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 1-3 of 3 results.

A167895 Primes in A175040.

Original entry on oeis.org

3, 5, 7, 11, 29, 41, 47, 73, 107, 137, 167, 173, 193, 211, 263, 269, 277, 307, 421, 433, 509, 601, 607, 643, 647, 683, 733, 739, 751, 797, 809, 821, 853, 857, 863, 883, 907, 911, 937, 977, 1031, 1069, 1229, 1231, 1321, 1367, 1373, 1447, 1523, 1553, 1567
Offset: 1

Views

Author

Giovanni Teofilatto, Nov 15 2009

Keywords

Crossrefs

Cf. A175040.

Extensions

More terms from R. J. Mathar, Dec 14 2009

A071562 Numbers n such that the sum of the middle divisors of n (A071090) is not zero.

Original entry on oeis.org

1, 2, 4, 6, 8, 9, 12, 15, 16, 18, 20, 24, 25, 28, 30, 32, 35, 36, 40, 42, 45, 48, 49, 50, 54, 56, 60, 63, 64, 66, 70, 72, 77, 80, 81, 84, 88, 90, 91, 96, 98, 99, 100, 104, 108, 110, 112, 117, 120, 121, 126, 128, 130, 132, 135, 140, 143, 144, 150, 153, 154, 156, 160
Offset: 1

Views

Author

Robert G. Wilson v, May 30 2002

Keywords

Comments

Numbers n such that A067742(n) is nonzero.
Numbers of the form m*k with m <= k <= 2m. - Vladeta Jovovic, May 07 2005
Numbers occurring in A100345 (except 0). - Franklin T. Adams-Watters, Apr 04 2010
This sequence is closed under multiplication. If n = a*b with a <= b <= 2a, and m = c*d with c <= d <= 2c, then min(a*d,b*c)*max(a*d,b*c) is a factorization of m*n with the specified property. - Franklin T. Adams-Watters, Apr 07 2010
Also numbers n with the property that the number of parts in the symmetric representation of sigma(n) is odd. - Michel Marcus and Omar E. Pol, Apr 25 2014. (For a proof see the link in A071561.) - Hartmut F. W. Hoft, Sep 09 2015
Among these numbers, those with sigma(n) also odd are 1, 2, 4, 8, 9, 16, ..., that is, probably A028982 (squares and twice squares). - Michel Marcus, Jun 21 2014
Records in A244367. - Omar E. Pol, Jul 27 2014
Starting from a(5), the sequence is a subset of a sequence formed out of the prime factorization of A129912(m), m >= 5; see associated PARI code in Prog section. - Bill McEachen, Jan 25 2018
For numbers n = 2^m * q, m >= 0, q odd, and where r(n) = floor( (sqrt(8n+1) - 1)/2 ), the symmetric representation of sigma(n) has an odd number of parts precisely when there exists an odd divisor d of n satisfying d <= r(n) and d*2^(m+1) > r(n); see the link for a proof and see the associated Mathematica code. - Hartmut F. W. Hoft, Feb 12 2018
All hexagonal numbers A000384 > 0 are in the sequence. - Omar E. Pol, Aug 28 2018

Examples

			From _Hartmut F. W. Hoft_, Feb 12 2018: (Start)
63 = 3^2*7 is in the sequence since 7*2^1 > r(63) = 10.
80 = 2^4*5 is in the sequence since 1*2^5 > r(80) = 12. (End)

Links

Crossrefs

Cf. A067742.
The complement is A071561.
Cf. A000203, A028982, A071090, A100345, A175040, A176039 (primitive elements), A237270, A237271, A237593, A244367.

