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 1-7 of 7 results.

A348997 a(n) = A348733(A276086(n)), where A348733(n) = gcd(A003959(n), A034448(n)), and A276086 gives the prime product form of primorial base expansion of n.

Original entry on oeis.org

1, 3, 4, 12, 2, 6, 6, 18, 24, 72, 12, 36, 2, 6, 8, 24, 4, 12, 18, 54, 72, 216, 36, 108, 2, 6, 8, 24, 4, 12, 8, 24, 32, 96, 16, 48, 48, 144, 192, 576, 96, 288, 16, 48, 64, 192, 32, 96, 144, 432, 576, 1728, 288, 864, 16, 48, 64, 192, 32, 96, 2, 6, 8, 24, 4, 12, 12, 36, 48, 144, 24, 72, 4, 12, 16, 48, 8, 24, 36, 108, 144
Offset: 0

Views

Author

Antti Karttunen, Nov 07 2021

Keywords

Links

Crossrefs

Cf. A003959, A034448, A276086, A348733, A348949, A348996.
Cf. also A346471 for similar construction. (Compare the scatter plots).

Programs

  • PARI
    A348997(n) = { my(m1=1, m2=1, p=2); while(n, if(n%p, m1 *= ((1+p)^(n%p)); m2 *= (1+(p^(n%p)))); n = n\p; p = nextprime(1+p)); gcd(m1, m2); };

Formula

a(n) = A348733(A276086(n)) = gcd(A348949(n), A348996(n)).

A348740 Positions k where A348733(k) is not multiplicative.

Original entry on oeis.org

1444, 3249, 3364, 4332, 4563, 6498, 7220, 7569, 9126, 10092, 10108, 12996, 13924, 15138, 15884, 16245, 16820, 17689, 18252, 18772, 21125, 21660, 22743, 22815, 23104, 23548, 24548, 24964, 25992, 27436, 30276, 30324, 31329, 31684, 31941, 32490, 33212, 35378, 35739, 36100, 36504, 37004, 37845, 38988, 41209, 41772
Offset: 1

Views

Author

Antti Karttunen, Nov 05 2021

Keywords

Comments

Numbers k with a factorization into coprime x and k/x with A348733(x) * A348733(k/x) <> A348733(k).

Crossrefs

Cf. A003959, A034448, A348733, A348734, A348735.
Cf. also A344702.

Programs

  • Mathematica
    f1[p_, e_] := (p + 1)^e; f2[p_, e_] := p^e + 1; a1[1] = 1; a1[n_] := GCD[Times @@ f1 @@@ (f = FactorInteger[n]), Times @@ f2 @@@ f]; f3[p_, e_] := a1[p^e]; a2[n_] := Times @@ f3 @@@ FactorInteger[n]; Position[Table[a2[n] - a1[n], {n, 1, 42000}], ?(# != 0 &)] // Flatten (* _Amiram Eldar, Nov 05 2021 *)
  • PARI
    A003959(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]++); factorback(f); };
A034448(n) = { my(f = factor(n)); prod(k=1, #f~, 1+(f[k, 1]^f[k, 2])); };
A348733(n) = gcd(A003959(n), A034448(n));
A348733mult(n) = { my(f = factor(n)); prod(k=1, #f~, A348733(f[k, 1]^f[k, 2])); };
isA348740(n) = (A348733(n)!=A348733mult(n));

A348734 Numerator of Product((p+1)^e / ((p^e)+1)), when n = Product(p^e), with p primes, and e their exponents.

Original entry on oeis.org

1, 1, 1, 9, 1, 1, 1, 3, 8, 1, 1, 9, 1, 1, 1, 81, 1, 8, 1, 9, 1, 1, 1, 3, 18, 1, 16, 9, 1, 1, 1, 81, 1, 1, 1, 72, 1, 1, 1, 3, 1, 1, 1, 9, 8, 1, 1, 81, 32, 18, 1, 9, 1, 16, 1, 3, 1, 1, 1, 9, 1, 1, 8, 729, 1, 1, 1, 9, 1, 1, 1, 24, 1, 1, 18, 9, 1, 1, 1, 81, 128, 1, 1, 9, 1, 1, 1, 3, 1, 8, 1, 9, 1, 1, 1, 81, 1, 32, 8
Offset: 1

Views

Author

Antti Karttunen, Nov 05 2021

Keywords

Comments

This is not multiplicative. The first point where a(m*n) = a(m)*a(n) does not hold for coprime m and n is 1444 = 2^2 * 19^2, where a(1444) = 360 != 1800 = 9*200 = a(4)*a(361). See A348740 for the list of such positions.

Links

Crossrefs

Cf. A003959, A034448, A348733, A348735 (denominators), A348740.

