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.

A353790 a(n) = A353749(sigma(A003961(n))), where A353749(k) = phi(k) * A064989(k), and A064989 shifts the prime factorization one step towards lower primes, while A003961 shifts the factorization one step towards higher primes.

Original entry on oeis.org

1, 2, 4, 132, 4, 16, 8, 48, 870, 16, 30, 528, 24, 32, 32, 5390, 24, 1740, 16, 528, 96, 120, 48, 384, 1224, 96, 1056, 1056, 16, 128, 306, 7920, 240, 96, 64, 114840, 120, 64, 288, 384, 140, 384, 32, 3960, 3480, 192, 144, 21560, 9180, 2448, 192, 3168, 96, 4224, 240, 768, 192, 64, 870, 4224, 416, 1224, 6960, 1191372, 192
Offset: 1

Views

Author

Antti Karttunen, May 10 2022

Keywords

Comments

See comments in A353789.

Links

Crossrefs

Cf. A003961, A003973, A064989, A326042, A353750, A353767, A353789.
Cf. also A353794.

Programs

  • 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>>valuation(n, 2))); for(i=1, #f~, f[i, 1] = precprime(f[i, 1]-1)); factorback(f); };
A353790(n) = { my(s=sigma(A003961(n))); (eulerphi(s)*A064989(s)); };

Formula

a(n) = A353750(A003961(n)) = A326042(n) * A353767(n).
a(n) = A000010(A003973(n)) * A064989(A003973(n)).

A353764 Numbers k for which A353749(sigma(k)) is a multiple of A353749(k), where A353749(k) = phi(k) * A064989(k), and A064989 shifts the prime factorization one step towards lower primes.

Original entry on oeis.org

1, 2, 4, 6, 8, 10, 18, 20, 24, 28, 30, 32, 40, 60, 72, 84, 90, 108, 120, 128, 200, 216, 224, 234, 252, 360, 384, 496, 600, 640, 672, 864, 936, 1080, 1120, 1152, 1170, 1488, 1800, 1920, 2016, 2176, 3200, 3360, 3456, 4320, 4464, 4680, 5148, 5600, 5760, 6048, 6528, 6552, 8128, 9600, 10080, 10880, 14976, 16800, 17280
Offset: 1

Views

Author

Antti Karttunen, May 10 2022

Keywords

Comments

Question: Are there any odd terms after the initial one? See A353789, A353796, A353797.

Links

Crossrefs

Cf. A000010, A000203, A006872, A062401, A064989, A336549, A336550, A353757, A353761, A353766 (characteristic function), A353789, A353796, A353797.
Positions of 1's in A353762. Cf. also A353765.
Subsequence of A353759. Cf. A007691 (a subsequence).

Programs

  • Mathematica
    f[p_, e_] := (p - 1)*p^(e - 1)*If[p == 2, 1, NextPrime[p, -1]^e]; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; Select[Range[20000], Divisible[s[DivisorSigma[1, #]], s[#]] &] (* Amiram Eldar, May 10 2022 *)
  • PARI
    A064989(n) = { my(f=factor(n>>valuation(n,2))); for(i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f); };
A353749(n) = (eulerphi(n)*A064989(n));
isA353764(n) = { my(s=sigma(n)); !(A353749(s)%A353749(n)); };

A353793 Multiplicative with a(p^e) = ((q-1)*p)^e, where q is the least prime larger than p.

Original entry on oeis.org

1, 4, 12, 16, 30, 48, 70, 64, 144, 120, 132, 192, 208, 280, 360, 256, 306, 576, 418, 480, 840, 528, 644, 768, 900, 832, 1728, 1120, 870, 1440, 1116, 1024, 1584, 1224, 2100, 2304, 1480, 1672, 2496, 1920, 1722, 3360, 1978, 2112, 4320, 2576, 2444, 3072, 4900, 3600, 3672, 3328, 3074, 6912, 3960, 4480, 5016, 3480, 3540
Offset: 1

Views

Author

Antti Karttunen, May 11 2022

Keywords

Links

Crossrefs

Cf. A003961, A151800, A339905, A353791, A353792, A353794.
Cf. also A353789.

Programs

  • Mathematica
    f[p_, e_] := ((NextPrime[p] - 1)*p)^e; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 60] (* Amiram Eldar, Dec 31 2022 *)
  • PARI
    A353793(n) = { my(f=factor(n)); for(i=1, #f~, f[i,1] = f[i,1]*(nextprime(f[i,1]+1)-1)); factorback(f); };

Formula

a(n) = A353791(A003961(n)).
a(n) = n * A339905(n).
Sum_{k=1..n} a(k) ~ c * n^3, where c = (1/3) * Product_{p prime} ((p^2-p)/(p^2-q(p)+1)) = 0.49154782..., where q(p) = nextprime(p) = A151800(p). - Amiram Eldar, Dec 31 2022

A353797 Numbers k such that k*A003557(A003961(k)) divides A353790(k), where A353790(n) = phi(A003973(n)) * A064989(A003973(n)).

Original entry on oeis.org

1, 2, 4, 44, 132, 220, 396, 660, 1980, 3920, 4400, 8800, 11484, 13200, 13328, 22000, 26400, 30800, 39984, 57420, 66640, 74800, 92400, 119952, 149600, 199920, 224400, 269892, 277200, 448800, 523600, 599760, 673200, 771012, 1063692, 1345792, 1346400, 1570800, 3478608, 4037376, 4712400, 5664400, 6344448, 8038800, 10574080
Offset: 1

Views

Author

Antti Karttunen, May 12 2022

Keywords

Comments

Note that A003557(A003961(n)) [= A003961(A003557(n))] is a divisor of A003972(n), therefore the set of k such that A353789(k) divides A353790(k) is a subset of this sequence.
Of 101 initial terms (terms < 2^32) all others apart from a(1) = 1 and a(2) = 2 are multiples of 4.

Links

Crossrefs

Subsequence of A353796.
Cf. A000010, A000203, A003557, A003961, A003972, A003973, A353789, A353790.

Programs

  • PARI
    A003557(n) = (n/factorback(factorint(n)[, 1]));
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>>valuation(n, 2))); for(i=1, #f~, f[i, 1] = precprime(f[i, 1]-1)); factorback(f); };
A353790(n) = { my(s=sigma(A003961(n))); (eulerphi(s)*A064989(s)); };
isA353797(n) = !(A353790(n)%(n*A003557(A003961(n))));
Showing 1-4 of 4 results.

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