Programs

  • Mathematica
    f[n_] := Plus @@ Select[ Divisors[n], Sqrt[n/2] <= # < Sqrt[n*2] &]; Select[ Range[175], f[ # ] != 0 &]
(* Related to the symmetric representation of sigma *)
(* subsequence of odd parts of number k for m <= k <= n *)
(* Function a237270[] is defined in A237270 *)
(* Using Wilson's Mathematica program (see above) I verified the equality of both for numbers k <= 10000 *)
a071562[m_,n_]:=Select[Range[m,n],OddQ[Length[a237270[#]]]&]
a071562[1,160] (* data *)
(* Hartmut F. W. Hoft, Jun 23 2014 *)
(* implementation using the odd divisor property *)
evenExp[n_] := First[NestWhile[{#[[1]]+1, #[[2]]/2}&, {0, n}, EvenQ[Last[#]]&]]
oddSRQ[n_] := Module[{e=2^evenExp[n], Floor[(Sqrt[8n+1]-1)/2]}, Select[Divisors[n/e], #<=r&&2 e #>r&]!={}]
a071562D[m_, n_] := Select[Range[m, n], oddSRQ]
a071562D[1, 160] (* data *) (* Hartmut F. W. Hoft, Feb 12 2018 *)
  • PARI
    is(n)=fordiv(n,d, if(d^2>=n/2 && d^2<2*n, return(1))); 0 \\ Charles R Greathouse IV, Aug 01 2016
  • PARI
    is(n,f=factor(n))=my(t=(n+1)\2); fordiv(f,d, if(d^2>=t, return(d^2<2*n))); 0 \\ Charles R Greathouse IV, Jan 22 2018
  • PARI
    list(lim)=my(v=List(),t); forfactored(n=1,lim\1, t=(n[1]+1)\2; fordiv(n[2],d, if(d^2>=t, if(d^2<2*n[1], listput(v,n[1])); break))); Vec(v) \\ Charles R Greathouse IV, Jan 22 2018
  • PARI
    /* functional code associated to the A129912 comment above */ for(j5=5, length(A129912), a=Mat(); a=factor(A129912[j5]); sum2=0; for(i5=1, length(a[,2]), sum2=sum2+a[i5,2]); listput(final,length(a[,1])*sum2)); v=Set(final); \\ Bill McEachen, Jan 25 2018

A100345 Triangle read by rows: T(n,k) = n*(n+k), 0 <= k <= n.

Original entry on oeis.org

0, 1, 2, 4, 6, 8, 9, 12, 15, 18, 16, 20, 24, 28, 32, 25, 30, 35, 40, 45, 50, 36, 42, 48, 54, 60, 66, 72, 49, 56, 63, 70, 77, 84, 91, 98, 64, 72, 80, 88, 96, 104, 112, 120, 128, 81, 90, 99, 108, 117, 126, 135, 144, 153, 162, 100, 110, 120, 130, 140, 150, 160, 170, 180, 190
Offset: 0

Views

Author

Reinhard Zumkeller, Nov 18 2004

Keywords

Comments

Distinct members (except 0) are in A071562. Numbers occurring at least twice are in A175040. - Franklin T. Adams-Watters, Apr 04 2010

Examples

			Triangle begins:
   0
   1   2
   4   6   8
   9  12  15  18
  16  20  24  28  32
  25  30  35  40  45  50
  36  42  48  54  60  66  72
  49  56  63  70  77  84  91  98
  64  72  80  88  96 104 112 120 128

Crossrefs

Cf. A000290, A000384, A001105, A002378, A005563, A014107, A028347, A028552, A028557, A028560, A028563, A028566, A028569, A046092, A054000, A071355, A071562, A085789, A098603, A175040.

Programs

  • Mathematica
    Table[n(n+k),{n,0,10},{k,0,n}]//Flatten (* Harvey P. Dale, Aug 16 2018 *)
  • PARI
    row(n) = vector(n+1, k, n*(n+k-1)); \\ Amiram Eldar, May 09 2025

Formula

T(n,0) = A000290(n).
T(n,1) = A002378(n) for n > 0.
T(n,2) = A005563(n) for n > 1.
T(n,3) = A028552(n) for n > 2.
T(n,4) = A028347(n+2) for n > 3.
T(n,5) = A028557(n) for n > 4.
T(n,6) = A028560(n) for n > 5.
T(n,7) = A028563(n) for n > 6.
T(n,8) = A028566(n) for n > 7.
T(n,9) = A028569(n) for n > 8.
T(n,10) = A098603(n) for n > 9.
T(n,n-5) = A071355(n-4) for n > 4.
T(n,n-4) = A054000(n-1) for n > 3.
T(n,n-3) = A014107(n) for n > 2.
T(n,n-2) = A046092(n-1) for n > 1.
T(n,n-1) = A000384(n) for n > 0.
T(n,n) = A001105(n).
Row sums give A085789 for n > 0.
G.f.: x*(1 + 2*y + 6*x^3*y^2 - 3*x^2*y*(1 + 2*y) + x*(1 - 3*y + 2*y^2))/((1 - x)^3*(1 - x*y)^3). - Stefano Spezia, Jul 03 2025
Showing 1-3 of 3 results.

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