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.

A348930 a(n) = A038502(sigma(n)), where A038502 is fully multiplicative with a(3) = 1, and a(p) = p for any other prime p.

Original entry on oeis.org

1, 1, 4, 7, 2, 4, 8, 5, 13, 2, 4, 28, 14, 8, 8, 31, 2, 13, 20, 14, 32, 4, 8, 20, 31, 14, 40, 56, 10, 8, 32, 7, 16, 2, 16, 91, 38, 20, 56, 10, 14, 32, 44, 28, 26, 8, 16, 124, 19, 31, 8, 98, 2, 40, 8, 40, 80, 10, 20, 56, 62, 32, 104, 127, 28, 16, 68, 14, 32, 16, 8, 65, 74, 38, 124, 140, 32, 56, 80, 62, 121, 14, 28
Offset: 1

Views

Author

Antti Karttunen, Nov 04 2021

Keywords

Comments

Note that a(A005820(4)) = A005820(4) and a(A005820(6)) = A005820(6), i.e., the fourth and sixth 3-perfect numbers, 459818240 and 51001180160 are among the fixed points of this sequence, precisely because they are also terms of A323653. As the former factorizes as 459818240 = 256 * 5 * 7 * 19 * 37 * 73, it must follow that a(256)/256 * a(5)/5 * a(7)/7 * a(19)/19 * a(37)/37 * a(73)/73 = 1, because ratio a(n)/n is multiplicative. See also comments in A348738.

Links

Crossrefs

Cf. A000203, A000265, A005820, A038502, A323653, A348738, A348931, A348932.
Cf. also A161942, A336457.

Programs

  • Mathematica
    s[n_] := n / 3^IntegerExponent[n, 3]; Table[s[DivisorSigma[1, n]], {n, 1, 100}] (* Amiram Eldar, Nov 04 2021 *)
  • PARI
    A038502(n) = (n/3^valuation(n, 3));
A348930(n) = A038502(sigma(n));

Formula

Multiplicative with a(p^e) = A038502(1 + p + p^2 + ... + p^e).
a(n) = A038502(A000203(n)).
For all n >= 1, A000265(a(n)) = A336457(n).

A348754 Numbers k congruent to 1 or 5 mod 6, for which A064989(A064989(sigma(k))) > A064989(A064989(k)), where A064989 shifts the prime factorization one step towards lower primes, and sigma is the sum of divisors function.

Original entry on oeis.org

25, 49, 121, 169, 175, 275, 289, 325, 625, 841, 925, 1225, 1445, 1525, 1675, 1681, 1825, 2401, 3025, 3125, 3481, 3757, 3925, 4075, 4225, 4375, 4825, 5041, 5275, 5929, 6125, 6875, 6925, 7075, 7225, 7825, 7921, 8125, 8275, 8281, 9025, 9925, 10201, 10525, 10625, 10693, 11425, 11875, 12005, 12025, 13075, 13225, 13475
Offset: 1

Views

Author

Antti Karttunen, Nov 04 2021

Keywords

Comments

Sequence A003961(A003961(A348752(n))), n=1.., sorted into ascending order.
Not a subsequence of A348749. The first terms that occur here but not there are: 169, 175, 275, 1675, 3757, 4075, 5275, 7075, 8275, 10693, 12025, ...

Links

Crossrefs

Cf. A003961, A007310, A064989, A348750, A348752, A348753.
Cf. also A348749, A348932, A348936 (square roots of squares present).

Programs

  • Mathematica
    f[2, e_] := 1; f[p_, e_] := NextPrime[p, -1]^e; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; Select[Range[15000], MemberQ[{1, 5}, Mod[#, 6]] && s[s[DivisorSigma[1, #]]] > s[s[#]] &] (* Amiram Eldar, Nov 04 2021 *)
  • PARI
    A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
isA348754(n) = ((n%2)&&(n%3)&&(A064989(A064989(sigma(n))) > A064989(A064989(n))));

A348931 Numbers k congruent to 1 or 5 mod 6, for which A348930(k) < k.

Original entry on oeis.org

5, 11, 17, 23, 29, 35, 41, 47, 49, 53, 55, 59, 65, 71, 77, 83, 85, 89, 95, 101, 107, 113, 115, 119, 125, 131, 137, 143, 145, 149, 155, 161, 167, 169, 173, 179, 185, 187, 191, 197, 203, 205, 209, 215, 221, 227, 233, 235, 239, 245, 251, 253, 257, 263, 265, 269, 275, 281, 287, 293, 295, 299, 305, 311, 317, 319, 323
Offset: 1

Views

Author

Antti Karttunen, Nov 04 2021

Keywords

Comments

See comments in A348930.

Links

Crossrefs

Cf. A007310, A348930, A348932, A348933.
Cf. also A348741, A348753.

Programs

  • Mathematica
    s[n_] := n / 3^IntegerExponent[n, 3]; Select[Range[350], MemberQ[{1, 5}, Mod[#, 6]] && s[DivisorSigma[1, #]] < # &] (* Amiram Eldar, Nov 04 2021 *)
  • PARI
    A038502(n) = (n/3^valuation(n, 3));
A348930(n) = A038502(sigma(n));
isA348931(n) = ((n%2)&&(n%3)&&(A348930(n)

A348934 Numbers k congruent to 1 or 5 mod 6, for which A348930(k^2) > k^2.

Original entry on oeis.org

5, 11, 17, 23, 25, 29, 41, 47, 49, 53, 55, 59, 71, 83, 85, 89, 101, 107, 113, 115, 121, 125, 131, 137, 145, 149, 167, 169, 173, 179, 187, 191, 197, 205, 227, 233, 235, 239, 245, 251, 253, 257, 263, 265, 269, 275, 281, 289, 293, 295, 311, 317, 319, 343, 347, 353, 355, 359, 361, 383, 389, 391, 401, 415, 419, 425, 431
Offset: 1

Views

Author

Antti Karttunen, Nov 04 2021

Keywords

Comments

See comments in A348933.

Links

Crossrefs

Cf. A348930, A348932, A348933, A348936.

Programs

  • Mathematica
    s[n_] := n / 3^IntegerExponent[n, 3]; Select[Range[450], MemberQ[{1, 5}, Mod[#, 6]] && s[DivisorSigma[1, #^2]] > #^2 &] (* Amiram Eldar, Nov 04 2021 *)
  • PARI
    A038502(n) = (n/3^valuation(n, 3));
A348930(n) = A038502(sigma(n));
isA348934(n) = ((n%2)&&(n%3)&&(A348930(n^2)>(n^2)));
Showing 1-4 of 4 results.

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