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 1-10 of 21 results. Next

A062094 a(1) = 2, a(n) = concatenation of two closest factors of a(n-1) whose product equals a(n-1) or if a(n-1) is a prime then the concatenation of 1 and a(n-1).

Original entry on oeis.org

2, 12, 34, 217, 731, 1743, 2183, 3759, 21179, 121179, 931303, 1931303, 11175573, 11379829, 17896361, 18419721, 96919009, 889910891, 1889910891, 2742368917, 25741106537, 110203233579, 679231622473, 1679231622473, 7921256096487
Offset: 1

Views

Author

Amarnath Murthy, Jun 16 2001

Keywords

Examples

			a(1) = 2 is a prime hence a(2) = 12; a(3) = 34, 3*4 = 12 and 3<4.

Links

Crossrefs

Cf. A003681, A062095.

Programs

  • Mathematica
    f[n_Integer] := (d = Divisors[n]; l = Length[d]; If[ EvenQ[l], ToExpression[ ToString[ d[[l/2]] ] <> ToString[ d[[l/2 + 1]] ]], ToExpression[ ToString[d[[l/2 + .5]] ] <> ToString[ d[[l/2 + .5]] ]]] ); NestList[f, 2, 25]

Extensions

More terms from Robert G. Wilson v, Aug 08 2001

A062095 a(1) = 1, a(n) = concatenation of two closest factors of a(n-1) whose product equals a(n-1) or if a(n-1) is a prime then the concatenation of 1 and a(n-1).

Original entry on oeis.org

1, 11, 111, 337, 1337, 7191, 51141, 317047, 1317047, 2814687, 9312743, 25193697, 30981533, 51496017, 192326779, 1427134777, 4987286171, 6471777063, 61653104971, 259323776747, 737046253821, 7171027958513, 31727922601647
Offset: 1

Views

Author

Amarnath Murthy, Jun 16 2001

Keywords

Examples

			a(3) = 111 hence a(4) = 337, as 3*37 = 111 and 3 < 37.

Links

Crossrefs

Cf. A062094, A003681.

Programs

  • Mathematica
    f[n_Integer] := (d = Divisors[n]; l = Length[d]; If[ EvenQ[l], ToExpression[ ToString[ d[[l/2]] ] <> ToString[ d[[l/2 + 1]] ]], ToExpression[ ToString[d[[l/2 + .5]] ] <> ToString[ d[[l/2 + .5]] ]]] ); NestList[f, 1, 25]

Extensions

More terms from Robert G. Wilson v, Aug 08 2001

A082120 Smallest difference > 1 between d and n/d for any divisor d of n.

Original entry on oeis.org

2, 3, 4, 5, 6, 2, 8, 3, 10, 4, 12, 5, 2, 6, 16, 3, 18, 8, 4, 9, 22, 2, 24, 11, 6, 3, 28, 7, 30, 4, 8, 15, 2, 5, 36, 17, 10, 3, 40, 11, 42, 7, 4, 21, 46, 2, 48, 5, 14, 9, 52, 3, 6, 10, 16, 27, 58, 4, 60, 29, 2, 12, 8, 5, 66, 13, 20, 3, 70, 6, 72, 35, 10, 15, 4, 7, 78, 2, 24, 39, 82, 5
Offset: 3

Views

Author

Ralf Stephan, Apr 04 2003

Keywords

Comments

a(p) = p-1 for prime p.

Links

Crossrefs

Cf. A082119, A056737, A003681, A082121.

Programs

  • Maple
    F:= proc(n) local p;
  p:= min(select(t -> t - n/t > 1, numtheory:-divisors(n)));
  p - n/p
end proc:
map(F, [$3..100]); # Robert Israel, Aug 12 2015
  • Mathematica
    sd[n_]:=Min[Select[Abs[#-n/#]&/@Divisors[n],#>1&]]; Array[sd,90,3] (* Harvey P. Dale, Sep 28 2013 *)
  • PARI
    for(n=3, 100, v=divisors(n); r=sqrt(n); t=0; for(k=1, length(v), if(v[k]>=r, t=k; break)); if(v[t]^2==n, u=t, u=t-1); if(v[t]-v[u]<2, u=u-1; t=t+1); print1(v[t]-v[u]", "))

A082125 Smallest difference>1 between d and p/d for any divisor d of the partial product p of the sequence, starting with 4.

Original entry on oeis.org

4, 3, 4, 2, 4, 8, 16, 64, 512, 16384, 2097152, 2147483648, 140737488355328, 1180591620717411303424, 40564819207303340847894502572032, 365375409332725729550921208179070754913983135744
Offset: 0

Views

Author

Ralf Stephan, Apr 04 2003

Keywords

Comments

a(n) is a power of two for n>1 and log_2(a(n))=A073941(n) for n>2. - Ralf Stephan, Apr 16 2003

Links

Crossrefs

Cf. A082120, A003681 (starts with 2, 3), A082126.
Cf. A029744.