Programs

  • Mathematica
    f[p_, e_] := (p + 1)^e/(p^e + 1); a[1] = 1; a[n_] := Numerator[Times @@ f @@@ FactorInteger[n]]; Array[a, 100] (* Amiram Eldar, Nov 05 2021 *)
  • PARI
    A003959(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]++); factorback(f); };
A034448(n) = { my(f = factor(n)); prod(k=1, #f~, 1+(f[k, 1]^f[k, 2])); }; \\ After code in A034448
A348734(n) = { my(u=A003959(n)); (u/gcd(u, A034448(n))); };
  • PARI
    A348734(n) = { my(f = factor(n)); numerator(prod(k=1, #f~, ((1+f[k, 1])^f[k, 2])/(1+(f[k, 1]^f[k, 2])))); };

Formula

a(n) = A003959(n) / gcd(A003959(n), A034448(n)).

A348735 Denominator of Product((p+1)^e / ((p^e)+1)), when n = Product(p^e), with p primes, and e their exponents.

Original entry on oeis.org

1, 1, 1, 5, 1, 1, 1, 1, 5, 1, 1, 5, 1, 1, 1, 17, 1, 5, 1, 5, 1, 1, 1, 1, 13, 1, 7, 5, 1, 1, 1, 11, 1, 1, 1, 25, 1, 1, 1, 1, 1, 1, 1, 5, 5, 1, 1, 17, 25, 13, 1, 5, 1, 7, 1, 1, 1, 1, 1, 5, 1, 1, 5, 65, 1, 1, 1, 5, 1, 1, 1, 5, 1, 1, 13, 5, 1, 1, 1, 17, 41, 1, 1, 5, 1, 1, 1, 1, 1, 5, 1, 5, 1, 1, 1, 11, 1, 25, 5, 65
Offset: 1

Views

Author

Antti Karttunen, Nov 05 2021

Keywords

Comments

This is not multiplicative. The first point where a(m*n) = a(m)*a(n) does not hold for coprime m and n is 1444 = 2^2 * 19^2, where a(1444) = 181 <> 5*181 = a(4)*a(361). See A348740 for the list of such positions.

Links

Crossrefs

Cf. A003959, A034448, A348733, A348734 (numerators), A348740.

Programs

  • Mathematica
    f[p_, e_] := (p + 1)^e/(p^e + 1); a[1] = 1; a[n_] := Denominator[Times @@ f @@@ FactorInteger[n]]; Array[a, 100] (* Amiram Eldar, Nov 05 2021 *)
  • PARI
    A348735(n) = { my(f = factor(n)); denominator(prod(k=1, #f~, ((1+f[k, 1])^f[k, 2])/(1+(f[k, 1]^f[k, 2])))); };
  • PARI
    A003959(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]++); factorback(f); };
A034448(n) = { my(f = factor(n)); prod(k=1, #f~, 1+(f[k, 1]^f[k, 2])); };
A348735(n) = { my(u=A034448(n)); (u/gcd(u, A003959(n))); };

Formula

a(n) = A034448(n) / A348733(n) = A034448(n) / gcd(A003959(n), A034448(n)).

A348946 a(n) = gcd(sigma(n), A348944(n)), where A348944 is the arithmetic mean of A003959 and A034448, and sigma is the sum of divisors function.

Original entry on oeis.org

1, 3, 4, 7, 6, 12, 8, 3, 13, 18, 12, 28, 14, 24, 24, 1, 18, 39, 20, 42, 32, 36, 24, 12, 31, 42, 2, 56, 30, 72, 32, 3, 48, 54, 48, 1, 38, 60, 56, 18, 42, 96, 44, 84, 78, 72, 48, 4, 57, 93, 72, 98, 54, 6, 72, 24, 80, 90, 60, 168, 62, 96, 104, 1, 84, 144, 68, 126, 96, 144, 72, 3, 74, 114, 124, 140, 96, 168, 80, 6, 1, 126
Offset: 1

Views

Author

Antti Karttunen, Nov 05 2021

Keywords

Comments

This is not multiplicative. The first point where a(m*n) = a(m)*a(n) does not hold for coprime m and n is 36 = 2^2 * 3^2, where a(36) = 1 <> 91 = 7*13 = a(4)*a(9).

Links

Crossrefs

Cf. A000203, A003959, A034448, A348944, A348945, A348947, A348948.
Cf. also A344695, A348047, A348503, A348733.

