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 1-5 of 5 results.

A351457 a(n) = A351456(n) - A339905(n).

Original entry on oeis.org

0, -1, -2, 8, -5, -6, -8, -4, 14, -11, -6, 8, -12, -18, -22, 84, -14, -2, -20, -12, -36, -18, -20, -24, 0, -28, -40, -16, -29, -46, -18, 40, -36, -32, -58, 296, -28, -42, -56, -44, -32, -76, -44, 24, -66, -48, -44, 136, 8, -36, -64, -16, -50, -104, -66, -72, -84, -59, -30, -72, -50, -54, -100, 1028, -92, -84, -66
Offset: 1

Views

Author

Antti Karttunen, Feb 12 2022

Keywords

Links

Crossrefs

Cf. A000203, A003958, A003961, A003973, A339905, A351442, A351456.
Cf. also A348736.

Programs

  • PARI
    A339905(n) = if(1==n,n,my(f=factor(n)); for(i=1,#f~,f[i,1] = nextprime(1+f[i,1])-1); factorback(f));
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); };
A351456(n) = A003958(sigma(A003961(n)));
A351457(n) = (A351456(n) - A339905(n));

Formula

a(n) = A351445(A003961(n)) = A351456(n) - A339905(n).

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).

A351456 a(n) = A003958(sigma(A003961(n))), where A003958 is multiplicative with a(p^e) = (p-1)^e, A003961 multiplicative with a(prime(k)^e) = prime(1+k)^e, and sigma is the sum of divisors function.

Original entry on oeis.org

1, 1, 2, 12, 1, 2, 2, 4, 30, 1, 6, 24, 4, 2, 2, 100, 4, 30, 2, 12, 4, 6, 8, 8, 36, 4, 24, 24, 1, 2, 18, 72, 12, 4, 2, 360, 12, 2, 8, 4, 10, 4, 2, 72, 30, 8, 8, 200, 108, 36, 8, 48, 8, 24, 6, 8, 4, 1, 30, 24, 16, 18, 60, 1092, 4, 12, 4, 48, 16, 2, 36, 120, 4, 12, 72, 24, 12, 8, 12, 100, 700, 10, 16, 48, 4, 2, 2, 24
Offset: 1

Views

Author

Antti Karttunen, Feb 12 2022

Keywords

Links

Crossrefs

Cf. A000203, A003958, A003961, A003973, A151800, A339905, A351442, A351457.
Cf. also A326042.

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); };
A351456(n) = A003958(sigma(A003961(n)));

Formula

Multiplicative with a(p^e) = A003958(1 + q + ... + q^e), where q = nextPrime(p) = A151800(p).
a(n) = A003958(A003973(n)) = A351442(A003961(n)) = A003958(A000203(A003961(n))).
a(n) = A351457(n) + A339905(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

A351448 Odd 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

8181, 400869, 1507005, 3918213, 11151837, 22002273, 26669007, 47319957, 58170393, 73843245, 75825981, 83488077, 94338513, 108277641, 119656197, 126889821, 137740257, 163057941, 184758813, 191992437, 199226061, 202842873, 204768225, 220926933, 228160557, 258457473, 264328677, 277602471, 300496797
Offset: 1

Views

Author

Antti Karttunen, Feb 12 2022

Keywords

Comments

Odd numbers k such that A351442(k) = 2*A003958(k).
Any hypothetical odd term of A005820, if such a term exists, should appear in this sequence, in A347391, and in A016754 (odd squares).
None of the first 33 terms is a square, and all of them except 75825981 and 204768225 are multiples of 81. Note that 81 is one of the terms of A008848 (and of A231484), squares whose sum of divisors is also square (with A000203(81) = 121).

Links

Crossrefs

Odd terms in A351447.
Cf. A000203, A003958, A005820, A008848, A016754, A231484, A339905, A347391, A351442.

Programs

  • PARI
    A003958(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]--); factorback(f); };
isA351448(n) = (n%2 && (A003958(sigma(n)) == 2*A003958(n)));
Showing 1-5 of 5 results.

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