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 1-5 of 5 results.

A333317 Partial sums of A248577.

Original entry on oeis.org

0, 2, 4, 7, 9, 17, 19, 23, 26, 34, 36, 48, 50, 58, 66, 71, 73, 85, 87, 99, 107, 115, 117, 133, 136, 144, 148, 160, 162, 186, 188, 194, 202, 210, 218, 236, 238, 246, 254, 270, 272, 296, 298, 310, 322, 330, 332, 352, 355, 367, 375, 387, 389, 405, 413, 429, 437
Offset: 1

Views

Author

Amiram Eldar, Mar 14 2020

Keywords

References

  • Jean-Marie De Koninck and Aleksandar Ivić, Topics in Arithmetical Functions: Asymptotic Formulae for Sums of Reciprocals of Arithmetical Functions and Related Fields, North-Holland, 1980, pp. 233-235.
  • Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 161.

Links

Crossrefs

Cf. A000005, A001221, A077761, A085548, A248577.

Programs

  • Mathematica
    f[n_] := DivisorSigma[0, n] * PrimeNu[n]; Accumulate @ Array[f, 100]
  • PARI
    a(n) = sum(k=1, n, numdiv(k)*omega(k)); \\ Michel Marcus, Dec 22 2020

Formula

a(n) = Sum_{k=1..n} A248577(k) = Sum_{k=1..n} A000005(k) * A001221(k).
a(n) ~ 2 * n * log(n) * log(log(n)) + 2 * B * n * log(n), where B = M - 1 - S/2 = -0.9646264971..., M is Mertens's constant (A077761) and S = Sum_{p prime} 1/p^2 (A085548).
Empirical: a(n) = Sum_{i*j <= n} omega(lcm(i, j)). See Heyman. - Michel Marcus, Dec 26 2020

A332085 Number of ordered pairs of divisors of n, (d1,d2), such that d1 is prime and d1 <= d2.

Original entry on oeis.org

0, 1, 1, 2, 1, 5, 1, 3, 2, 5, 1, 9, 1, 5, 5, 4, 1, 9, 1, 8, 5, 5, 1, 13, 2, 5, 3, 8, 1, 18, 1, 5, 5, 5, 5, 15, 1, 5, 5, 12, 1, 17, 1, 8, 9, 5, 1, 17, 2, 9, 5, 8, 1, 13, 5, 12, 5, 5, 1, 29, 1, 5, 9, 6, 5, 17, 1, 8, 5, 18, 1, 21, 1, 5, 9, 8, 5, 17, 1, 16, 4, 5, 1, 28, 5, 5, 5, 11, 1, 30
Offset: 1

Views

Author

Wesley Ivan Hurt, Aug 22 2020

Keywords

Examples

			a(7) = 1; There are two divisors of 7: {1,7}. If we list the ordered pairs of divisors of n, (d1,d2) where d1 is prime and d1 <= d2, we get (7,7). So a(7) = 1.
a(8) = 3; There are 4 divisors of 8: {1,2,4,8}. If we list the ordered pairs of divisors of n, (d1,d2) where d1 is prime and d1 <= d2, we get (2,2), (2,4) and (2,8). So a(8) = 3.
a(9) = 2; There are three divisors of 9: {1,3,9}. If we list the ordered pairs of divisors of n, (d1,d2) where d1 is prime and d1 <= d2, we get (3,3) and (3,9). So a(9) = 2.
a(10) = 5; There are four divisors of 10: {1,2,5,10}. If we list the ordered pairs of divisors of n, (d1,d2) where d1 is prime and d1 <= d2, we get (2,2), (2,5), (2,10), (5,5) and (5,10). So a(10) = 5.

Crossrefs

Cf. A001221 (omega), A135539, A337228, A337320, A337322, A248577.

Programs

  • Mathematica
    Table[Sum[Sum[(PrimePi[i] - PrimePi[i - 1]) (1 - Ceiling[n/k] + Floor[n/k]) (1 - Ceiling[n/i] + Floor[n/i]), {i, k}], {k, n}], {n, 100}]
  • PARI
    row(n) = my(d=divisors(n)); vector(n, k, #select(x->(x>=k), d)); \\ A135539
a(n) = my(v=row(n)); sumdiv(n, d, if (isprime(d), v[d])); \\ Michel Marcus, May 24 2025

Formula

a(n) = Sum_{d1|n, d2|n, d1 is prime, d1 <= d2} 1.
a(n) = A337320(n) + omega(n).
a(n) = Sum_{p|n, p prime} A135539(n,p). - Ridouane Oudra, May 24 2025
a(n) = A248577(n) - A337322(n). - Ridouane Oudra, May 30 2025

A337322 Number of ordered pairs of divisors of n, (d1,d2), such that d2 is prime and d1 < d2.

