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.

Previous Showing 11-16 of 16 results.

A347690 Values of k such that (product of first k squarefree numbers) / (sum of first k squarefree numbers) is an integer.

Original entry on oeis.org

1, 3, 8, 10, 12, 14, 16, 21, 22, 24, 30, 31, 34, 36, 37, 38, 39, 42, 43, 46, 51, 52, 55, 59, 60, 61, 62, 66, 67, 68, 72, 73, 75, 77, 78, 82, 83, 87, 91, 93, 96, 97, 98, 99, 101, 102, 105, 110, 115, 121, 124, 125, 129, 135, 136, 139, 140, 143, 145, 147, 151
Offset: 1

Views

Author

N. J. A. Sloane, Sep 19 2021

Keywords

Comments

a(n) is the number of terms in the numerator (or denominator) when A322609(n) is found.

Links

Crossrefs

Cf. A005117, A111059, A173143, A322607, A322608.

Programs

  • Mathematica
    Module[{nn=4000,sf},sf=Select[Range[nn],SquareFreeQ];Table[ If[ IntegerQ[ Times@@Take[sf,n]/Total[Take[sf,n]]],n,Nothing],{n,Length[sf]}]] (* Harvey P. Dale, Sep 19 2021 *)

Extensions

More terms from Harvey P. Dale, Sep 19 2021

A358038 Partial sums of the cubefree numbers.

Original entry on oeis.org

1, 3, 6, 10, 15, 21, 28, 37, 47, 58, 70, 83, 97, 112, 129, 147, 166, 186, 207, 229, 252, 277, 303, 331, 360, 390, 421, 454, 488, 523, 559, 596, 634, 673, 714, 756, 799, 843, 888, 934, 981, 1030, 1080, 1131, 1183, 1236, 1291, 1348, 1406, 1465, 1525, 1586, 1648, 1711
Offset: 1

Views

Author

Amiram Eldar, Oct 29 2022

Keywords

Comments

First differs from A025706 and A025730 at n = 14.
The analogous sequence with squarefree numbers is A173143.

Links

Crossrefs

Cf. A002117, A004709, A173143.
Cf. A025706, A025730.

Programs

  • Mathematica
    Accumulate[Select[Range[100], Max[FactorInteger[#][[;; , 2]]] < 3 &]]

Formula

a(n) = Sum_{k=1..n} A004709(k).
a(n) = (zeta(3)/2)*n^2 + O(n^(3/2+eps)) (Weiyi, 2004).

Extensions

Corrected by Amiram Eldar, Jan 02 2024

A194128 (Partial sums of the squarefree integers) that are prime.

Original entry on oeis.org

3, 11, 17, 521, 647, 1327, 9437, 14153, 16141, 16831, 17299, 18013, 21839, 24029, 28429, 30293, 31247, 32213, 35201, 36929, 37991, 42013, 46643, 51481, 55259, 61837, 65951, 67349, 70663, 72109, 78553, 89017, 90647, 91193
Offset: 1

Views

Author

Robert G. Wilson v, Aug 15 2011

Keywords

Comments

Select from the partial sums of the squarefree integers those partial sums that are prime.

Links

Crossrefs

Cf. A005117, A173143.

Programs

  • Mathematica
    Select[ Accumulate@ Select[ Range@ 550, SquareFreeQ], PrimeQ]

A355544 Numbers k such that the arithmetic mean of the first k squarefree numbers is an integer.

Original entry on oeis.org

1, 3, 6, 37, 75, 668, 1075, 37732, 742767, 1811865, 3140083, 8937770, 108268896, 282951249, 633932500, 1275584757, 60455590365
Offset: 1

Views

Author

Amiram Eldar, Jul 06 2022

Keywords

Comments

Numbers k such that A173143(k) is divisible by k.
The corresponding quotients A173143(k)/k are 1, 2, 4, 29, ..., and the corresponding values of A005117(k) are 1, 3, 7, 59, ... (see the link for more values).

Examples

			3 is a term since the arithmetic mean of the first 3 squarefree numbers, (1+2+3)/3 = 2, is an integer.

Links

Crossrefs

Cf. A005117, A173143.
Similar sequences: A045345, A050226, A056550, A064605, A064606, A064607, A064610, A064611, A064612, A048290.

Programs

  • Mathematica
    s = Select[Range[10^6], SquareFreeQ]; r = Accumulate[s]/Range[Length[s]]; ind = Position[r, _?IntegerQ] // Flatten
  • PARI
    upto(n) = my(s=0,k=0); forsquarefree(m=1, n, s+=m[1]; k+=1; if(s%k == 0, print1(k, ", "))); \\ Daniel Suteu, Jul 06 2022

Extensions

a(17) from Daniel Suteu, Jul 06 2022

A382664 Partial sums of the exponentially odd numbers (A268335).

Original entry on oeis.org

1, 3, 6, 11, 17, 24, 32, 42, 53, 66, 80, 95, 112, 131, 152, 174, 197, 221, 247, 274, 303, 333, 364, 396, 429, 463, 498, 535, 573, 612, 652, 693, 735, 778, 824, 871, 922, 975, 1029, 1084, 1140, 1197, 1255, 1314, 1375, 1437, 1502, 1568, 1635, 1704, 1774, 1845, 1918
Offset: 1

Views

Author

Amiram Eldar, Apr 02 2025

Keywords

Links

Crossrefs

Cf. A065489, A268335.
Similar sequences: A173143, A174172, A358038, A362971.

Programs

  • Mathematica
    Accumulate[Select[Range[100], AllTrue[FactorInteger[#][[;; , 2]], OddQ] &]]
  • PARI
    isexpodd(n) = {my(f = factor(n)); for(i=1, #f~, if(!(f[i, 2] % 2), return (0))); 1;}
list(lim) = {my(s = 0); for(k = 1, lim, if(isexpodd(k), s += k; print1(s, ", "))); }

Formula

a(n) = Sum_{k=1..n} A268335(k).
a(n) ~ c * n^2 / 2, where c = Product_{p prime} (1 + 1/(p^2+p-1)) = 1.419562... (A065489).

A194080 Partial sums of the squarefree integers that are triangular.

Original entry on oeis.org

1, 3, 6, 45, 2775, 239778, 3293461, 3815703, 8142176137578, 25025955346953, 27262684236385, 2430563686654005, 17543349622599580, 95966329187365806, 69152680331115315990, 130831703828369011153, 10553338813847920825078, 120456843371021570368956
Offset: 1

Views

Author

Robert G. Wilson v, Aug 19 2011

Keywords

Comments

For the partial sum to the squarefree integer: 1, 2, 3, 11, 94, 890, 3290, 3538, 5175587, 9073717, 9470490, 89421695, 240240143, ..., .
No more terms in the first 4*10^11 partial sums. - Donovan Johnson, Feb 20 2012

Crossrefs

Cf. A005117, A173143.

Programs

  • Mathematica
    triQ[n_] := IntegerQ@ Sqrt[ 8n + 1]; k = 1; s = 0; lst = {}; While[k < 3*10^8, If[ SquareFreeQ@ k, s = s + k; If[ triQ@ s, AppendTo[lst, s]; Print[{k, s}]]]; k++]

Extensions

a(14)-a(16) from Donovan Johnson, Aug 24 2011
a(17)-a(18) from Donovan Johnson, Feb 20 2012
Previous Showing 11-16 of 16 results.

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

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