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.

A344587 Deficiency of prime-shifted n: a(n) = 2*A003961(n) - sigma(A003961(n)).

Original entry on oeis.org

1, 2, 4, 5, 6, 6, 10, 14, 19, 10, 12, 12, 16, 18, 22, 41, 18, 26, 22, 22, 38, 22, 28, 30, 41, 30, 94, 42, 30, 18, 36, 122, 46, 34, 58, 47, 40, 42, 62, 58, 42, 42, 46, 52, 102, 54, 52, 84, 109, 66, 70, 72, 58, 126, 70, 114, 86, 58, 60, 6, 66, 70, 178, 365, 94, 54, 70, 82, 110, 78, 72, 110, 78, 78, 148, 102, 118, 78
Offset: 1

Views

Author

Antti Karttunen, May 28 2021

Keywords

Comments

First negative value occurs as a(120) = -30.
Questions: Which subsets of natural numbers generate the "cut sigmoid" graph(s) that cross the X-axis in the (lowermost) scatter plot?

Links

Crossrefs

Cf. A000203, A003961, A003973, A033879, A153881, A336851, A337386 (positions of terms <= 0), A346246 (Dirichlet inverse), A349387, A378216, A378231 [= a(n^2)].
Inverse Möbius transform of A337544.

Programs

  • PARI
    A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A344587(n) = { my(u=A003961(n)); (u+u - sigma(u)); };

Formula

a(n) = A033879(A003961(n)) = 2*A003961(n) - A003973(n).
a(n) = Sum_{d|n} A337544(d).
From Antti Karttunen, Nov 23 2024: (Start)
a(n) = Sum_{d|n} A003961(d)*A153881(n/d) = A003961(n) - A336851(n).
a(n) = Sum_{d|n} A033879(d)*A349387(n/d).
a(n) = Sum_{d|n} A003972(d)*A378216(n/d).
(End)

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).

A377879 Deficiency of squares: a(n) = 2n^2 - sigma(n^2).

Original entry on oeis.org

1, 1, 5, 1, 19, -19, 41, 1, 41, -17, 109, -115, 155, -7, 47, 1, 271, -199, 341, -161, 141, 37, 505, -499, 469, 71, 365, -199, 811, -1021, 929, 1, 449, 163, 683, -1159, 1331, 221, 663, -737, 1639, -1659, 1805, -251, 299, 361, 2161, -2035, 2001, -467, 1211, -265, 2755, -1819, 1927, -967, 1545, 631, 3421, -5293, 3659, 737
Offset: 1

Views

Author

Antti Karttunen, Nov 23 2024

Keywords

Comments

It is conjectured that 1's occur only when n is two's power (A000079), and that there are no -1's in this sequence. See comments in A033879 and in A337339.

Links

Crossrefs

Cf. A000290, A000079 (conjectured to give positions of all 1's), A033879, A378231 [= a(A003961(n))].
Cf. A000012, A083884.
Cf. also square array A083064.

Programs

Formula

a(n) = A033879(A000290(n)).

A379501 a(n) = (3/2)*A019565((2n-1)^2) - A019565(A001065((2n-1)^2)), where A019565 is the base-2 exp-function, and A001065 is the sum of proper divisors of n.

Original entry on oeis.org

2, 16, 216, 422, 470, 51016, 5082, 4446, 864, 106688, 1301846, 880, 204182, 1985872, 236964, 646310, 1030, 176778, 2799756, 96178962, 563400, 62092576, 1566805968, 27274, 559406, -16252236, 1040774592, 263042394, 7794826, 115781204, 13256922, -16386856, -1230440, 376172, -67188814, 222905278, 13547232, 28352541646
Offset: 1

Views

Author

Antti Karttunen, Jan 05 2025

Keywords

Links

Crossrefs

Cf. A001065, A016754, A019565, A379495, A379496.
Cf. also A378231, A337339.

Programs

  • PARI
    A019565(n) = { my(m=1, p=1); while(n>0, p = nextprime(1+p); if(n%2, m *= p); n >>= 1); (m); };
A379501(n) = { my(osq=((2*n)-1)^2); ((3/2)*A019565(osq) - A019565(sigma(osq)-osq)); };

Formula

a(n) = A379496(A016754(n)) = A019565(1+A016754(n)) - A379495(A016754(n)).
a(n) = (3/2)*A019565(A016754(n)) - A379495(A016754(n)).
Showing 1-5 of 5 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.