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 1-7 of 7 results.

A243284 a(n) = the number of distinct ways of writing such products m = k^2 * j, 0 < j <= k, (j and k natural numbers) that m is in range [1,n]; Partial sums of A102354.

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 2, 3, 4, 4, 4, 4, 4, 4, 4, 5, 5, 6, 6, 6, 6, 6, 6, 6, 7, 7, 8, 8, 8, 8, 8, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 11, 12, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 15, 15, 15, 15, 15, 15, 15, 15, 16, 16, 16, 17
Offset: 1

Views

Author

Antti Karttunen, Jun 02 2014

Keywords

Comments

a(n) = the number of distinct ways of writing such products m = k^2 * j, 0 < j <= k, (j and k natural numbers) that m is in range [1,n].
Different ways to write product for the same m are counted separately, e.g. for 64, both 8^2 * 1 and 4^2 * 4 are counted, so a(64) = a(63)+2 = 13+2 = 15.
Differs from A243283 for the first time at n=48, where a(48)=11, while A243283(48)=10. This is because 48 = 2*2*2*2*3 is the first integer which can be represented in the form k^2 * j, 0 < j <= k (namely as 48 = 4^2 * 3), even though it is not a member of A070003.

Crossrefs

Partial sums of A102354.
Cf. A243283, A013928, A057627.

A102448 a(n) is the number of ways to write n = k^2 * j, j <= k, gcd(k,j) = 1, where j and k are positive integers.

Original entry on oeis.org

1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 0, 0, 0, 0
Offset: 1

Views

Author

Leroy Quet, Feb 23 2005

Keywords

Comments

Sum_{n>0} a(n)/n = 2.

Examples

			a(18) = 1 because 18 = k^2 * j, j <= k, gcd(k,j)=1, in one way: k=3, j=2.

Links

Crossrefs

Cf. A102354, A104021, A104023, A104025.

Programs

  • Mathematica
    t = Sort[ Flatten[ Table[ If[ GCD[j, k] == 1, k^2*j, {}], {k, 11}, {j, k}]]]; Table[ Count[t, n], {n, 105}] (* Robert G. Wilson v, Feb 25 2005 *)
  • PARI
    A102448(n) = sumdiv(n,d,((1==gcd(d,(n/d))) && issquare(d) && (sqrtint(d) >= (n/d)))); \\ Antti Karttunen, Aug 27 2017

Formula

a(n) <= A102354(n). - Antti Karttunen, Aug 27 2017

Extensions

More terms from Robert G. Wilson v, Feb 24 2005

A104020 Numbers that can be represented as k^2*j, 0 < j <= k.

Original entry on oeis.org

4, 9, 16, 18, 25, 32, 36, 48, 49, 50, 64, 72, 75, 81, 98, 100, 108, 121, 128, 144, 147, 162, 169, 180, 192, 196, 200, 225, 242, 243, 245, 256, 288, 289, 294, 300, 320, 324, 338, 361, 363, 384, 392, 400, 405, 432, 441, 448, 450, 484, 486, 500, 507, 512, 529, 567
Offset: 1

Views

Author

Leroy Quet and Robert G. Wilson v, Feb 24 2005

Keywords

Crossrefs

Cf. A104021, A102354, A104022, A104024, A104026.

Programs

  • Mathematica
    Take[ Union[ Flatten[ Table[ k^2*j, {k, 25}, {j, k - 1}]]], 56]

A243283 One more than the partial sums of the characteristic function of A070003.

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 2, 3, 4, 4, 4, 4, 4, 4, 4, 5, 5, 6, 6, 6, 6, 6, 6, 6, 7, 7, 8, 8, 8, 8, 8, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 11, 12, 12, 12, 12, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 14, 14, 14, 14, 14, 14, 14, 14, 15, 15, 15, 16
Offset: 1

Views

Author

Antti Karttunen, Jun 02 2014

Keywords

Comments

a(n) tells how many positive integers <= n are divisible by the square of their largest noncomposite divisor. (This definition includes 1 as it is divisible by 1^2.)
a(n) = n - A243285(n).
a(1) = 1 and for all n > 1, a(A070003(n-1)) = n, thus this sequence works as an inverse function for the injection {a(1) = 1, a(n>1) = A070003(n-1)} (a sequence which is the union of {1} and A070003).

Links

Crossrefs

One more than A243282.
Differs from A243284 for the first time at n=48. Here a(48)=10.
Cf. A070003, A102354, A243285, A057627.

A104022 Numbers that have more than one way of being written as k^2*j, 0 < j <= k.

Original entry on oeis.org

64, 100, 144, 196, 256, 324, 400, 484, 512, 576, 648, 676, 729, 784, 800, 900, 968, 1024, 1089, 1152, 1156, 1296, 1352, 1444, 1521, 1568, 1600, 1728, 1764, 1800, 1936, 2025, 2028, 2048, 2116, 2304, 2312, 2352, 2500, 2592, 2601, 2700, 2704, 2888, 2916
Offset: 1

Views

Author

Leroy Quet and Robert G. Wilson v, Feb 23 2005

Keywords

Crossrefs

Cf. A104023, A102354, A104020, A104024.

Programs

  • Mathematica
    t = Sort[ Flatten[ Table[k^2*j, {k, 55}, {j, k}]]]; u = Table[ Count[t, n], {n, 3000}]; Select[ Range[3000], u[[ # ]] > 1 &]

A104024 Least number k which can be written as k^2 * j, 0 < j <= k in n ways.

Original entry on oeis.org

2, 1, 64, 900, 5184, 32400, 57600, 176400, 705600, 1166400, 3240000, 6350400, 14288400, 37454400, 25401600, 87609600
Offset: 0

Views

Author

Leroy Quet and Robert G. Wilson v, Feb 23 2005

Keywords

Crossrefs

Cf. A102354, A104020, A104022.

Programs

  • Mathematica
    t = Sort[ Flatten[ Table[n = k^2*j; If[n < 10^7, n, {}], {k, 10000}, {j, k}]]]; l = Length[t]; t[[Select[ Range[l - 11], t[[ # ]] == t[[ # + 11]] &]]]

A104025 Least number k that can be written as k^2 * j with 0 < j <= k and gcd(k, j) = 1 in n ways.

Original entry on oeis.org

2, 1, 100, 900, 44100, 108900, 1232100, 11492100, 5336100, 12744900, 97416900
Offset: 0

Views

Author

Leroy Quet and Robert G. Wilson v, Feb 23 2005

Keywords

Crossrefs

Cf. A102354, A104021, A104023.

Programs

  • Mathematica
    t = Sort[ Flatten[ Table[n = k^2*j; If[ GCD[k, j] == 1 && 10^5 < n < 10^8, n, {}], {k, 15000}, {j, k}]]]; l = Length[t]; t[[Select[ Range[l - 11], t[[ # ]] == t[[ # + 9]] &]]]
Showing 1-7 of 7 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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