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.

A348749 Odd numbers k for which A064989(sigma(k)) > 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

9, 25, 45, 49, 75, 81, 117, 121, 225, 243, 289, 325, 333, 405, 441, 529, 549, 605, 625, 657, 675, 729, 841, 925, 1053, 1089, 1125, 1215, 1225, 1413, 1445, 1521, 1525, 1575, 1665, 1681, 1737, 1825, 1875, 2025, 2205, 2401, 2475, 2493, 2601, 2817, 2825, 2925, 2997, 3025, 3033, 3125, 3249, 3481, 3573, 3645, 3675, 3789
Offset: 1

Views

Author

Antti Karttunen, Nov 02 2021

Keywords

Comments

Sequence obtained when A003961 is applied to A348739 and the terms are sorted into ascending order.
From Robert Israel, Nov 12 2024: (Start)
If a and b are terms and are coprime, then a * b is a term.
If p > 2 is in A053182, Legendre's conjecture implies p^2 is in this sequence. (End)

Links

Crossrefs

Cf. A000203, A003961, A053182, A064989, A326042, A348739, A348748, A348939 (terms of A228058 that occur here).
Cf. also A348742, A348754.

Programs

  • Maple
    g:= prevprime: g(2):= 1:
f:= proc(n) local F,t;
  F:= ifactors(n)[2];
  mul(g(t[1])^t[2],t=F)
end proc:
select(t -> f(numtheory:-sigma(t)) > f(t), [seq(i,i=1..4000,2)]); # Robert Israel, Nov 12 2024
  • Mathematica
    f[2, e_] := 1; f[p_, e_] := NextPrime[p, -1]^e; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; Select[Range[1, 4000, 2], s[DivisorSigma[1, #]] > 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)};
isA348749(n) = ((n%2)&&(A064989(sigma(n)) > A064989(n)));

A348753 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

5, 7, 11, 13, 17, 19, 23, 29, 31, 35, 37, 41, 43, 47, 53, 55, 59, 61, 65, 67, 71, 73, 77, 79, 83, 85, 89, 91, 95, 97, 101, 103, 107, 109, 113, 115, 119, 125, 127, 131, 133, 137, 139, 143, 145, 149, 151, 155, 157, 161, 163, 167, 173, 179, 181, 185, 187, 191, 193, 197, 199, 203, 205, 209, 211, 215, 217, 221, 223, 227
Offset: 1

Views

Author

Antti Karttunen, Nov 04 2021

Keywords

Comments

Sequence A003961(A003961(A348751(n))), n>=1, sorted into ascending order.
a(38) = 125 is the first term not in A276378.
Not a subsequence of A348748. The first terms that occur here but not there are: 529, 605, 2825, 6425, 7025, 8425, 10825, 15425, 16025, 16325, 16925, 17689, ...
The first squares in this sequence are: 361, 529, 961, 1369, 1849, 2209, 2809, 3721, etc., see A348935 for their square roots.
Of the natural numbers < 2^20, 347712 are in this sequence and only 1812 in A348754.

Links

Crossrefs

Cf. A003961, A007310, A064989, A276378, A348750, A348751, A348754.
Cf. also A348748, A348931, A348935.

Programs

  • Mathematica
    f[2, e_] := 1; f[p_, e_] := NextPrime[p, -1]^e; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; Select[Range[250], 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)};
isA348753(n) = ((n%2)&&(n%3)&&(A064989(A064989(sigma(n))) < A064989(A064989(n))));

A348932 Numbers k congruent to 1 or 5 mod 6, for which A348930(k) > k.

Original entry on oeis.org

7, 13, 19, 25, 31, 37, 43, 61, 67, 73, 79, 91, 97, 103, 109, 121, 127, 133, 139, 151, 157, 163, 175, 181, 193, 199, 211, 217, 223, 229, 241, 247, 259, 271, 277, 283, 289, 301, 307, 313, 325, 331, 337, 343, 349, 367, 373, 379, 397, 403, 409, 421, 427, 433, 439, 457, 463, 469, 475, 481, 487, 499, 511, 523, 529, 541
Offset: 1

Views

Author

Antti Karttunen, Nov 04 2021

Keywords

Comments

See comments in A348930.

Links

Crossrefs

Cf. A007310, A348930, A348931.
Cf. also A348742, A348754.

Programs

  • Mathematica
    s[n_] := n / 3^IntegerExponent[n, 3]; Select[Range[550], 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));
isA348932(n) = ((n%2)&&(n%3)&&(A348930(n)>n));

A348936 Numbers k congruent to 1 or 5 mod 6, for which A064989(A064989(sigma(k^2))) > A064989(A064989(k^2)), where A064989 shifts the prime factorization one step towards lower primes, and sigma is the sum of divisors function.

Original entry on oeis.org

5, 7, 11, 13, 17, 25, 29, 35, 41, 49, 55, 59, 65, 71, 77, 85, 89, 91, 95, 101, 115, 119, 121, 125, 131, 143, 145, 155, 161, 167, 169, 173, 175, 185, 187, 203, 205, 209, 215, 221, 227, 235, 245, 253, 265, 275, 287, 289, 293, 295, 305, 319, 323, 325, 329, 343, 355, 361, 365, 377, 383, 385, 391, 413, 415, 425, 445, 451
Offset: 1

Views

Author

Antti Karttunen, Nov 04 2021

Keywords

Comments

Square roots of squares present in A348754.
See comments in A348935.

Links

Crossrefs

Cf. A000203, A003961, A007310, A064989, A348750, A348754, A348934, A348935.

Programs

  • Mathematica
    f[2, e_] := 1; f[p_, e_] := NextPrime[p, -1]^e; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; Select[Range[450], MemberQ[{1, 5}, Mod[#, 6]] && s[s[DivisorSigma[1, #^2]]] > s[s[#^2]] &] (* 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)};
isA348936(n) = ((n%2)&&(n%3)&&(A064989(A064989(sigma(n^2))) > A064989(A064989(n^2))));

A348752 Numbers k for which A348750(k) > k.

Original entry on oeis.org

4, 9, 12, 16, 20, 25, 28, 32, 36, 48, 49, 64, 72, 80, 81, 100, 112, 116, 121, 128, 144, 162, 176, 180, 192, 196, 200, 208, 212, 225, 236, 240, 242, 243, 252, 256, 268, 272, 288, 300, 304, 320, 324, 336, 361, 384, 400, 405, 432, 441, 448, 450, 464, 468, 484, 496, 500, 512, 560, 567, 576, 588, 592, 596, 604, 625, 640
Offset: 1

Views

Author

Antti Karttunen, Nov 02 2021

Keywords

Crossrefs

Cf. A003961, A064989, A348750, A348751.
Cf. A348754 (corresponding 5-rough numbers, terms of A007310).

Programs

Showing 1-5 of 5 results.

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