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 1-8 of 8 results.

A134333 Numbers n whose number of prime factors (counted with multiplicity) is a prime factor of n.

Original entry on oeis.org

4, 6, 10, 12, 14, 18, 22, 26, 27, 30, 34, 38, 42, 45, 46, 58, 62, 63, 66, 74, 75, 78, 80, 82, 86, 94, 99, 102, 105, 106, 114, 117, 118, 120, 122, 134, 138, 142, 146, 147, 153, 158, 165, 166, 171, 174, 178, 180, 186, 194, 195, 200, 202, 206, 207, 214, 218, 222, 226
Offset: 1

Views

Author

Hieronymus Fischer, Oct 23 2007

Keywords

Examples

			a(1) = 4, since 4 has 2 prime factors and 2 is a prime factor of 4.
a(4) = 12, since 12 = 2*2*3 has 3 prime factors, and 3 is a prime factor of 12.
a(21) = 75, since 75 = 3*3*5 has 3 prime factors. and 3 is a prime factor of 75.
9 = 3*3 is not a term, since the number of prime factors (=2) is not a divisor of 9.
28 = 2*2*7 is not a term, since the number of prime factors (=3) is not a divisor of 28.

Links

Crossrefs

Cf. A000040, A001222, A100118, A046363, A133620, A133621, A133880, A133890, A133900, A133910, A133911, A046346, A134331, A134332, A134334, A134344, A134376, A063989.

Programs

  • Mathematica
    fQ[n_] := Module[{d = Total[Transpose[FactorInteger[n]][[2]]]}, PrimeQ[d] && Mod[n, d] == 0]; Select[Range[2, 226], fQ] (* T. D. Noe, Apr 05 2013 *)
  • PARI
    a(n)=my(t=bigomega(n)); n%t==0 && isprime(t) \\ Charles R Greathouse IV, Sep 14 2015

Formula

a(n) << n log n/(log log n)^k for any fixed k. - Charles R Greathouse IV, Sep 14 2015

Extensions

Sequence definition corrected and examples added by Hieronymus Fischer, Apr 05 2013

A134344 Composite numbers such that the arithmetic mean of their prime factors (counted with multiplicity) is prime.

Original entry on oeis.org

4, 8, 9, 16, 20, 21, 25, 27, 32, 33, 44, 49, 57, 60, 64, 68, 69, 81, 85, 93, 105, 112, 116, 121, 125, 128, 129, 133, 145, 156, 169, 177, 180, 188, 195, 205, 212, 213, 217, 220, 231, 237, 243, 249, 253, 256, 265, 272, 275, 289, 297, 309, 332, 336, 343, 356, 361
Offset: 1

Views

Author

Hieronymus Fischer, Oct 23 2007

Keywords

Comments

Originally, the definition started with "Nonprime numbers ...". This may be misleading, since 1 is also nonprime, but has no prime factors. - Hieronymus Fischer, May 05 2013

Examples

			a(1) = 4, since 4 = 2*2 and the arithmetic mean (2+2)/2 = 2 is prime.
a(5) = 20, since 20 = 2*2*5 and the arithmetic mean (2+2+5)/3 = 3 is prime.

Links

Crossrefs

Cf. A000040, A001222, A100118, A046363, A133620, A133621, A133880, A133890, A133900, A133910, A133911, A046346, A134331, A134332, A134333, A134334.

Programs

  • Mathematica
    ampfQ[n_]:=PrimeQ[Mean[Flatten[Table[#[[1]],{#[[2]]}]&/@FactorInteger[ n]]]]; nn=400;Select[Complement[Range[nn],Prime[Range[ PrimePi[nn]]]], ampfQ] (* Harvey P. Dale, Nov 06 2012 *)
  • PARI
    is(n)=if(n<4,return(0)); my(f=factor(n),s=sum(i=1,#f~,f[i,1]*f[i,2])/sum(i=1,#f~,f[i,2])); (#f~>1 || f[1,2]>1) && denominator(s)==1 && isprime(s) \\ Charles R Greathouse IV, Sep 14 2015

Extensions

Definition clarified by Hieronymus Fischer, May 05 2013

A134376 Numbers whose sum of prime factors (counted with multiplicity) is not prime.

