry on oeis.org

211082, 2364062, 2774165, 3379802, 3743573, 4390682, 5651042, 5845442, 6708578, 7326122, 7371482, 8566394, 8839202, 9056282, 10154642, 10301333, 10325621, 10446242, 10540202, 11238341, 11719562, 11978762, 12377282, 12871058, 13456202, 16840058, 16954562, 17155141
Offset: 1

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Author

Jaroslav Krizek, Dec 29 2020

Keywords

Comments

Numbers m such that tau(m) = tau(m + 1)/2 = tau(m + 2)/3 = tau(m + 3)/4 = tau(m + 4)/5, where tau(k) = the number of divisors of k (A000005).
Quintuples of [tau(a(n)), tau(a(n) + 1), tau(a(n) + 2), tau(a(n) + 3), tau(a(n) + 4)] = [tau(a(n)), 2*tau(a(n)), 3*tau(a(n)), 4*tau(a(n)), 5*tau(a(n))]: [4, 8, 12, 16, 20], [4, 8, 12, 16, 20], [4, 8, 12, 16, 20], [8, 16, 24, 32, 40], [4, 8, 12, 16, 20], [4, 8, 12, 16, 20], ...
Corresponding values of numbers k: 4, 4, 4, 8, 4, 4, 4, 4, 4, 8, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 8, 4, 4, 4, 4, 4, 4, 4, ...
1524085621 is the smallest prime term (see A294528).
Subsequence of A063446, A339778 and A340157.

Examples

			tau(211082) = 4, tau(211083) = 8, tau(211084) = 12, tau(211085) = 16, tau(211086) = 20.

Crossrefs

Cf. A000005, A063446, A294528, A339778, A340157, A340159.

Programs

  • Magma
    [m: m in [1..10^6] | #Divisors(m) eq #Divisors(m + 1)/2 and #Divisors(m) eq #Divisors(m + 2)/3 and #Divisors(m) eq #Divisors(m + 3)/4 and #Divisors(m) eq #Divisors(m + 4)/5]
  • Mathematica
    Select[Range[5*10^6], Equal @@ (DivisorSigma[0, # + {0, 1, 2, 3, 4}]/{1, 2, 3, 4, 5}) &] (* Amiram Eldar, Dec 30 2020 *)
  • PARI
    isok(m) = my(k = numdiv(m)); (numdiv(m+1) == 2*k) && (numdiv(m+2) == 3*k) && (numdiv(m+3) == 4*k) && (numdiv(m+4) == 5*k); \\ Michel Marcus, Jan 16 2021

A340157 Numbers m such that numbers m, m + 1, m + 2 and m + 3 have k, 2k, 3k and 4k divisors respectively.

Original entry on oeis.org

421, 3013, 5029, 5223, 5245, 5893, 6487, 10533, 11911, 14677, 17173, 23077, 23573, 24613, 25141, 25213, 27637, 27973, 28357, 30661, 32407, 34117, 37477, 38282, 39751, 43495, 45973, 47365, 48423, 50821, 50965, 53413, 53989, 54421, 55141, 56103, 57877, 58165
Offset: 1

Views

Author

Jaroslav Krizek, Dec 29 2020

Keywords

Comments

Numbers m such that tau(m) = tau(m + 1)/2 = tau(m + 2)/3 = tau(m + 3)/4, where tau(k) = the number of divisors of k (A000005).
Quadruplets of [tau(a(n)), tau(a(n) + 1), tau(a(n) + 2), tau(a(n) + 3)] = [tau(a(n)), 2*tau(a(n)), 3*tau(a(n)), 4*tau(a(n))]: [2, 4, 6, 8], [4, 8, 12, 16], [4, 8, 12, 16], [4, 8, 12, 16], [4, 8, 12, 16], [4, 8, 12, 16], [4, 8, 12, 16], ...
Corresponding values of tau(a(n)): 2, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 2, 4, 4, 4, 4, 4, 4, 4, 4, 4, ...
Subsequence of A063446 and A339778. Supersequence of A340158.
Prime terms (primes p such that p, p + 1, p + 2 and p + 3 have 2, 4, 6 and 8 divisors respectively): 421, 30661, 50821, 54421, 130021, 195541, 423781, 635461, 1003381, 1577941, 1597381, 1883941, ...

Examples

			tau(421) = 2, tau(422) = 4, tau(423) = 6, tau(424) = 8.

Crossrefs

Cf. A000005, A063446, A294528, A303578, A339778, A340158, A340159.

Programs

  • Magma
    [m: m in [1..10^5] | #Divisors(m) eq #Divisors(m + 1)/2 and #Divisors(m) eq #Divisors(m + 2)/3 and #Divisors(m) eq #Divisors(m + 3)/4]
  • Mathematica
    Select[Range[60000], Equal @@ (DivisorSigma[0, # + {0, 1, 2, 3}]/{1, 2, 3, 4}) &] (* Amiram Eldar, Dec 30 2020 *)
  • PARI
    isok(m, n=4) = {my(k=numdiv(m)); for (i=1, n-1, if (numdiv(m+i) != (i+1)*k, return (0));); return(1);} \\ Michel Marcus, Dec 30 2020

A341213 a(n) is the smallest number m such that numbers m, m - 1, m - 2, ..., m - n + 1 have k, 2*k, 3*k, ..., n*k divisors respectively.

