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 1-7 of 7 results.

A064623 Primitive subsequence of A066031: terms of A066031 which are not a multiple of some previous terms.

Original entry on oeis.org

30, 70, 84, 105, 231, 234, 260, 286, 456, 528, 532, 627, 646, 805, 897, 1116, 1122, 1275, 1364, 1581, 1665, 1798, 1924, 2064, 2150, 2275, 2565, 2632, 2958, 2967, 3055, 3213, 3339, 3526, 3861, 4070, 4543, 4824, 5025, 5152, 5487, 5658, 5840, 6164, 6461
Offset: 1

Views

Author

Robert G. Wilson v and Vladeta Jovovic, Dec 12 2001

Keywords

Links

Programs

  • Mathematica
    a = {30}; Do[ If[ IntegerQ[ n / Apply[ Plus, First[ Transpose[ FactorInteger[n]]]]] && Mod[ n, n - EulerPhi[n]] != 0 && First[ Union[ Mod[n, a]]] != 0, a = Append[a, n]], {n, 1, 10000} ]; a
  • PARI
    f(n)=my(f=factor(n)[,1]); #f[,1]>1 && n%sum(i=1,#f,f[i])==0
list(lim)=my(v=List(),p=2);forprime(q=p+1,nextprime(lim),for(n=p+1,q-1, if(f(n),for(i=1,#v,if(n%v[i]==0,next(2)));listput(v,n)));p=q);Vec(v) \\ Charles R Greathouse IV, Sep 08 2012

A175650 a(n)=A066031(n)/sopf(A066031(n)).

Original entry on oeis.org

3, 6, 5, 7, 9, 7, 12, 10, 15, 14, 18, 11, 13, 24, 21, 13, 27, 20, 11, 30, 21, 28, 25, 36, 45, 19, 26, 48, 35, 42, 26, 35, 33, 19, 54, 40, 22, 49, 60, 19, 17, 56, 33, 50, 39, 72, 49, 75, 63, 23, 81, 23, 90, 38, 52, 63, 96, 70, 84, 52, 66, 38, 108, 31, 80, 34, 44, 98, 120, 51, 34
Offset: 1

Views

Author

Zak Seidov, Aug 02 2010

Keywords

Crossrefs

A008472, A066031.

Programs

  • Mathematica
    Reap[Do[PFL=PrimeFactorList[n];sopf=Plus@@PFL;
If[Length[PFL]>1&&Mod[n,sopf]==0,Sow[n/sopf]],{n,2,2000}]][[2,1]]

A086486 Numbers k such that the sum of the distinct prime divisors divides rad(k)=A007947(k).

Original entry on oeis.org

2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, 25, 27, 29, 30, 31, 32, 37, 41, 43, 47, 49, 53, 59, 60, 61, 64, 67, 70, 71, 73, 79, 81, 83, 89, 90, 97, 101, 103, 105, 107, 109, 113, 120, 121, 125, 127, 128, 131, 137, 139, 140, 149, 150, 151, 157, 163, 167
Offset: 1

Views

Author

Amarnath Murthy, Jul 28 2003

Keywords

Comments

Every prime power is a member.
Numbers with exactly two distinct prime divisors are not members of the sequence. - Victoria A Sapko (vsapko(AT)canes.gsw.edu), Sep 23 2003
Numbers k such that A008472(k) divides A007947(k).

Examples

			30 is a member. The prime divisors of 30 are 2, 3 and 5 and 2+3+5 = 10, divides 30.
84, however, is not a member because the sum of its distinct prime divisors (2+3+7=12) does not divide the product of its distinct prime divisors (2*3*7=42), even though 12 does divide 84. - _Harvey P. Dale_, Nov 26 2011, based on a comment from _Ray Chandler_

Links

Crossrefs

Cf. A086487, A066031. A proper subset of A089352.

Programs

  • Mathematica
    sdpQ[n_]:=Module[{dpds=Transpose[FactorInteger[n]][[1]]}, Divisible[ Times@@dpds,Total[dpds]]]; Select[Range[2,200],sdpQ] (* Harvey P. Dale, Nov 26 2011 *)

Extensions

More terms from Victoria A Sapko (vsapko(AT)canes.gsw.edu), Sep 23 2003
Edited by Franz Vrabec, Sep 03 2005

A190882 Numbers other than prime powers divisible by the sum of the squares of their prime divisors.

Original entry on oeis.org

46206, 72105, 73346, 92412, 96096, 97440, 98098, 99528, 113883, 117040, 127680, 134805, 138618, 143520, 146692, 150024, 165880, 165886, 184824, 192192, 194880, 196196, 199056, 216315, 234080, 255360, 269192, 276640, 277236, 287040, 288288, 292320, 293384, 294216, 298584, 300048, 331760
Offset: 1

Views

Author

Michel Lagneau, May 23 2011

Keywords

Comments

The number of distinct prime divisors of n is >= 3, because if n = p^a * q^b where p and q are distinct primes, p^2+q^2 | n => p+q == 0 (mod p) or 0 (mod q), but p==0 (mod q), or q==0 (mod p) is impossible.
Koninck & Luca show that this sequence is infinite. - Charles R Greathouse IV, Sep 08 2012

Examples

			46206 is in the sequence because the prime distinct divisors of this number are {2, 3, 17, 151} and 2^2 + 3^2 + 17^2 + 151^2 = 23103, then 46206 = 23103*2.