Programs

  • PARI
    p=4; print1(p, ", "); for(n=1, 50, v=divisors(p); r=sqrt(p); t=0; for(k=1, matsize(v)[2], if(v[k]>=r, t=k; break)); if(v[t]^2==p, u=t, u=t-1); if(v[t]-v[u]<2, u=u-1; t=t+1); print1(v[t]-v[u]", "); p=p*(v[t]-v[u]))

A082123 Smallest difference > 1 between d and p/d for any divisor d of the partial product p of the sequence, starting with 17.

Original entry on oeis.org

17, 16, 26, 36, 76, 94, 432, 37220, 996768, 158267352, 973348166592, 8429202561226344, 419324164827901536306744, 339991740461303603766175692597227316, 12025891484499365294341150949542442100059557280661504
Offset: 1

Views

Author

Ralf Stephan, Apr 04 2003

Keywords

Crossrefs

Cf. A082120, A003681 (starts with 2, 3), A082124.

Programs

  • Maple
    branch:= proc(m,dm, bestyet)
    local t,x, nby,r;
    nby:= bestyet;
    for t from F[m][2] by -1 to 0 do
      x:= dm*F[m][1]^t;
      if x >= nby then next
      elif x >= c then nby:= x
      elif (x*R[m] < c) or (m=nF) then break
      else nby:= branch(m+1,x,nby);
      fi
    od;
    nby
end proc:
P:= 17: A[1]:= 17:
for n from 2 to 15 do
  c:= ceil(1/2+1/2*sqrt(5+4*P));
  while not type(c,integer) do Digits:= 2*Digits; c:= eval(c) od:
  F:= ifactors(P)[2]; nF:= nops(F);
  F:= sort(F,(s,t)->s[1]>t[1]);
  R:= [seq(mul(F[i][1]^F[i][2],i=j+1..nF),j=1..nF)];
  d:= branch(1,1,P);
  A[n]:= d - P/d;
  P:= P*A[n]
od:
seq(A[n],n=1..15); # Robert Israel, May 20 2015
  • PARI
    p=17; print1(p,","); for(n=1,13,r=floor(sqrt(p)); d1=1; d2=1; nE=omega(p); P=factor(p); E=P[,2]; P=P[,1]; forvec(v=vector(nE,i,[0,E[i]]),x=prod(k=1,nE,P[k]^v[k]); if(x<=r && x>=d2,d1=d2; d2=x,if(x<=d2 && x>=d1,d1=x))); difer=p/d2-d2; if(difer<=1,difer=p/d1-d1); print1(difer","); p*=difer)

Extensions

a(12) from Herman Jamke (hermanjamke(AT)fastmail.fm), Nov 02 2006
a(13) from Herman Jamke (hermanjamke(AT)fastmail.fm), Nov 14 2006
a(14) and a(15) from Robert Israel, May 20 2015

A082126 Smallest difference>1 between d and p/d for any divisor d of the partial product p of the sequence, starting with 19.

Original entry on oeis.org

19, 18, 29, 27, 9, 27, 2187, 6561, 531441, 387420489, 7625597484987, 328256967394537077627, 381520424476945831628649898809, 235655016338368235499067731945871638181119123
Offset: 0

Views

Author

Ralf Stephan, Apr 04 2003

Keywords

Comments

Except for the first three, the members are all powers of 3. Proved by Luke Pebody, pers. comm.

Crossrefs

Cf. A082120, A003681 (starts with 2, 3), A082128.

Programs

  • PARI
    p=19; print1(p, ", "); for(n=1, 50, v=divisors(p); r=sqrt(p); t=0; for(k=1, matsize(v)[2], if(v[k]>=r, t=k; break));  if(v[t]^2==p, u=t, u=t-1);  if(v[t]-v[u]<2, u=u-1; t=t+1); print1(v[t]-v[u]", "); p=p*(v[t]-v[u]))

A082128 Smallest difference>1 between d and p/d for any divisor d of the partial product p of the sequence, starting with 21.

Original entry on oeis.org

21, 4, 5, 13, 8, 2, 32, 16, 64, 512, 131072, 4194304, 8589934592, 9007199254740992, 75557863725914323419136, 20769187434139310514121985316880384
Offset: 0

Views

Author

Ralf Stephan, Apr 04 2003

Keywords

Comments

For n>3, the members are all powers of two. Proved by Luke Pebody, pers. comm.