Programs

  • Mathematica
    f1[p_, e_] := (p^(e + 1) - 1)/(p - 1); f2[p_, e_] := (p + 1)^e; f3[p_, e_] := p^e + 1; a[1] = 1; a[n_] := GCD[Times @@ f1 @@@ (f = FactorInteger[n]), (Times @@ f2 @@@ f + Times @@ f3 @@@ f)/2]; Array[a, 100] (* Amiram Eldar, Nov 05 2021 *)
  • PARI
    A003959(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]++); factorback(f); };
A034448(n) = { my(f = factor(n)); prod(k=1, #f~, 1+(f[k, 1]^f[k, 2])); };
A348946(n) = gcd(sigma(n), ((1/2)*(A003959(n)+A034448(n))));

Formula

a(n) = gcd(A000203(n), A348944(n)).
a(n) = gcd(A000203(n), A348945(n)) = gcd(A348944(n), A348945(n));
a(n) = A348944(n) / A348947(n) = A000203(n) / A348948(n).

A348944 a(n) = (1/2) * (A003959(n)+A034448(n)), where A003959 is multiplicative with a(p^e) = (p+1)^e and A034448 (usigma) is multiplicative with a(p^e) = (p^e)+1.

Original entry on oeis.org

1, 3, 4, 7, 6, 12, 8, 18, 13, 18, 12, 28, 14, 24, 24, 49, 18, 39, 20, 42, 32, 36, 24, 72, 31, 42, 46, 56, 30, 72, 32, 138, 48, 54, 48, 97, 38, 60, 56, 108, 42, 96, 44, 84, 78, 72, 48, 196, 57, 93, 72, 98, 54, 138, 72, 144, 80, 90, 60, 168, 62, 96, 104, 397, 84, 144, 68, 126, 96, 144, 72, 261, 74, 114, 124, 140, 96
Offset: 1

Views

Author

Antti Karttunen, Nov 05 2021

Keywords

Comments

This is not multiplicative. The first point where a(m*n) = a(m)*a(n) does not hold for coprime m and n is 36 = 2^2 * 3^2, where a(36) = 97 != 91 = 7*13 = a(4)*a(9).

Links

Crossrefs

Arithmetic mean of A003959 and A034448.
Cf. A348732, A348733, A348945.
Cf. also A325973.

Programs

  • Mathematica
    f1[p_, e_] := (p + 1)^e; f2[p_, e_] := p^e + 1; a[1] = 1; a[n_] := (Times @@ f1 @@@ (f = FactorInteger[n]) + Times @@ f2 @@@ f) / 2; Array[a, 100] (* Amiram Eldar, Nov 05 2021 *)
  • PARI
    A003959(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]++); factorback(f); };
A034448(n) = { my(f = factor(n)); prod(k=1, #f~, 1+(f[k, 1]^f[k, 2])); }; \\ After code in A034448
A348944(n) = ((1/2)*(A003959(n)+A034448(n)));

A348984 a(n) = gcd(sigma(n), A325973(n)), where A325973 is the arithmetic mean of {sum of squarefree divisors} and {sum of unitary divisors}.

Original entry on oeis.org

1, 3, 4, 1, 6, 12, 8, 3, 1, 18, 12, 4, 14, 24, 24, 1, 18, 3, 20, 6, 32, 36, 24, 12, 1, 42, 8, 8, 30, 72, 32, 9, 48, 54, 48, 1, 38, 60, 56, 18, 42, 96, 44, 12, 6, 72, 48, 4, 1, 3, 72, 14, 54, 24, 72, 24, 80, 90, 60, 24, 62, 96, 8, 1, 84, 144, 68, 18, 96, 144, 72, 3, 74, 114, 4, 20, 96, 168, 80, 6, 1, 126, 84, 32, 108
Offset: 1

Views

Author

Antti Karttunen, Nov 06 2021

Keywords

Comments

This is not multiplicative. The first point where a(m*n) = a(m)*a(n) does not hold for coprime m and n is 108 = 4*27, where a(108) = 4, although a(4) = 1 and a(27) = 8.

Links

Crossrefs

Cf. A000203, A048250, A034448, A325973, A325974, A348985, A348986.
Differs from A348047 for the first time at n=108, where a(108) = 4, while A348047(108) = 8.
Cf. also A348733, A348946.

Programs

  • Mathematica
    f1[p_, e_] := p + 1; f2[p_, e_] := p^e + 1; s[1] = 1; s[n_] := (Times @@ f1 @@@ (f = FactorInteger[n]) + Times @@ f2 @@@ f)/2; a[n_] := GCD[s[n], DivisorSigma[1, n]]; Array[a, 100] (* Amiram Eldar, Nov 06 2021 *)
  • PARI
    A325973(n) = (1/2)*sumdiv(n, d, d*(issquarefree(d) + (1==gcd(d, n/d))));
A348984(n) = gcd(sigma(n), A325973(n));

Formula

a(n) = gcd(A000203(n), A325973(n)).
a(n) = gcd(A000203(n), A325974(n)) = gcd(A325973(n), A325974(n)).
a(n) = A000203(n) / A348985(n) = A325973(n) / A348986(n).
Showing 1-7 of 7 results.

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