A379482 -id:A379482 - cp's OEIS frontend

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

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

Antti Karttunen, Dec 27 2024

Antti Karttunen, Table of n, a(n) for n = 1..20000
Index entries for sequences related to prime indices in the factorization of n.

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

{1}~Join~Array[Apply[Times, Map[NextPrime[#1]^#2 & @@ # &, FactorInteger[#]] ]^2 &, 53, 2] (* Michael De Vlieger, Dec 27 2024 *)

A379481(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1); f[i, 2] *= 2); factorback(f); };

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)

A379484 a(n) is the highest power of 3 dividing 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, 1, 1, 1, 3, 1, 1, 1, 1, 3, 3, 1, 1, 1, 3, 1, 3, 1, 1, 3, 1, 3, 1, 1, 1, 1, 1, 1, 3, 3, 3, 1, 3, 3, 3, 1, 1, 1, 1, 3, 3, 1, 1, 3, 3, 1, 1, 1, 1, 1, 3, 1, 1, 1, 9, 1, 1, 3, 3, 3, 3, 3, 1, 1, 3, 3, 1, 3, 1, 3, 3, 1, 3, 1, 1, 1, 3, 1, 1, 3, 1, 3, 1, 1, 9, 1, 3, 3, 3, 3, 1, 1, 3, 1, 3, 1, 1, 1, 3, 1, 3, 3, 1, 1, 3

Offset: 1

Antti Karttunen, Dec 27 2024

Cf. A038500, A151800, A379473, A379481, A379482.

{1}~Join~Array[3^IntegerExponent[#, 3] &[
  DivisorSigma[1,
    Apply[Times, Map[NextPrime[#1]^#2 & @@ # &, FactorInteger[#]]]^2]] &,
105, 2] (* Michael De Vlieger, Dec 27 2024 *)

A379484(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1); f[i, 2] *= 2); 3^valuation(sigma(factorback(f)),3); };

Multiplicative with a(p^e) = A038500((q^(2e+1) - 1)/(q-1)), where q = nextprime(p) = A151800(p).
a(n) = A038500(A379482(n)).
a(n) = A379473(A379481(n)).

A379483 a(n) is the number of trailing 1-bits in the binary representation of 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, 1, 5, 1, 1, 2, 1, 1, 1, 1, 3, 3, 2, 1, 3, 1, 1, 1, 1, 1, 2, 2, 3, 2, 1, 3, 2, 1, 1, 2, 7, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 5, 1, 7, 1, 2, 4, 4, 1, 1, 2, 2, 1, 4, 6, 1, 3, 1, 3, 4, 1, 2, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 7, 4, 1, 2, 1, 1, 1, 1, 1, 2, 2, 1, 1, 6, 2, 2, 1, 7, 4, 1, 2, 1, 2, 6, 1, 2, 1, 1, 3, 1, 6, 2

Offset: 1

Antti Karttunen, Dec 27 2024

nonn

Cf. A003961, A003973, A007814, A048673, A379222, A379482, A379483.

Cf. also A336700.

{1}~Join~Array[Length@ Last@ Split[IntegerDigits[#, 2]][[1 ;; -1 ;; 2]] &[
DivisorSigma[1,
  Apply[Times, Map[NextPrime[#1]^#2 & @@ # &, FactorInteger[#]] ]^2] ] &,
    105, 2] (* Michael De Vlieger, Dec 27 2024 *)

A379483(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1); f[i, 2] *= 2); valuation(1+sigma(factorback(f)),2); };

a(n) = A007814(1+A379482(n)).

a(n) = A379222(A048673(n)).

cp's OEIS Frontend

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

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A379484 a(n) is the highest power of 3 dividing sigma(A003961(n^2)), where A003961 is fully multiplicative with a(prime(i)) = prime(i+1), and sigma is the sum of divisors function.

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Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A379483 a(n) is the number of trailing 1-bits in the binary representation of sigma(A003961(n^2)), where A003961 is fully multiplicative with a(prime(i)) = prime(i+1), and sigma is the sum of divisors function.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula