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.

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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)); };

A353795 Numbers k such that k divides A353794(k), where A353794(n) = A003958(A003973(n)) * A064989(A003973(n)).

Original entry on oeis.org

1, 4, 12, 24, 36, 44, 72, 96, 112, 132, 180, 220, 360, 384, 396, 400, 480, 560, 660, 784, 832, 864, 1044, 1056, 1188, 1200, 1344, 1920, 1980, 2088, 2352, 2376, 2496, 2800, 3168, 3600, 3920, 4320, 4736, 5220, 5280, 5376, 5824, 5940, 6800, 6912, 7056, 7200, 7488, 8400, 8800, 9504, 9900, 10000, 10440, 10800, 11484
Offset: 1

Views

Author

Antti Karttunen, May 12 2022

Keywords

Comments

Of 2608 initial terms, only 188 are not in A353796. The first three of these are: 400, 784, 832.

Links

Crossrefs

Cf. A000010, A003958, A003961, A003973, A064989, A353794.
Cf. also A353796.

Programs

  • PARI
    A003958(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]--); factorback(f); };
A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); }; \\ From A003961
A064989(n) = { my(f=factor(n>>valuation(n,2))); for(i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f); };
A353794(n) = { my(s=sigma(A003961(n))); (A003958(s)*A064989(s)); };
isA353795(n) = !(A353794(n)%n);

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 1-3 of 3 results.

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