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 1-6 of 6 results.

A351442 a(n) = A003958(sigma(n)), where A003958 is multiplicative with a(p^e) = (p-1)^e and sigma is the sum of divisors function.

Original entry on oeis.org

1, 2, 1, 6, 2, 2, 1, 8, 12, 4, 2, 6, 6, 2, 2, 30, 4, 24, 4, 12, 1, 4, 2, 8, 30, 12, 4, 6, 8, 4, 1, 24, 2, 8, 2, 72, 18, 8, 6, 16, 12, 2, 10, 12, 24, 4, 2, 30, 36, 60, 4, 36, 8, 8, 4, 8, 4, 16, 8, 12, 30, 2, 12, 126, 12, 4, 16, 24, 2, 4, 4, 96, 36, 36, 30, 24, 2, 12, 4, 60, 100, 24, 12, 6, 8, 20, 8
Offset: 1

Views

Author

Antti Karttunen, Feb 12 2022

Keywords

Comments

Question: Are there more fixed points than 1, 2, 8, 128, 288, 720, 32768, 29719872, ..., 2147483648 ?

Links

Crossrefs

Cf. A000203, A003958, A322582, A339905, A351443, A351444, A351445, A351446, A351447, A351448, A351456.
Cf. also A348512.

Programs

  • PARI
    A003958(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]--); factorback(f); };
A351442(n) = A003958(sigma(n));

Formula

Multiplicative with a(p^e) = A003958(1 + p + ... + p^e).
a(n) = A003958(A000203(n)).
a(n) = A351444(n) - A322582(n) = A351445(n) + A003958(n).

A351446 Numbers k for which A003958(sigma(k)) = A003958(k), where A003958 is multiplicative with a(p^e) = (p-1)^e and sigma is the sum of divisors function.

Original entry on oeis.org

1, 6, 10, 26, 28, 49, 54, 74, 122, 126, 146, 294, 314, 386, 408, 490, 496, 554, 626, 680, 794, 842, 914, 1082, 1226, 1232, 1274, 1322, 1346, 1466, 1514, 1560, 1754, 1768, 1994, 2186, 2306, 2402, 2426, 2474, 2642, 2646, 2762, 2906, 3242, 3314, 3360, 3506, 3626, 3672, 3746, 3808, 3866, 3986, 4034
Offset: 1

Views

Author

Antti Karttunen, Feb 12 2022

Keywords

Comments

Numbers k for which A351442(k) = A003958(k), or equally, for which k = A351444(k) = A322582(k) + A351442(k).

Links

Crossrefs

Cf. A000203, A003958, A322582, A351442, A351447.
Fixed points of A351444, positions of zeros in A351445.
Subsequences: A000396, A351443 (odd terms), A351440, A336702 (numbers k for which A064989(sigma(k)) = A064989(k)).

Programs

A351445 a(n) = A003958(sigma(n)) - A003958(n), where A003958 is multiplicative with a(p^e) = (p-1)^e and sigma is the sum of divisors function.

Original entry on oeis.org

0, 1, -1, 5, -2, 0, -5, 7, 8, 0, -8, 4, -6, -4, -6, 29, -12, 20, -14, 8, -11, -6, -20, 6, 14, 0, -4, 0, -20, -4, -29, 23, -18, -8, -22, 68, -18, -10, -18, 12, -28, -10, -32, 2, 8, -18, -44, 28, 0, 44, -28, 24, -44, 0, -36, 2, -32, -12, -50, 4, -30, -28, -12, 125, -36, -16, -50, 8, -42, -20, -66, 92, -36, 0
Offset: 1

Views

Author

Antti Karttunen, Feb 12 2022

Keywords

Links

Crossrefs

Cf. A000203, A003958, A351442, A351444, A351457 [= a(A003961(n))].
Cf. A351446 (positions of zeros), A351443 (odd terms there).
Cf. also A348736.

Programs

Formula

a(n) = A351442(n) - A003958(n) = A351444(n) - n.

