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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.

A353794 a(n) = A353791(sigma(A003961(n))), where A353791(n) = A003958(n) * A064989(n).

Original entry on oeis.org

1, 1, 4, 132, 1, 4, 4, 12, 870, 1, 30, 528, 16, 4, 4, 4900, 12, 870, 4, 132, 16, 30, 48, 48, 1224, 16, 528, 528, 1, 4, 306, 3960, 120, 12, 4, 114840, 120, 4, 64, 12, 70, 16, 4, 3960, 870, 48, 64, 19600, 9180, 1224, 48, 2112, 48, 528, 30, 48, 16, 1, 870, 528, 208, 306, 3480, 1191372, 16, 120, 16, 1584, 192, 4, 1116
Offset: 1

Views

Author

Antti Karttunen, May 11 2022

Keywords

Comments

It is conjectured that a(n) is not a multiple of A353793(n) on any other n except on n=1. See also A353795.

Links

Crossrefs

Cf. A000203, A003958, A003961, A003973, A064989, A326042, A351456, A353791, A353792, A353793, A353795 [numbers k such that k divides a(k)].
Cf. also A353790.

Programs

  • PARI
    A003958(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]--); factorback(f); };
A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); }; \\ From A003961
A064989(n) = { my(f=factor(n>>valuation(n,2))); for(i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f); };
A353794(n) = { my(s=sigma(A003961(n))); (A003958(s)*A064989(s)); };

Formula

Multiplicative with a(p^e) = A003958(1 + q + ... + q^e) * A064989(1 + q + ... + q^e), where q is the least prime larger than p.
a(n) = A353791(A003973(n)) = A353792(A003961(n)).
a(n) = A326042(n) * A351456(n) = A064989(A003973(n)) * A003958(A003973(n)).

A353789 Multiplicative with a(p^e) = (q - 1) * q^(e-1) * p^e, where q is the least prime greater than p.

Original entry on oeis.org

1, 4, 12, 24, 30, 48, 70, 144, 180, 120, 132, 288, 208, 280, 360, 864, 306, 720, 418, 720, 840, 528, 644, 1728, 1050, 832, 2700, 1680, 870, 1440, 1116, 5184, 1584, 1224, 2100, 4320, 1480, 1672, 2496, 4320, 1722, 3360, 1978, 3168, 5400, 2576, 2444, 10368, 5390, 4200, 3672, 4992, 3074, 10800, 3960, 10080, 5016, 3480
Offset: 1

Views

Author

Antti Karttunen, May 10 2022

Keywords

Comments

Question: Does a(n) divide A353790(n) only when n=1? Compare to A353764.

Links

Crossrefs

Cf. A003961, A003972, A151800, A353749, A353764, A353790.
Cf. also A353793.

Programs

  • Mathematica
    f[p_, e_] := ((q = NextPrime[p]) - 1) * q^(e - 1) * p^e; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, May 10 2022 *)
  • PARI
    A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A353789(n) = (n * eulerphi(A003961(n)));
  • Python
    from math import prod
from sympy import nextprime, factorint
def A353789(n): return prod((q:= nextprime(p))**(e-1)*p**e*(q-1) for p, e in factorint(n).items()) # Chai Wah Wu, May 10 2022

Formula

Multiplicative with a(p^e) = (q - 1) * q^(e-1) * p^e, where q is the least prime greater than p.
a(n) = A353749(A003961(n)) = n * A003972(n).
Sum_{k=1..n} a(k) ~ c * n^3, where c = (1/3) * Product_{p prime} ((p^3-p^2-p+1)/(p^3 - p*q)) = 0.836506229..., where q(p) = nextprime(p) = A151800(p). - Amiram Eldar, Dec 31 2022

A353792 a(n) = A003958(sigma(n)) * A064989(sigma(n)).

Original entry on oeis.org

1, 4, 1, 30, 4, 4, 1, 48, 132, 16, 4, 30, 30, 4, 4, 870, 16, 528, 12, 120, 1, 16, 4, 48, 870, 120, 12, 30, 48, 16, 1, 480, 4, 64, 4, 3960, 306, 48, 30, 192, 120, 4, 70, 120, 528, 16, 4, 870, 1224, 3480, 16, 900, 64, 48, 16, 48, 12, 192, 48, 120, 870, 4, 132, 14238, 120, 16, 208, 480, 4, 16, 16, 6336, 1116, 1224, 870
Offset: 1

Views

Author

Antti Karttunen, May 11 2022

Keywords

Links

Crossrefs

Cf. A003958, A064989, A350073, A351442, A353791, A353793, A353794 [= a(A003961(n))].
Cf. A046528 (positions of 1's).
Cf. also A353750.

Programs

  • PARI
    A003958(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]--); factorback(f); };
A064989(n) = { my(f=factor(n>>valuation(n,2))); for(i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f); };
A353792(n) = { my(s=sigma(n)); (A003958(s)*A064989(s)); };
  • PARI
    A353792(n) = { my(f=factor(n),s); prod(i=1, #f~, s = sigma(f[i,1]^f[i,2]); A003958(s)*A064989(s)); };

Formula

Multiplicative with a(p^e) = A003958(1 + p + ... + p^e) * A064989(1 + p + ... + p^e).
a(n) = A353791(A000203(n)).
a(n) = A351442(n) * A350073(n) = A003958(A000203(n)) * A064989(A000203(n)).

A353791 Multiplicative with a(p^e) = ((p-1)*q)^e, where q is the largest prime less than p, and 1 if p = 2.

Original entry on oeis.org

1, 1, 4, 1, 12, 4, 30, 1, 16, 12, 70, 4, 132, 30, 48, 1, 208, 16, 306, 12, 120, 70, 418, 4, 144, 132, 64, 30, 644, 48, 870, 1, 280, 208, 360, 16, 1116, 306, 528, 12, 1480, 120, 1722, 70, 192, 418, 1978, 4, 900, 144, 832, 132, 2444, 64, 840, 30, 1224, 644, 3074, 48, 3540, 870, 480, 1, 1584, 280, 4026, 208, 1672, 360
Offset: 1

Views

Author

Antti Karttunen, May 11 2022

Keywords

Links

Crossrefs

Cf. A003958, A064989, A151799, A353792 [= a(A000203(n))], A353793 [= a(A003961(n))], A353794 [= a(sigma(A003961(n)))].
Cf. also A353749.

Programs

  • Mathematica
    f[p_, e_] := (If[p == 2, 1, NextPrime[p, -1]]*(p-1))^e; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 70] (* Amiram Eldar, Dec 31 2022 *)
  • PARI
    A353791(n) = { my(f=factor(n>>valuation(n,2))); for(i=1, #f~, f[i,1] = (f[i,1]-1)*precprime(f[i,1]-1)); factorback(f); };

Formula

a(n) = A003958(n) * A064989(n).
a(n) = a(2*n) = a(A000265(n)).
Sum_{k=1..n} a(k) ~ c * n^3, where c = (1/3) * Product_{p prime} ((p^3-p^2)/(p^3-p*q+q)) = 0.1075035014..., where q(p) = prevprime(p) = A151799(p) if p > 2 and q(2) = 1. - Amiram Eldar, Dec 31 2022
Showing 1-4 of 4 results.

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