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.

A329328 The squares in A329472.

Original entry on oeis.org

4, 49, 1024, 11881, 15876, 29241, 23530332816, 90070213689, 165698657721, 233002186209, 8246098046404, 363533405168704, 24015392820628036, 48563553937960000, 6648251155785800089, 497199122464645742436, 749745222626569665409, 10925409774976373110009
Offset: 1

Views

Author

Metin Sariyar, Nov 15 2019

Keywords

Comments

The first 18 terms are the sums of the first 1, 5, 27, 95, 110, 150, 135833, 265758, 360459, 427441, 2542860, 16883814, 137228168, 195143291, 2283242905, 19745293160, 24246846494, 92558706480 nonsquarefree numbers.
a(19) > 2*10^22. - Giovanni Resta, Nov 17 2019

Examples

			49 is a term because sum of first five nonsquarefree numbers is a square 4 + 8 + 9 + 12 + 16 = 49.

Crossrefs

Cf. A000290, A262783, A329472.

Programs

  • Mathematica
    p=0; Do[If[!SquareFreeQ[n],p=p+n; If[IntegerQ[p^(1/2)], Print[p]]], {n,1,10^8}]
  • PARI
    lista(nn) = {my(s=0); for(k=1, nn, if(omega(k)!=bigomega(k), s+=k; if(issquare(s), print1(s, ", ")))); } \\ Jinyuan Wang, Nov 17 2019

Formula

Equals A000290 intersection A329472.

Extensions

a(13)-a(18) from Giovanni Resta, Nov 17 2019

A373412 Sum of the n-th maximal antirun of nonsquarefree numbers differing by more than one.

Original entry on oeis.org

12, 99, 52, 180, 93, 49, 335, 279, 156, 629, 99, 540, 237, 245, 125, 521, 567, 450, 963, 340, 347, 728, 1386, 1080, 1637, 243, 244, 1511, 1610, 555, 852, 1171, 2142, 960, 985, 1689, 343, 1042, 351, 1068, 724, 732, 1116, 1905, 1980, 2898, 424, 2161, 3150, 2339
Offset: 1

Views

Author

Gus Wiseman, Jun 06 2024

Keywords

Comments

The length of this antirun is given by A373409.
An antirun of a sequence (in this case A013929) is an interval of positions at which consecutive terms differ by more than one.

Examples

			Row-sums of:
   4   8
   9  12  16  18  20  24
  25  27
  28  32  36  40  44
  45  48
  49
  50  52  54  56  60  63
  64  68  72  75
  76  80
  81  84  88  90  92  96  98
  99

Links

Crossrefs

The partial sums are a subset of A329472.
Functional neighbors: A068781, A373404, A373405, A373409, A373410, A373411, A373414.
A005117 lists the squarefree numbers, first differences A076259.
A013929 lists the nonsquarefree numbers, first differences A078147.
Cf. A025157, A049093, A049094, A061399, A101836, A112925, A143658, A294242, A350842, A372683, A373197.

Programs

  • Mathematica
    Total/@Split[Select[Range[100],!SquareFreeQ[#]&],#1+1!=#2&]//Most

A373414 Sum of the n-th maximal run of nonsquarefree numbers differing by one.

Original entry on oeis.org

4, 17, 12, 16, 18, 20, 49, 55, 32, 36, 40, 89, 147, 52, 54, 56, 60, 127, 68, 72, 151, 161, 84, 88, 90, 92, 96, 297, 104, 108, 112, 233, 241, 375, 128, 132, 271, 140, 144, 295, 150, 305, 156, 160, 162, 164, 337, 343, 351, 180, 184, 377, 192, 196, 198, 200, 204
Offset: 1

Views

Author

Gus Wiseman, Jun 06 2024

Keywords

Comments

The length of this run is given by A053797.
A run of a sequence (in this case A013929) is an interval of positions at which consecutive terms differ by one.

