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.

A349164 a(n) = A064989(A003961(n) / gcd(sigma(n), A003961(n))), where A003961 shifts the prime factorization of n one step towards larger primes, while A064989 shifts it back towards smaller primes, and sigma is the sum of divisors function.

Original entry on oeis.org

1, 1, 3, 4, 5, 3, 7, 4, 9, 5, 11, 12, 13, 7, 15, 16, 17, 9, 19, 2, 21, 11, 23, 4, 25, 13, 9, 28, 29, 15, 31, 8, 33, 17, 35, 36, 37, 19, 39, 10, 41, 21, 43, 22, 45, 23, 47, 48, 49, 25, 51, 52, 53, 9, 55, 28, 19, 29, 59, 6, 61, 31, 63, 64, 13, 33, 67, 17, 69, 35, 71, 12, 73, 37, 75, 76, 77, 39, 79, 40, 81, 41, 83, 84
Offset: 1

Views

Author

Antti Karttunen, Nov 09 2021

Keywords

Links

Crossrefs

Cf. A000203, A003961, A319630, A326042, A336702, A342671, A348993, A349161, A349162, A349169, A349174.
Cf. A349144 and A349168 [positions where a(n) is / is not relatively prime with A349163(n) = n/a(n)].

Programs

  • Mathematica
    Array[Times @@ Map[If[#1 <= 2, 1, NextPrime[#1, -1]]^#2 & @@ # &, FactorInteger[#2/GCD[##]]] & @@ {DivisorSigma[1, #], Times @@ Map[NextPrime[#1]^#2 & @@ # &, FactorInteger[#]]} &, 84] (* Michael De Vlieger, Nov 11 2021 *)
  • PARI
    A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A064989(n) = { my(f=factor(n)); if((n>1 && f[1,1]==2), f[1,2] = 0); for(i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f); };
A349164(n) = { my(u=A003961(n)); A064989(u/gcd(u,sigma(n))); };

Formula

a(n) = A064989(A349161(n)).
a(n) = n / A349163(n).

A349166 Numbers k such that sigma(k) and A003961(k) share a prime factor, where A003961(n) is fully multiplicative function with a(prime(k)) = prime(k+1).

Original entry on oeis.org

2, 6, 8, 10, 14, 18, 20, 22, 24, 26, 27, 30, 32, 34, 38, 40, 42, 44, 46, 50, 54, 56, 57, 58, 60, 62, 65, 66, 68, 70, 72, 74, 78, 80, 82, 86, 87, 88, 90, 92, 94, 96, 98, 99, 100, 102, 104, 106, 108, 110, 114, 116, 118, 120, 122, 126, 128, 130, 132, 134, 135, 136, 138, 140, 142, 146, 150, 152, 154, 158, 160, 162, 164
Offset: 1

Views

Author

Antti Karttunen, Nov 09 2021

Keywords

Comments

The only prime term is 2. A prime power prime(j)^k with k > 1 is a term if and only if k+1 is divisible by the multiplicative order of prime(j) mod prime(j+1). - Robert Israel, May 22 2025

Examples

			For n = 2, A000203(2) = A003961(2) = 3, therefore they share a prime factor 3, and 2 is included in this sequence.
For n = 10 = 2*5, sigma(10) = 18 = 2 * 3^2, while A003961(10) = 21 = 3*7, therefore 10 is included, as there is a shared prime factor (3).

Links

Crossrefs

Cf. A000203, A003961.
Positions of terms larger than ones in A342671, and also in A349163.
Positions of zeros in A349167.
Cf. A349165 (complement), A349168 (subsequence).

Programs

  • Maple
    filter:= proc(n) local F,a,b,t;
   F:= ifactors(n)[2];
   b:= convert(map(nextprime,F[..,1]),`*`);
   a:= mul((t[1]^(t[2]+1)-1)/(t[1]-1),t=F);
   igcd(a,b) <> 1
end proc;
select(filter, [$1..1000]); # Robert Israel, May 21 2025
  • PARI
    A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
isA349166(n) = (1!=gcd(sigma(n), A003961(n)));

A349163 a(n) = A064989(gcd(sigma(n), A003961(n))), where A003961 shifts the prime factorization of n one step towards larger primes, while A064989 shifts it back towards smaller primes, and sigma is the sum of divisors function.

Original entry on oeis.org

1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 10, 1, 2, 1, 6, 1, 2, 3, 1, 1, 2, 1, 4, 1, 2, 1, 1, 1, 2, 1, 4, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 1, 6, 1, 2, 3, 2, 1, 10, 1, 2, 1, 1, 5, 2, 1, 4, 1, 2, 1, 6, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 3, 4, 1, 2, 1, 2, 1, 2, 1, 4, 1, 2, 11, 5, 1, 2, 1, 2, 1
Offset: 1

Views

Author

Antti Karttunen, Nov 09 2021

Keywords

Links

Crossrefs

Cf. A000203, A003961, A342671, A349161, A349162, A349165 (positions of 1's), A349166 (of terms > 1).
Cf. A349144 and A349168 [positions where a(n) is / is not relatively prime with A349164(n) = n/a(n)].

Programs

  • Mathematica
    Array[Times @@ Map[If[#1 <= 2, 1, NextPrime[#1, -1]]^#2 & @@ # &, FactorInteger@ GCD[##]] & @@ {DivisorSigma[1, #], Times @@ Map[NextPrime[#1]^#2 & @@ # &, FactorInteger[#]]} &, 105] (* Michael De Vlieger, Nov 11 2021 *)
  • PARI
    A003961(n) = { my(f=factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A064989(n) = { my(f=factor(n)); if((n>1 && f[1,1]==2), f[1,2] = 0); for(i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f); };
A349163(n) = A064989(gcd(sigma(n),A003961(n)));

Formula

a(n) = A064989(A342671(n)).
a(n) = n / A349164(n).

A349144 Numbers k for which A342671(k) [= gcd(sigma(k), A003961(k))] and A349161(k) [= A003961(k)/A342671(k)] are relatively prime, where A003961(n) is fully multiplicative with a(prime(k)) = prime(k+1), and sigma is the sum of divisors function.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 21, 22, 23, 25, 26, 28, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 45, 46, 47, 48, 49, 50, 51, 52, 53, 55, 57, 58, 59, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 81, 82, 83, 84, 85, 86, 87, 89, 90, 91, 93, 94, 95
Offset: 1

Views

Author

Antti Karttunen, Nov 11 2021

Keywords

Comments

Numbers k for which A349163(k) and A349164(k) are coprime, i.e., k such that A349163(k) and A349164(k) are unitary divisors of k.

Links

Crossrefs

Cf. A000203, A003961, A342671, A349161, A349163, A349164.
Complement of A349168.
Cf. A349165 (subsequence).

Programs

  • Mathematica
    Select[Range[95], GCD[#2, #1/#2] == 1 & @@ {#2, #2/GCD[##]} & @@ {DivisorSigma[1, #], If[# == 1, 1, Times @@ Map[NextPrime[#1]^#2 & @@ # &, FactorInteger[#]]]} &] (* Michael De Vlieger, Nov 11 2021 *)
  • PARI
    A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
isA349144(n) = { my(u=A003961(n), x=gcd(u,sigma(n))); (1==gcd(x,u/x)); };
Showing 1-4 of 4 results.

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