Original entry on oeis.org

0, 1, 1, 1, 1, 3, 1, 1, 1, 3, 1, 3, 1, 3, 3, 1, 1, 3, 1, 4, 3, 3, 1, 3, 1, 3, 1, 4, 1, 6, 1, 1, 3, 3, 3, 3, 1, 3, 3, 4, 1, 7, 1, 4, 3, 3, 1, 3, 1, 3, 3, 4, 1, 3, 3, 4, 3, 3, 1, 7, 1, 3, 3, 1, 3, 7, 1, 4, 3, 6, 1, 3, 1, 3, 3, 4, 3, 7, 1, 4, 1, 3, 1, 8, 3, 3, 3, 5, 1, 6, 3, 4, 3, 3, 3, 3
Offset: 1

Views

Author

Wesley Ivan Hurt, Aug 23 2020

Keywords

Examples

			a(39) = 3; There are 4 divisors of 39, {1,3,13,39}. There are three ordered pairs of divisors, (d1,d2), such that d2 is prime and d1 < d2. They are: (1,3), (1,13) and (3,13). So a(39) = 3.
a(40) = 4; There are 8 divisors of 40, {1,2,4,5,8,10,20,40}. There are four ordered pairs of divisors, (d1,d2), such that d2 is prime and d1 < d2. They are: (1,2), (1,5), (2,5) and (4,5). So a(40) = 4.
a(41) = 1; There are 2 divisors of 41, {1,41}. There is one ordered pair of divisors, (d1,d2), such that d2 is prime and d1 < d2. It is (1,41). So a(41) = 1.
a(42) = 7; There are 8 divisors of 42, {1,2,3,6,7,14,21,42}. There are seven ordered pairs of divisors, (d1,d2), such that d2 is prime and d1 < d2. They are: (1,2), (1,3), (1,7), (2,3), (2,7), (3,7) and (6,7). So a(42) = 7.

Crossrefs

Cf. A001221 (omega), A332085, A337228, A337320, A248577, A332085.

Programs

  • Mathematica
    Table[Sum[Sum[(PrimePi[k] - PrimePi[k - 1]) (1 - Ceiling[n/k] + Floor[n/k]) (1 - Ceiling[n/i] + Floor[n/i]), {i, k - 1}], {k, n}], {n, 100}]

Formula

a(n) = Sum_{d1|n, d2|n, d2 is prime, d1 < d2} 1.
a(n) = A337228(n) - omega(n).
a(n) = A248577(n) - A332085(n). - Ridouane Oudra, May 28 2025

A325938 a(n) = omega(n)^tau(n), where omega=A001221 and tau=A000005.

Original entry on oeis.org

0, 1, 1, 1, 1, 16, 1, 1, 1, 16, 1, 64, 1, 16, 16, 1, 1, 64, 1, 64, 16, 16, 1, 256, 1, 16, 1, 64, 1, 6561, 1, 1, 16, 16, 16, 512, 1, 16, 16, 256, 1, 6561, 1, 64, 64, 16, 1, 1024, 1, 64, 16, 64, 1, 256, 16, 256, 16, 16, 1, 531441, 1, 16, 64, 1, 16, 6561, 1, 64
Offset: 1

Views

Author

Hauke Löffler, Sep 09 2019

Keywords

Examples

			a(5) = 1; 5 has one distinct prime divisor {5} and two divisors {1,5}, so a(5) = 1^2 = 1.
a(6) = 16; 6 has two distinct prime divisors {2,3} and four divisors {1,2,3,6}, so a(6) = 2^4 = 16.

Links

Crossrefs

Cf. A000005(tau), A001221(omega), A110088, A248577.

Programs

  • PARI
    a(n) = {omega(n)^numdiv(n)} \\ Andrew Howroyd, Sep 09 2019
  • SageMath
    [ len(prime_divisors(x))^(len(divisors(x))) for x in range(1,20) ]

Formula

a(n) = A001221(n) ^ A000005(n).

A351746 a(n) = Sum_{p|n, p prime} (p-1) * tau(n/p).

Original entry on oeis.org

0, 1, 2, 2, 4, 6, 6, 3, 4, 10, 10, 10, 12, 14, 12, 4, 16, 11, 18, 16, 16, 22, 22, 14, 8, 26, 6, 22, 28, 28, 30, 5, 24, 34, 20, 18, 36, 38, 28, 22, 40, 36, 42, 34, 20, 46, 46, 18, 12, 19, 36, 40, 52, 16, 28, 30, 40, 58, 58, 44, 60, 62, 26, 6, 32, 52, 66, 52, 48, 44, 70, 25, 72, 74
Offset: 1

Views

Author

Wesley Ivan Hurt, Feb 17 2022

Keywords

Examples

			a(12) = 10; a(12) = Sum_{p|12, p prime} (p-1) * tau(12/p) = (2-1)*tau(12/2) + (3-1)*tau(12/3) = tau(6) + 2*tau(4) = 4 + 2*3 = 10.

Crossrefs

Cf. A000005 (tau), A001221 (omega), A248577, A351711.

Programs

  • PARI
    a(n) = my(f=factor(n)); sum(k=1, #f~, (f[k,1]-1)*numdiv(n/f[k,1])); \\ Michel Marcus, Feb 18 2022

Formula

a(n) = A351711(n) - A248577(n).
Showing 1-5 of 5 results.

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