Crossrefs

Cf. A082120, A003681 (starts with 2, 3).

Programs

  • PARI
    p=21; print1(p, ", "); for(n=1, 50, v=divisors(p); r=sqrt(p); t=0; for(k=1, matsize(v)[2], if(v[k]>=r, t=k; break)); if(v[t]^2==p, u=t, u=t-1); if(v[t]-v[u]<2, u=u-1; t=t+1); print1(v[t]-v[u]", "); p=p*(v[t]-v[u]))

A060772 Minimal Thompson primes: a(n) is the smallest prime expressible as p1*p2*...*pk-q1*q2*...*qj, where k+j=n and {p1,...,qj} are the first n primes.

Original entry on oeis.org

7, 11, 13, 17, 107, 41, 157, 1811, 1579, 18859, 98411, 17659, 1995293, 3517693, 2396687, 211899707, 1537380473, 2507890183, 3952306763, 341777053, 599742417059, 1733348865733, 3316679648071, 33504570543263, 29075165225531, 853500418442743, 6016564035665531, 63537798773018491, 8517623042250013
Offset: 3

Views

Author

Len Smiley, Apr 24 2001

Keywords

Examples

			a(9)=157=(3*5*7*11*13)-(2*17*19*23)

Links

Crossrefs

Slight similarity with A003681.

Extensions

More terms from Naohiro Nomoto, Jun 24 2001
[Incorrect values a(23)-a(26) submitted by Jud McCranie, Jan 12 2016]
a(23)-a(60) (corrected and extended) from Don Reble, Jul 11 2020, added by N. J. A. Sloane, Jul 11 2020

A063383 a(1) = 6, a(n) = concatenation of two closest divisors of a(n-1) whose product equals a(n-1) or if a(n-1) is a prime then the concatenation of 1 and a(n-1).

Original entry on oeis.org

6, 23, 123, 341, 1131, 2939, 12939, 57227, 89643, 329881, 1073083, 1197553, 7171079, 17171079, 57301247, 208327509, 1171780577, 1219684137, 1478297171, 2587571433, 2795835979, 8663322733, 13666409441, 113666409441, 1030771102733, 2114885171103, 6993025586797
Offset: 1

Views

Author

Robert G. Wilson v, Aug 08 2001

Keywords

Links

Crossrefs

Cf. A003681, A062094 and A062095.

Programs

  • Mathematica
    f[ n_Integer ] := (d = Divisors[ n ]; l = Length[ d ]; If[ EvenQ[ l ], ToExpression[ ToString[ d[[ l/2 ] ] ] <> ToString[ d[[ l/2 + 1 ] ] ] ], ToExpression[ ToString[ d[[ l/2 + .5 ] ] ] <> ToString[ d[[ l/2 + .5 ] ] ] ] ] ); NestList[ f, 6, 25 ]
tcf[n_]:=Module[{d=Divisors[n],len},len=Length[d]/2;FromDigits[Flatten[ IntegerDigits/@Take[d,{len,len+1}]]]]; ctc[n_]:=If[PrimeQ[ n], 10^IntegerLength[ n]+n,tcf[n]]; NestList[ctc,6,30] (* Harvey P. Dale, May 19 2019 *)
  • Python
    from sympy import divisors, isprime
def aupton(terms):
    alst = [6]
    for n in range(2, terms+1):
        if isprime(alst[-1]): alst.append(int('1' + str(alst[-1])))
        else:
            divs = divisors(alst[-1])
            d1 = divs[(len(divs)-1)//2]
            d2 = alst[-1]//d1
            alst.append(int(str(d1) + str(d2)))
    return alst
print(aupton(27)) # Michael S. Branicky, Jun 23 2021

Extensions

Definition clarified by Harvey P. Dale, May 19 2019

A082121 Smallest difference>1 between d and p/d for any divisor d of the partial product p of the sequence, starting with 7.

Original entry on oeis.org

7, 6, 11, 19, 37, 109, 515, 3301, 267271, 130914197, 209015618081, 887384060899271, 58605404461258015758293
Offset: 0

Views

Author

Ralf Stephan, Apr 04 2003

Keywords

Crossrefs

Cf. A082120, A003681 (starts with 2, 3), A082122.

Programs

  • PARI
    p=7; print1(p, ", "); for(n=1, 50, v=divisors(p); r=sqrt(p); t=0; for(k=1, matsize(v)[2], if(v[k]>=r, t=k; break)); if(v[t]^2==p, u=t, u=t-1); if(v[t]-v[u]<2, u=u-1; t=t+1); print1(v[t]-v[u]", "); p=p*(v[t]-v[u]))
Showing 1-10 of 21 results. Next

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

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