Links

Crossrefs

Cf. A066031.

Programs

  • Maple
    with(numtheory):for n from 1 to 200000 do:x:=factorset(n):n1:=nops(x):s:=0:for
  p from 1 to n1 do: s:=s+x[p]^2:od:if n1 >= 2 and irem(n,s)=0 then printf(`%d,`,n):else fi:od:
  • Mathematica
    Select[Range[2,332000],!PrimePowerQ[#]&&Divisible[#,Total[Select[ Divisors[#],PrimeQ]^2]]&] (* Harvey P. Dale, May 24 2022 *)
  • PARI
    is(n)=my(f=factor(n)[,1]);#f>2&n%sum(i=1,#f,f[i]^2)==0 \\ Charles R Greathouse IV, May 23 2011
  • PARI
    is(n)=n>9 && !isprimepower(n) && n%norml2(factor(n)[,1])==0 \\ Charles R Greathouse IV, Feb 03 2016

A268373 Numbers other than prime powers divisible by the sum of the cubes of their prime divisors.

Original entry on oeis.org

378, 480, 756, 960, 1134, 1440, 1512, 1920, 2268, 2400, 2548, 2646, 2880, 3024, 3402, 3840, 4320, 4536, 4800, 5096, 5292, 5760, 6048, 6804, 7200, 7680, 7938, 8640, 9072, 9600, 10192, 10206, 10584, 11520, 12000, 12096, 12960, 13608, 14400, 15360, 15876, 17280, 17836, 18144, 18522, 18711
Offset: 1

Views

Author

Michel Marcus, Feb 03 2016

Keywords

Comments

Koninck & Luca prove that this set is infinite. - Charles R Greathouse IV, Feb 03 2016

Examples

			The prime factors of 480 are 2, 3 and 5. The sum of their cubes is 2^3+3^3+5^3=160, and 480 is divisible by 160.

Links

Crossrefs

Cf. A066031, A190882.

Programs

  • Mathematica
    Select[Range[10^4], Length[(p = FactorInteger[#][[;;,1]])] > 1 && Divisible[#, Total[p^3]] &] (* Amiram Eldar, Sep 05 2019 *)
  • PARI
    isok(n) = my(f = factor(n)[,1]) ; (#f>2) && ((n % sum(k=1, #f, f[k]^3)) == 0);

A268417 Numbers other than prime powers divisible by the sum and the sum of squares of their prime divisors.

Original entry on oeis.org

99528, 117040, 143520, 199056, 234080, 287040, 288288, 294216, 298584, 349440, 357357, 383040, 398112, 430560, 468160, 574080, 576576, 585200, 588432, 597168, 631488, 698880, 717600, 766080, 796224, 819280, 861120, 864864, 870870, 882648, 895752, 901824, 936320, 957000
Offset: 1

Views

Author

Michel Marcus, Feb 04 2016

Keywords

Comments

Intersection of A066031 and A190882.
Prime divisors taken without multiplicity. - Harvey P. Dale, Dec 27 2018

Links

Crossrefs

Cf. A066031, A190882.

Programs

  • Mathematica
    dssQ[n_]:=Module[{pf=FactorInteger[n][[All,1]]},!PrimePowerQ[ n] && Divisible[ n, Total[pf]]&&Divisible[n,Total[pf^2]]]; Select[ Range[ 960000],dssQ] (* Harvey P. Dale, Dec 27 2018 *)
  • PARI
    isok(n) = my(f = factor(n)[,1]); (#f>2) && ((n % vecsum(f)) == 0) && ((n % sum(k=1, #f, f[k]^2)) == 0);

A268418 Numbers other than prime powers divisible by the sum, the sum of squares and the sum of cubes of their prime divisors.

Original entry on oeis.org

12192180, 15724800, 24384360, 31449600, 36576540, 47174400, 48768720, 60960900, 62899200, 73153080, 78624000, 85345260, 94348800, 97537440, 109729620, 110073600, 121921800, 125798400, 134113980, 141523200, 146306160, 157248000, 158498340, 170690520, 182882700, 188697600, 195074880
Offset: 1

Views

Author

Michel Marcus, Feb 04 2016

Keywords

Links

Crossrefs

Intersection of A268373 and A066031 and A190882.
Intersection of A268373 and A268417.

Programs

  • PARI
    isok(n) = my(f = factor(n)[,1]); (#f>2) && ((n % vecsum(f)) == 0) && ((n % sum(k=1, #f, f[k]^2)) == 0) && ((n % sum(k=1, #f, f[k]^3)) == 0);
Showing 1-7 of 7 results.

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