Original entry on oeis.org

1, 7, 47, 1019, 154379, 59423129, 3100501318, 126544656838
Offset: 1

Views

Author

Jaroslav Krizek, Feb 07 2021

Keywords

Comments

a(n) is the smallest number m such that tau(m) = tau(m - 1)/2 = tau(m - 2)/3 = tau(m - 3)/4 = ... = tau(m - n + 1)/n, where tau(k) = the number of divisors of k (A000005).
Corresponding values of numbers k: 1, 2, 2, 2, 4, 4, 4, 4, ...

Examples

			a(3) = 47 because 45, 46 and 47 have 6, 4, and 2 divisors respectively and there is no smaller number having this property.

Crossrefs

Cf. A341214 (similar sequence with primes).
Cf. A000005, A340158, A340159, A341212.

Programs

  • Python
    def tau(n): # A000005
    d, t = 1, 0
    while d*d < n:
        if n%d == 0:
            t = t+2
        d = d+1
    if d*d == n:
        t = t+1
    return t
n, a = 1, 1 # corrected by Martin Ehrenstein, Apr 14 2021
while n > 0:
    nn, t1 = 1, tau(a)
    while nn < n and tau(a-nn) == (nn+1)*t1:
        nn = nn+1
    if nn == n:
        print(n,a)
        n = n+1
    a = a+1 # A.H.M. Smeets, Feb 07 2021

Extensions

a(6) from Amiram Eldar, Feb 07 2021
a(7) from Jinyuan Wang, Feb 08 2021
a(1) corrected and extended with a(8) by Martin Ehrenstein, Apr 14 2021

A341212 Numbers m such that m, m - 1, m - 2, m - 3 and m - 4 have k, 2k, 3k, 4k and 5k divisors respectively.

Original entry on oeis.org

154379, 1075198, 4211518, 4700758, 4745227, 5954379, 6036043, 6330235, 6485998, 6524878, 6851227, 7846798, 8536027, 8556358, 11718598, 12100027, 12126838, 13584838, 14869379, 15320587, 16934998, 17074379, 18154379, 18904027, 19013129, 19774379, 19779995
Offset: 1

Views

Author

Jaroslav Krizek, Feb 07 2021

Keywords

Comments

Numbers m such that tau(m) = tau(m - 1)/2 = tau(m - 2)/3 = tau(m - 3)/4 = tau(m - 4)/5, where tau(k) = the number of divisors of k (A000005).
Corresponding values of numbers k: 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, ...
First prime term 55414379 (= A341214(5)) of this sequence is the smallest prime p such that p, p - 1, p - 2, p - 3 and p - 4 have 2, 4, 6, 8 and 10 divisors respectively.

Examples

			tau(154375) = 20, tau(154376) = 16, tau(154377) = 12, tau(154378) = 8, tau(154379) = 4.

Links

Crossrefs

Cf. A000005, A340158, A340159, A341213, A341214.

Programs

  • Magma
    [m: m in [5..10^6] | #Divisors(m - 1) eq 2*#Divisors(m) and #Divisors(m - 2) eq 3*#Divisors(m) and #Divisors(m - 3) eq 4*#Divisors(m) and #Divisors(m - 4) eq 5*#Divisors(m)]
  • Mathematica
    seq[max_, n_] := Module[{d = DivisorSigma[0, Range[n]], s = {}}, Do[If[Length @ Union[d/Range[n, 1, -1]] == 1, AppendTo[s, k - 1]]; d = Join[Rest@d, {DivisorSigma[0, k]}], {k, n + 1, max}]; s]; seq[5*10^6, 5] (* Amiram Eldar, Feb 08 2021 *)
  • PARI
    isok(m) = if (m>5, my(nb=numdiv(m)); (numdiv(m-1) == 2*nb) && (numdiv(m-2) == 3*nb) && (numdiv(m-3) == 4*nb) && (numdiv(m-4) == 5*nb)); \\ Michel Marcus, Apr 01 2021
  • Python
    def tau(n): # A000005
    d, t = 1, 0
    while d*d < n:
        if n%d == 0:
            t = t+2
        d = d+1
    if d*d == n:
        t = t+1
    return t
n, a = 1, 2
while n <= 27:
    nn, t1 = 1, tau(a)
    while nn < 5 and tau(a-nn) == (nn+1)*t1:
        nn = nn+1
    if nn == 5:
        print(n,a)
        n = n+1
    a = a+1 # A.H.M. Smeets, Feb 07 2021
Showing 1-4 of 4 results.

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