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.

A337339 Denominator of (1+sigma(s)) / ((s+1)/2), where s is the square of n prime-shifted once (s = A003961(n)^2 = A003961(n^2)).

Original entry on oeis.org

1, 5, 13, 41, 25, 113, 61, 365, 313, 221, 85, 1013, 145, 109, 613, 3281, 181, 2813, 265, 1985, 1513, 761, 421, 9113, 1201, 1301, 7813, 377, 481, 5513, 685, 29525, 2113, 1625, 2965, 25313, 841, 2381, 3613, 17861, 925, 13613, 1105, 6845, 15313, 3785, 1405, 82013, 7321, 10805, 4513, 11705, 1741, 70313, 4141, 8821, 6613, 865
Offset: 1

Views

Author

Antti Karttunen, Aug 24 2020

Keywords

Comments

All terms are members of A007310, because all terms of A337336 and A337337 are.
No 1's after the initial one at a(1) => No quasiperfect numbers. See comments in A336700 & A337342.
If any quasiperfect numbers qp exist, they must occur also in A325311.
Question: Is there any reliable lower bound for this sequence? See A337340, A337341.
Duplicate values are rare, but at least two cases exist: a(21) = a(74) = 1513 and a(253) = a(554) = 71065. - Antti Karttunen, Jan 03 2024

Links

Crossrefs

Cf. A003961, A003973, A007310, A048673, A074627, A074630, A209922, A324899, A325311, A336697, A336700, A336844, A337194, A337336, A337337, A337340, A337341, A337342.
Cf. A337338 (numerators).
Cf. also A336848, A336849.

Programs

  • PARI
    A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A337339(n) = { my(s=(A003961(n)^2),u=(s+1)/2); (u/gcd(1+sigma(s), u)); };
\\ Or alternatively as:
A337339(n) = { my(s=A003961(n^2)); denominator((1+sigma(s))/((s+1)/2)); };

Formula

a(n) = A337336(n) / A337337(n) = A048673(n^2) / gcd(A048673(n^2), A336844(n^2)).
a(n) = A337336(n) / gcd(A337336(n), 1+A003973(n^2)).

A337337 a(n) = gcd(1+sigma(s), (s+1)/2), where s is the square of n once prime-shifted (s = A003961(n)^2 = A003961(n^2)).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 13, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 1, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 17, 1, 1, 1
Offset: 1

Views

Author

Antti Karttunen, Aug 24 2020

Keywords

Comments

All terms are in A007310, because all terms of A337336 are.

Links

Crossrefs

Cf. A003961, A003973, A007310, A048673, A336697, A336844, A337194, A337335, A337336, A337338, A337339.

Programs

  • PARI
    A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A337337(n) = { my(s=(A003961(n)^2)); gcd((s+1)/2, 1+sigma(s)); };
  • PARI
    A048673(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); (factorback(f)+1)/2; };
A336697(n) = { my(s=((n+n-1)^2)); gcd((s+1)/2, 1+sigma(s)); };
A337337(n) = A336697(A048673(n));

Formula

a(n) = gcd((s+1)/2, 1+sigma(s)), where s = A003961(n)^2 = A003961(n^2).
a(n) = gcd(A048673(n^2), 1+A003973(n^2)).
a(n) = gcd(A048673(n^2), A337194(A003961(n)^2)) = gcd(A337336(n), A336844(n^2)).
a(n) = A336697(A048673(n)).
a(n) = A337335(n^2).

A379481 Square of prime-shifted n, or equally, n squared, then prime-shifted one step towards larger primes.

Original entry on oeis.org

1, 9, 25, 81, 49, 225, 121, 729, 625, 441, 169, 2025, 289, 1089, 1225, 6561, 361, 5625, 529, 3969, 3025, 1521, 841, 18225, 2401, 2601, 15625, 9801, 961, 11025, 1369, 59049, 4225, 3249, 5929, 50625, 1681, 4761, 7225, 35721, 1849, 27225, 2209, 13689, 30625, 7569, 2809, 164025, 14641, 21609, 9025, 23409, 3481, 140625
Offset: 1

Views

Author

Antti Karttunen, Dec 27 2024

Keywords

Links

Crossrefs

Cf. A000290, A003961, A016754, A048673, A111003, A337336, A378231, A379482 [= sigma(a(n))], A379484 [= A379473(a(n))].

Programs

  • Mathematica
    {1}~Join~Array[Apply[Times, Map[NextPrime[#1]^#2 & @@ # &, FactorInteger[#]] ]^2 &, 53, 2] (* Michael De Vlieger, Dec 27 2024 *)
  • PARI
    A379481(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1); f[i, 2] *= 2); factorback(f); };

Formula

Fully multiplicative with a(prime(i)) = prime(i+1)^2.
a(n) = A003961(n^2) = A003961(n)^2.
a(n) = A016754(A048673(n)-1).
a(n) = (1/2)*(A378231(n)+A379482(n)).
From Amiram Eldar, Dec 28 2024: (Start)
Sum_{n>=1} 1/a(n) = Pi^2/8 (A111003).
Sum_{n>=1} (-1)^(n+1)/a(n) = 7*Pi^2/72. (End)

A379482 a(n) = sigma(A003961(n^2)), where A003961 is fully multiplicative with a(prime(i)) = prime(i+1), and sigma is the sum of divisors function.

Original entry on oeis.org

1, 13, 31, 121, 57, 403, 133, 1093, 781, 741, 183, 3751, 307, 1729, 1767, 9841, 381, 10153, 553, 6897, 4123, 2379, 871, 33883, 2801, 3991, 19531, 16093, 993, 22971, 1407, 88573, 5673, 4953, 7581, 94501, 1723, 7189, 9517, 62301, 1893, 53599, 2257, 22143, 44517, 11323, 2863, 305071, 16105, 36413, 11811, 37147, 3541
Offset: 1

Views

Author

Antti Karttunen, Dec 27 2024

Keywords

Links

Crossrefs

Cf. A000290, A003973, A048673, A151800, A378231, A379223, A379481, A379483, A379484 [= A038500(a(n))].
Cf. also A337336, A337337.

Programs

  • Mathematica
    {1}~Join~Array[DivisorSigma[1, #] &[Apply[Times, Map[NextPrime[#1]^#2 & @@ # &, FactorInteger[#]] ]^2] &, 52, 2] (* Michael De Vlieger, Dec 27 2024 *)
  • PARI
    A379482(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1); f[i, 2] *= 2); sigma(factorback(f)); };

Formula

Multiplicative with a(p^e) = (q^(2e+1) - 1)/(q-1), where q = nextprime(p) = A151800(p).
a(n) = A000203(A379481(n)) = A003973(A000290(n)).
a(n) = A379223(A048673(n)).
a(n) = 2*A379481(n) - A378231(n).
Showing 1-4 of 4 results.

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