Original entry on oeis.org

1, 4, 8, 9, 14, 15, 16, 18, 20, 21, 24, 25, 26, 27, 30, 32, 33, 35, 36, 38, 39, 42, 44, 46, 49, 50, 51, 55, 57, 60, 62, 64, 65, 66, 68, 69, 70, 72, 74, 77, 78, 81, 84, 85, 86, 87, 91, 92, 93, 94, 95, 98, 100, 102, 105, 106, 110, 111, 112, 114, 115, 116, 119, 120, 121, 122
Offset: 1

Views

Author

Hieronymus Fischer, Oct 23 2007

Keywords

Comments

The first term is 1, since 1 has no prime factors and so the sum of prime factors evaluates to zero.
Conjecture: a(n) ~ n. - Charles R Greathouse IV, Apr 28 2015

Examples

			a(2) = 4, since 4 = 2*2 and 2+2 = 4 is not prime.
a(5) = 14, since 14 = 2*7 and 2+7 = 9 is not prime.

Links

Crossrefs

Cf. A000040, A001222, A100118, A046363, A133620, A133621, A133880, A133890, A133900, A133910, A133911, A046346, A134331, A134332, A134333, A134334, A134344, A078177.

Programs

  • Mathematica
    Select[Range[150],!PrimeQ[Total[Flatten[Table[#[[1]],#[[2]]]&/@ FactorInteger[ #]]]]&] (* Harvey P. Dale, Jul 05 2021 *)
  • PARI
    sopfr(n)=my(f=factor(n)); sum(i=1,#f~,f[i,1]*f[i,2])
is(n)=!isprime(sopfr(n)) \\ Charles R Greathouse IV, Apr 28 2015

Extensions

Edited by the author at the suggestion of T. D. Noe, May 20 2013

A134334 Numbers which are not divisible by the number of their prime factors (counted with multiplicity).

Original entry on oeis.org

8, 9, 15, 20, 21, 25, 28, 32, 33, 35, 39, 44, 48, 49, 50, 51, 52, 54, 55, 57, 64, 65, 68, 69, 70, 72, 76, 77, 81, 85, 87, 90, 91, 92, 93, 95, 98, 108, 110, 111, 112, 115, 116, 119, 121, 123, 124, 125, 126, 128, 129, 130, 133, 135, 141, 143, 145, 148, 150, 154, 155, 159
Offset: 1

Views

Author

Hieronymus Fischer, Oct 23 2007

Keywords

Comments

The asymptotic density of this sequence is 1 (Erdős and Pomerance, 1990). - Amiram Eldar, Jul 10 2020

Examples

			a(1) = 8, since 8 = 2*2*2 has 3 prime factors and 8 is not divisible by 3.
a(3) = 15, since 15 = 3*5 has 2 prime factors and 15 is not divisible by 2.

Links

Crossrefs

Cf. A000040, A001222, A074946 (complement), A100118, A046363, A133620, A133621.
Cf. A133880, A133890, A133900, A133910, A133911, A046346, A134331, A134332, A134333, A134335, A134344, A134376.

Programs

  • Mathematica
    Select[Range[2,200],Mod[#,PrimeOmega[#]]!=0&] (* Harvey P. Dale, May 13 2023 *)
  • PARI
    isok(n) = (n % bigomega(n)) \\ Michel Marcus, Jul 15 2013

A133911 Number of prime factors (counted with multiplicity) of the period numbers defined by A133900.

Original entry on oeis.org

0, 2, 2, 4, 2, 5, 2, 6, 4, 6, 2, 8, 2, 6, 5, 8, 2, 9, 2, 8, 5, 7, 2, 10, 4, 7, 6, 8, 2, 12, 2, 10, 6, 8, 5, 12, 2, 8, 6, 11, 2, 12, 2, 9, 8, 8, 2, 13, 4, 10, 6, 9, 2, 12, 5, 11, 6, 8, 2, 14, 2, 8, 8, 12, 5, 13, 2, 10, 6, 13, 2, 14, 2, 9, 8, 10, 5, 13, 2, 13, 8, 10, 2, 17, 5, 9, 7, 11, 2, 16, 5, 10, 7, 9
Offset: 1

Views

Author

Hieronymus Fischer, Oct 20 2007

Keywords

Examples

			a(6)=5, since A133900(6)=72=2*2*2*3*3.
a(12)=8, since A133900(12)=864=2*2*2*2*2*3*3*3.

Crossrefs

Cf. A000040, A001222, A100118, A046363, A133620, A133621, A133623, A133630, A133635.
Cf. A133872, A133880, A133890, A133900, A133910, A134331.

Formula

a(n)=A001222(A133900(n)).

A134332 Integer part of the arithmetic mean of the prime factors (counted with multiplicity) of the period numbers defined by A133900.