A351444 a(n) = n - A003958(n) + A003958(sigma(n)), where A003958 is multiplicative with a(p^e) = (p-1)^e and sigma is the sum of divisors function.

Original entry on oeis.org

1, 3, 2, 9, 3, 6, 2, 15, 17, 10, 3, 16, 7, 10, 9, 45, 5, 38, 5, 28, 10, 16, 3, 30, 39, 26, 23, 28, 9, 26, 2, 55, 15, 26, 13, 104, 19, 28, 21, 52, 13, 32, 11, 46, 53, 28, 3, 76, 49, 94, 23, 76, 9, 54, 19, 58, 25, 46, 9, 64, 31, 34, 51, 189, 29, 50, 17, 76, 27, 50, 5, 164, 37, 74, 73, 82, 19, 66, 5, 136
Offset: 1

Views

Author

Antti Karttunen, Feb 12 2022

Keywords

Links

Crossrefs

Cf. A000203, A003958, A322582, A351442, A351445.
Cf. A351446 (fixed points), A351443 (odd terms there).

Programs

Formula

a(n) = A322582(n) + A351442(n) = n - A003958(n) + A003958(sigma(n)).
a(n) = n + A351445(n).

A351447 Numbers k for which A003958(sigma(k)) = 2*A003958(k), where A003958 is multiplicative with a(p^e) = (p-1)^e and sigma is the sum of divisors function.

Original entry on oeis.org

2, 98, 120, 136, 312, 520, 672, 888, 1080, 1120, 1464, 1480, 1752, 2440, 2520, 2808, 2912, 2920, 3420, 3768, 3848, 4632, 5880, 6048, 6280, 6344, 6552, 6648, 6664, 7512, 7592, 7720, 7992, 8181, 8288, 8892, 9528, 10104, 10968, 11080, 12464, 12520, 12984, 13176, 13664, 14712, 15288
Offset: 1

Views

Author

Antti Karttunen, Feb 12 2022

Keywords

Comments

Numbers k such that A351442(k) = 2*A003958(k).
In contrast, numbers x for which A064989(sigma(x)) = 2*A064989(x) seem to consist just of {2} followed by A005820: 2, 120, 672, 523776, ..., etc, which (also) contains as its subsequence all the odd terms of A336702 multiplied by 2.

Links

Crossrefs

Cf. A000203, A003958, A064989, A351442.
Subsequences: A005820 (3-perfect numbers), odd terms of A336702 doubled, the terms of A351443 doubled (2, 98, 81810, ...), A351448 (odd terms in this sequence).

Programs

A387159 Odd numbers k such that A173557(k) = A173557(sigma(k)), where A173557(n) is multiplicative with a(p^e) = p-1 and sigma is the sum of divisors function.

Original entry on oeis.org

1, 63, 135, 351, 875, 891, 999, 1647, 1859, 1971, 4239, 5211, 7479, 8451, 10719, 11367, 12339, 14607, 16317, 16551, 17847, 18171, 19791, 20439, 22103, 23679, 26919, 27951, 29511, 31131, 31407, 31487, 32427, 32751, 33399, 35667, 37287, 39231, 43767, 44739, 47331, 50571, 52191, 53811, 54459, 57319, 57699, 63207, 66771
Offset: 1

Views

Author

Antti Karttunen, Aug 19 2025

Keywords

Comments

Odd numbers k for which A173557(k) == A387157(k).

Links

Crossrefs

Odd terms of A387158.
Cf. A000203, A173557, A387157.
Cf. also A351443, A353679, A386425.

Programs

  • Mathematica
    A387159Q[k_] := OddQ[k] && #[k] == #[DivisorSigma[1, k]] & [Times @@ (FactorInteger[#][[All, 1]] - 1) &];
Select[Range[100000], A387159Q] (* Paolo Xausa, Aug 20 2025 *)
  • PARI
    A173557(n) = factorback(apply(p -> p-1,factor(n)[,1]));
is_A387159(n) = (n%2 && (A173557(sigma(n))==A173557(n)));
Showing 1-6 of 6 results.

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