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.

A305676 a(n) = A305671(A305672(n)).

Original entry on oeis.org

7, 24, 72, 144, 240, 336, 504, 720, 576, 720, 1440, 2016, 2880, 4320, 5760, 10080, 10080, 10080, 10080, 8640, 8640, 10080, 17280, 15120, 17280, 17280, 17280, 20160, 30240, 40320, 60480, 120960, 181440, 241920, 362880, 483840, 725760, 1451520, 2177280, 2419200, 2419200
Offset: 1

Views

Author

Felix Fröhlich, Jun 08 2018

Keywords

Comments

Do any numbers x exist that never occur in this sequence? Can this be proven for any particular value of x? For example, does 96 occur in this sequence? Note that 96 occurs four times as a value of sigma among the initial 100 composites.

Crossrefs

Cf. A145899, A305671, A305672, A305673, A305675, A305676.

Programs

  • PARI
    composite(n) = my(i=0); forcomposite(c=1, , i++; if(i==n, return(c)))
mcv(v) = my(w=vecsort(v, , 8), count=vector(#w), ind=0, i=0); for(x=1, #w, for(y=1, #v, if(w[x]==v[y], count[x]++))); for(k=1, #count, if(count[k]==vecmax(count), ind=k; i++)); if(i > 1, return(0), return(w[ind]))
a305671(n) = my(v=[]); for(k=1, n, v=concat(v, sigma(composite(k)))); mcv(v)
terms(n) = my(x=0, k=1, i=0); while(1, if(a305671(k) > 0, print1(a305671(k), ", "); i++); if(i==n, break); while(a305671(k) > 0, k++); while(a305671(k)==0, k++))
terms(11) \\ Print initial 11 terms of the sequence

A305672 Indices i where a run of nonzero values starts in A305671.

Original entry on oeis.org

1, 8, 35, 88, 162, 225, 363, 386, 427, 508, 998, 1633, 1901, 3091, 4816, 5905, 6179, 6297, 6674, 7221, 7293, 7715, 13100, 13130, 13238, 14927, 15012, 15520, 22413, 32219, 39561, 79864, 148622, 175500, 274750, 402174
Offset: 1

Views

Author

Felix Fröhlich, Jun 08 2018

Keywords

Comments

For n > 1, numbers i such that A305671(i) != 0 and A305671(i-1) = 0.

Crossrefs

Cf. A305671, A305673, A305674, A305675, A305676.

Programs

  • Mathematica
    Block[{c = Select[Range[10^4], CompositeQ], s}, s = DivisorSigma[1, c]; Prepend[1 + Accumulate[Total /@ #], #[[1, 1]]] &@ Partition[Length /@ SplitBy[#, # == 0 &], 2, 2] &@ Array[If[Length@ # == 1, #[[1, 1]], 0] &@ Last@ SplitBy[SortBy[Tally@ Take[s, #], Last], Last] &, Length@ s]] (* Michael De Vlieger, Jul 21 2018 *)
  • PARI
    my(v=[], i=0, b=0, idx=1); forcomposite(c=1, , my(sig=sigma(c)); if(#v < sig, v=concat(v, vector(sig-#v))); v[sig]++; for(k=1, #v, if(v[k]==vecmax(v), i++)); if(i < 2, if(b==0, print1(idx, ", "); b=1)); if(i > 1, b=0); idx++; i=0)

A305673 Indices i where a run of zeros starts in A305671.

Original entry on oeis.org

2, 15, 55, 108, 195, 256, 370, 419, 437, 782, 1616, 1857, 3055, 4806, 5851, 6142, 6275, 6487, 7161, 7278, 7591, 13041, 13122, 13179, 14904, 14979, 15451, 21767, 32056, 39478, 79649, 148518, 174716, 273952, 400581
Offset: 1

Views

Author

Felix Fröhlich, Jun 08 2018

Keywords

Comments

Numbers i such that A305671(i) = 0 and A305671(i-1) != 0.