Original entry on oeis.org

1, 2, 3, 2, 5, 2, 7, 2, 3, 3, 11, 2, 13, 3, 3, 2, 17, 2, 19, 2, 4, 4, 23, 2, 5, 5, 3, 3, 29, 3, 31, 2, 5, 5, 5, 2, 37, 6, 6, 2, 41, 3, 43, 4, 3, 7, 47, 2, 7, 3, 7, 4, 53, 2, 7, 3, 8, 8, 59, 2, 61, 9, 4, 2, 8, 3, 67, 5, 9, 3, 71, 2, 73, 9, 4, 5, 8, 4, 79, 2, 3, 9, 83, 3, 9, 11, 10, 3, 89, 2, 9, 6, 11, 12
Offset: 1

Views

Author

Hieronymus Fischer, Oct 23 2007

Keywords

Examples

			a(6)=2, since floor(A134331(6)/A133911(6))=floor(12/5)=2.
a(7)=7, since floor(A134331(7)/A133911(7))=floor(14/2)=7.

Crossrefs

Cf. A000040, A100118, A046363, A133620, A133621, A133623, A133630, A133635.
Cf. A133872, A133880, A133890, A133900, A133910, A133911, A134331.

Formula

a(n)=floor(A134331(n)/A133911(n)) for n>1, defining a(1):=1.
a(n)=n, if n is a prime or 1.

A078177 Composite numbers with an integer arithmetic mean of all prime factors.

Original entry on oeis.org

4, 8, 9, 15, 16, 20, 21, 25, 27, 32, 33, 35, 39, 42, 44, 49, 50, 51, 55, 57, 60, 64, 65, 68, 69, 77, 78, 81, 85, 87, 91, 92, 93, 95, 105, 110, 111, 112, 114, 115, 116, 119, 121, 123, 125, 128, 129, 133, 140, 141, 143, 145, 155, 156, 159, 161, 164, 169, 170, 177, 180
Offset: 1

Views

Author

Reinhard Zumkeller, Nov 20 2002

Keywords

Comments

That is, composite numbers such that the arithmetic mean of their prime factors (counted with multiplicity) is an integer.

Examples

			60 = 2*2*3*5: (2+2+3+5)/4 = 3, therefore 60 is a term.

Links

Crossrefs

Cf. A078175, A001222, A100118, A046363, A133620, A133621.
Cf. A133880, A133890, A133900, A133910, A133911, A046346, A134331, A134332, A134333, A134334, A134344, A134376.

Programs

  • Mathematica
    Select[Range[200], CompositeQ[#] && IntegerQ[Mean[Flatten[Table[#[[1]], #[[2]]]& /@ FactorInteger[#]]]]&] (* Jean-François Alcover, Aug 03 2018 *)
  • PARI
    lista(nn) = {forcomposite(n=1, nn, my(f = factor(n)); if (! (sum(k=1, #f~, f[k,1]*f[k,2]) % vecsum(f[,2])), print1(n, ", ")););} \\ Michel Marcus, Feb 22 2016

Formula

A001414(a(n)) == 0 modulo A001222(a(n)).

Extensions

Edited by N. J. A. Sloane, May 30 2008 at the suggestion of R. J. Mathar

A134335 Numbers such that the arithmetic mean of their prime factors (counted with multiplicity) is an integer, but not a prime.

Original entry on oeis.org

15, 35, 39, 42, 50, 51, 55, 65, 77, 78, 87, 91, 92, 95, 110, 111, 114, 115, 119, 123, 140, 141, 143, 155, 159, 161, 164, 170, 183, 185, 186, 187, 189, 201, 203, 204, 209, 215, 219, 221, 222, 225, 230, 235, 236, 242, 247, 258, 259, 264, 267, 284, 285, 287, 290
Offset: 1

Views

Author

Hieronymus Fischer, Oct 23 2007

Keywords

Examples

			a(1) = 15, since 15 = 3*5 and (3+5)/2 = 4 is not prime.
a(5) = 50, since 50 = 2*5*5 and (2+5+5)/3 = 4 is not prime.

Links

Crossrefs

Cf. A000040, A001222, A100118, A046363, A133620, A133621.
Cf. A133880, A133890, A133900, A133910, A133911, A046346, A134331, A134332, A134333, A134334.

Programs

  • Mathematica
    fp[{a_,b_}]:=a*b;s={};Do[If[q=Total[fp/@FactorInteger[n]]/Total[Last/@FactorInteger[n]];IntegerQ[q]&&!PrimeQ[q],AppendTo[s,n]],{n,2,290}];s (* James C. McMahon, Apr 05 2025 *)

Extensions

Definition clarified by the author, May 06 2013
Showing 1-8 of 8 results.

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

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