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.

A353750 a(n) = phi(sigma(n)) * A064989(sigma(n)), where A064989 shifts the prime factorization one step towards lower primes.

Original entry on oeis.org

1, 4, 2, 30, 4, 8, 4, 48, 132, 24, 8, 60, 30, 16, 16, 870, 24, 528, 24, 120, 16, 48, 16, 96, 870, 120, 48, 120, 48, 96, 16, 720, 32, 144, 32, 3960, 306, 96, 120, 288, 120, 64, 140, 240, 528, 96, 32, 1740, 1224, 3480, 96, 1050, 144, 192, 96, 192, 96, 288, 96, 480, 870, 64, 528, 14238, 240, 192, 416, 720, 64, 192, 96
Offset: 1

Views

Author

Antti Karttunen, May 07 2022

Keywords

Comments

In contrast to A353749, this is not multiplicative, except on positions given by A336547.
It seems that a(n) = A353749(n) only on n=1. This would then imply that the intersection of A006872 and A336702 = {1}.

Links

Crossrefs

Cf. A000010, A000203, A006872, A062401, A064989, A336547, A336548, A336549, A336702, A350073, A353747, A353748, A353749.
Cf. A353757, A353758 (where a(n) < A353749(n)), A353759 (where a(n) >= A353749(n)), A353760, A353790 [= a(A003961(n))].
Cf. also A353792.

Programs

  • PARI
    A064989(n) = { my(f=factor(n>>valuation(n,2))); for(i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f); };
A353750(n) = { my(s=sigma(n)); (eulerphi(s)*A064989(s)); };

Formula

a(n) = A353749(A000203(n)) = A062401(n) * A350073(n).
a(n) = A353749(n) + A353757(n).

Extensions

Dubious comment deleted by Antti Karttunen, Jan 26 2023

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)).

A353793 Multiplicative with a(p^e) = ((q-1)*p)^e, where q is the least prime larger than p.

Original entry on oeis.org

1, 4, 12, 16, 30, 48, 70, 64, 144, 120, 132, 192, 208, 280, 360, 256, 306, 576, 418, 480, 840, 528, 644, 768, 900, 832, 1728, 1120, 870, 1440, 1116, 1024, 1584, 1224, 2100, 2304, 1480, 1672, 2496, 1920, 1722, 3360, 1978, 2112, 4320, 2576, 2444, 3072, 4900, 3600, 3672, 3328, 3074, 6912, 3960, 4480, 5016, 3480, 3540
Offset: 1

Views

Author

Antti Karttunen, May 11 2022

Keywords

Links

Crossrefs

Cf. A003961, A151800, A339905, A353791, A353792, A353794.
Cf. also A353789.

Programs

  • Mathematica
    f[p_, e_] := ((NextPrime[p] - 1)*p)^e; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 60] (* Amiram Eldar, Dec 31 2022 *)
  • PARI
    A353793(n) = { my(f=factor(n)); for(i=1, #f~, f[i,1] = f[i,1]*(nextprime(f[i,1]+1)-1)); factorback(f); };

Formula

a(n) = A353791(A003961(n)).
a(n) = n * A339905(n).
Sum_{k=1..n} a(k) ~ c * n^3, where c = (1/3) * Product_{p prime} ((p^2-p)/(p^2-q(p)+1)) = 0.49154782..., where q(p) = nextprime(p) = A151800(p). - Amiram Eldar, Dec 31 2022

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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