Crossrefs

Cf. A305671, A305672, A305674, A305675, A305676.

Programs

  • Mathematica
    Block[{c = Select[Range[10^3], CompositeQ], s}, s = DivisorSigma[1, c]; Select[SplitBy[Partition[Position[Array[If[Length@ # == 1, #[[1, 1]], 0] &@ Last@ SplitBy[SortBy[Tally@ Take[s, #], Last], Last] &, Length@ s], 0][[All, 1]], 2, 1], Differences], Length@ # > 1 &][[All, 1, 1]]] (* Michael De Vlieger, Jun 14 2018 *)
  • PARI
    mcv(v) = my(w=vecsort(v, , 8), count=vector(#w), ind=0, i=0); for(x=1, #w, for(y=1, #v, if(w[x]==v[y], count[x]++))); for(k=1, #count, if(count[k]==vecmax(count), ind=k; i++)); if(i > 1, return(0), return(w[ind]))
my(v=[], i=1, t=0); forcomposite(c=1, , v=concat(v, [sigma(c)]); if(mcv(v)==0, if(t==0, print1(i, ", ")); t++, t=0); i++)

Extensions

More terms from Michael De Vlieger, Jun 14 2018

A305675 Run lengths of successive equal terms in A305671.

Original entry on oeis.org

1, 6, 7, 20, 20, 33, 20, 54, 33, 30, 31, 107, 7, 16, 33, 8, 10, 71, 274, 216, 618, 17, 224, 44, 1154, 36, 1715, 10, 1035, 54, 237, 37, 96, 22, 190, 187, 487, 60, 57, 15, 298, 124, 5326, 59, 22, 8, 49, 59, 1666, 23, 52, 33, 439, 69, 6247, 646, 9643, 163, 7259
Offset: 1

Views

Author

Felix Fröhlich, Jun 08 2018

Keywords

Comments

First differences of A305672 UNION A305673.

Crossrefs

Cf. A305671, A305672, A305673, A305674, A305676.

Programs

  • Mathematica
    Block[{c = Select[Range[10^4], CompositeQ], s}, s = DivisorSigma[1, c]; Most[Length /@ SplitBy[#, # == 0 &]] &@ Array[If[Length@ # == 1, #[[1, 1]], 0] &@ Last@ SplitBy[SortBy[Tally@ Take[s, #], Last], Last] &, Length@ s]] (* Michael De Vlieger, Jul 22 2018 *)
  • PARI
    composite(n) = my(i=0); forcomposite(c=1, , i++; if(i==n, return(c)))
mcv(v) = my(w=vecsort(v, , 8), count=vector(#w), ind=0, i=0); for(x=1, #w, for(y=1, #v, if(w[x]==v[y], count[x]++))); for(k=1, #count, if(count[k]==vecmax(count), ind=k; i++)); if(i > 1, return(0), return(w[ind]))
a305671(n) = my(v=[]); for(k=1, n, v=concat(v, sigma(composite(k)))); mcv(v)
terms(n) = my(i=0, j=1); for(k=1, oo, if(a305671(k)==a305671(k+1), j++, if(j > 0, print1(j, ", "); i++; j=1)); if(i==n, break))
terms(20) \\ Print initial 20 terms of the sequence

A305674 Indices i such that A305671(i) != A305671(i-1).

Original entry on oeis.org

2, 8, 15, 35, 55, 88, 108, 162, 195, 225, 256, 363, 370, 386, 419, 427, 437, 508, 782, 998, 1616, 1633, 1857, 1901, 3055, 3091, 4806, 4816, 5851, 5905, 6142, 6179, 6275, 6297, 6487, 6674, 7161, 7221, 7278, 7293, 7591, 7715, 13041, 13100, 13122, 13130, 13179
Offset: 1

Views

Author

Felix Fröhlich, Jun 08 2018

Keywords

Crossrefs

Cf. A305671, A305672, A305673, A305675, A305676.

Programs

Showing 1-5 of 5 results.

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