Examples

			Row-sums of:
   4
   8   9
  12
  16
  18
  20
  24  25
  27  28
  32
  36
  40
  44  45
  48  49  50

Links

Crossrefs

The partial sums are a subset of A329472.
Functional neighbors: A053797, A053806, A054265, A373406, A373412, A373413.
A005117 lists the squarefree numbers, first differences A076259.
A013929 lists the nonsquarefree numbers, first differences A078147.
Cf. A049093, A049094, A061398, A061399, A077641, A077643, A101836, A143658 A294242, A373123, A373197.

Programs

  • Mathematica
    Total/@Split[Select[Range[100],!SquareFreeQ[#]&],#1+1==#2&]//Most

A376164 Maximum of the n-th maximal run of nonsquarefree numbers (increasing by 1 at a time).

Original entry on oeis.org

4, 9, 12, 16, 18, 20, 25, 28, 32, 36, 40, 45, 50, 52, 54, 56, 60, 64, 68, 72, 76, 81, 84, 88, 90, 92, 96, 100, 104, 108, 112, 117, 121, 126, 128, 132, 136, 140, 144, 148, 150, 153, 156, 160, 162, 164, 169, 172, 176, 180, 184, 189, 192, 196, 198, 200, 204, 208
Offset: 1

Views

Author

Gus Wiseman, Sep 15 2024

Keywords

Examples

			The maximal runs of nonsquarefree numbers begin:
       4
     8   9
      12
      16
      18
      20
    24  25
    27  28
      32
      36
      40
    44  45
  48  49  50

Crossrefs

For length instead of maximum we have A053797 (firsts A373199).
For lengths of anti-runs we have A373409 (firsts A373573).
For sum instead of maximum we have A373414, anti A373412.
For minimum instead of maximum we have A053806, anti A373410.
For anti-runs instead of runs we have A068781.
For squarefree instead of nonsquarefree we have A373415, anti A007674.
For nonprime instead of nonsquarefree we have A006093 with 2 removed.
A005117 lists the squarefree numbers, first differences A076259.
A013929 lists the nonsquarefree numbers, differences A078147, sums A329472.
A061398 counts squarefree numbers between primes, nonsquarefree A061399.
A120992 gives squarefree run-lengths, anti A373127 (firsts A373128).
A373413 adds up each maximal run of squarefree numbers, min A072284.
A375707 counts squarefree numbers between consecutive nonsquarefree numbers.
Cf. A000040, A020754, A045882, A049094, A054265, A077641, A101836, A143658, A294242, A375709.

Programs

  • Mathematica
    Max/@Split[Select[Range[100],!SquareFreeQ[#]&],#1+1==#2&]//Most

A329598 Partial sums of the nontriangular numbers (A014132).

Original entry on oeis.org

2, 6, 11, 18, 26, 35, 46, 58, 71, 85, 101, 118, 136, 155, 175, 197, 220, 244, 269, 295, 322, 351, 381, 412, 444, 477, 511, 546, 583, 621, 660, 700, 741, 783, 826, 870, 916, 963, 1011, 1060, 1110, 1161, 1213, 1266, 1320, 1376, 1433, 1491, 1550, 1610, 1671, 1733
Offset: 1

Views

Author

Metin Sariyar and Robert G. Wilson v, Nov 17 2019

Keywords

Comments

Terms which are triangular: 6, 136, 351, 741, 2415, 3916, 5995, 12561, 17391, 23436, ..., .

Examples

			The nontriangular numbers begin 2, 4, 5, 7, ..., so their partial sums begin 2, 6, 11, 18, etc.

References

  • John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See p. 64.

Crossrefs

Cf. A329472, A014132, A086849.
Cf. A000217, A060432.

Programs

  • Mathematica
    triQ[n_] := IntegerQ @ Sqrt[8n + 1]; Accumulate@ Select[ Range@ 70, !triQ@# &]
  • Python
    from math import isqrt
def A329598(n): return (k:=(r:=isqrt(m:=n+1<<1))+int((m<<2)>(r<<2)*(r+1)+1)-1)*(k*(-k - 3) + 6*n - 2)//6 + (n*(n+3)>>1) # Chai Wah Wu, Jun 18 2024

Formula

a(n) = Sum_{i=1..n} A014132(i).
a(n) = A000217(n) + A060432(n). [corrected by Gerald Hillier, Jul 31 2022]
Showing 1-5 of 5 results.

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