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.

A360905 Starts of run of 3 consecutive integers that are all terms of A360902.

Original entry on oeis.org

7939375, 12799375, 20410623, 30466287, 56661199, 83365119, 105146991, 197479375, 235838223, 259360623, 293380623, 555499375, 657880623, 691579375, 871374591, 871720623, 953280495, 975079375, 996393391, 1032100623, 1047979375, 1096579375, 1348000623, 1355419375
Offset: 1

Views

Author

Amiram Eldar, Feb 25 2023

Keywords

Examples

			7939375 is a term since A034444(7939375) = A005361(7939375) = 4, A034444(7939376) = A005361(7939376) = 4, and A034444(7939377) = A005361(7939377) = 4.

Links

Crossrefs

Subsequence of A360902 and A360903.
Cf. A005361, A034444.

Programs

  • Mathematica
    q[n_] := Module[{e = FactorInteger[n][[;; , 2]]}, Times @@ e == 2^Length[e]]; q[1] = True; seq[kmax_] := Module[{tri = q /@ Range[3], s = {}, k = 4}, While[k < kmax, If[And @@ tri, AppendTo[s, k - 3]]; tri = Join[Rest[tri], {q[k]}]; k++]; s]; seq[10^7]
  • PARI
    is(k) = {my(e = factor(k)[,2]); prod(i = 1, #e, e[i]) == 2^#e; }
lista(kmax) = {my(tri = vector(3, i, is(i)), k = 4); while(k < kmax, if(vecsum(tri) == 3, print1(k-3, ", ")); tri = concat(vecextract(tri, "^1"), is(k)); k++); }

A360903 a(n) is the least number that has exactly 2^n squarefree divisors and exactly 2^n powerful divisors.

Original entry on oeis.org

1, 4, 36, 720, 25200, 1940400, 227026800, 42454011600, 10486140865200, 3858899838393600, 1902437620328044800, 1120535758373218387200, 953575930375608847507200, 977415328634999068694880000, 1218836914807843838662515360000, 1775845384875028472931284879520000
Offset: 0

Views

Author

Amiram Eldar, Feb 25 2023

Keywords

Comments

a(n) is the least term k of A360902 with A034444(k) = A005361(k) = 2^n.

Examples

			a(1) = 4 since 4 is the least number that has 2^1 = 2 squarefree divisors (1 and 2) and 2 powerful divisors (1 and 4).
a(2) = 36 since 36 is the least number that has 2^2 = 4 squarefree divisors (1, 2, 3 and 6) and 4 powerful divisors (1, 4, 9 and 36).

Links

Crossrefs

Subsequence of A025487 and A360902.
Cf. A005361, A034444.

Programs

  • Mathematica
    f1[p_, e_] := 2; s1[1] = 1; s1[n_] := Times @@ f1 @@@ FactorInteger[n]; f2[p_, e_] := e; s2[1] = 1; s2[n_] := Times @@ f2 @@@ FactorInteger[n]; v = Cases[Import["https://oeis.org/A025487/b025487.txt", "Table"], {, }][[;; , 2]]; With[{m = 9}, seq = Table[0, {m}]; Do[If[(s = s1[v[[k]]]) == s2[v[[k]]], e = IntegerExponent[s, 2] + 1; If[e <= m && seq[[e]] == 0, seq[[e]] = v[[k]]]], {k, 1, Length[v]}]; seq]

A360904 Numbers k such that k and k+1 both have the same number of squarefree divisors and powerful divisors.

Original entry on oeis.org

48, 2511, 5328, 6723, 7856, 10287, 15471, 15632, 16640, 18063, 20816, 28592, 33124, 36368, 38799, 39600, 40400, 40816, 54512, 57121, 60624, 67472, 75248, 79375, 83024, 88047, 93231, 101168, 119375, 126927, 134703, 137456, 146688, 147824, 148224, 154448, 160624
Offset: 1

Views

Author

Amiram Eldar, Feb 25 2023

Keywords

Comments

Numbers k such that k and k+1 are both terms of A360902.

Examples

			48 is a term since A034444(48) = A005361(48) = 4 and A034444(49) = A005361(49) = 2.

Links

Crossrefs

Subsequence of A360902.
A360905 is a subsequence.
Cf. A005361, A034444.

Programs

  • Mathematica
    q[n_] := Module[{e = FactorInteger[n][[;; , 2]]}, Times @@ e == 2^Length[e]]; q[1] = True; seq[kmax_] := Module[{s = {}, k = 1, q1 = q[1], q2}, Do[q2 = q[k]; If[q1 && q2, AppendTo[s, k-1]]; q1 = q2, {k, 2, kmax}]; s]; seq[2*10^5]
  • PARI
    is(k) = {my(e = factor(k)[,2]); prod(i = 1, #e, e[i]) == 2^#e; }
lista(kmax) = {my(is1 = is(1), i2); for(k=2, kmax, is2 = is(k); if(is1 && is2, print1(k-1, ", ")); is1 = is2); }

A360906 Numbers with the same number of cubefree divisors and 3-full divisors.

Original entry on oeis.org

1, 16, 81, 384, 625, 640, 896, 1296, 1408, 1664, 2176, 2401, 2432, 2944, 3712, 3968, 4374, 4736, 5248, 5504, 6016, 6784, 7552, 7808, 8576, 9088, 9216, 9344, 10000, 10112, 10624, 10935, 11392, 12416, 12928, 13184, 13696, 13952, 14464, 14641, 15309, 16256, 16768
Offset: 1

Views

Author

Amiram Eldar, Feb 25 2023

Keywords

Comments

Numbers k such that A073184(k) = A190867(k).
Numbers whose largest cubefree divisor (A007948) and cubefull part (A360540) have the same number of divisors (A000005).
If k and m are coprime terms, then k*m is also a term.
The characteristic function of this sequence depends only on prime signature.
1 is the only cubefree (A004709) term.
Includes the 4th powers of squarefree numbers (1 and A113849).
The 4th powers of primes (A030514) are the only terms that are prime powers (A246655).
Numbers of the for m*p^(3*2^k+1), where m is squarefree, p is prime, gcd(m, p) = 1 and omega(m) = k, are all terms. In particular, this sequence includes numbers of the form p^7*q, where p != q are primes (A179664), and numbers of the form p^13*q*r where p, q, and r are distinct primes.
The corresponding numbers of cubefree (or 3-full) divisors are 1, 3, 3, 6, 3, 6, 6, 9, 6, 6, 6, 3, 6, 6, ... .

Links

Crossrefs

Cf. A000005, A004709, A007948, A073184, A190867, A246655, A360540, A360902.
Subsequences: A030514, A113849, A179664, A360907.

Programs

  • Mathematica
    q[n_] := Module[{e = FactorInteger[n][[;; , 2]]}, Times @@ (Min[#, 3] & /@ (e + 1)) == Times @@ (Max[#, 1] & /@ (e - 1))]; q[1] = True; Select[Range[10^4], q]
  • PARI
    is(k) = {my(e = factor(k)[,2]); prod(i = 1, #e, min(e[i] + 1, 3)) == prod(i = 1, #e, max(e[i] - 1, 1)); }
Showing 1-4 of 4 results.

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