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 1-4 of 4 results.

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

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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

A082572 a(n) is the least number m such that the arithmetic mean of the distinct primes dividing m equals n.

Original entry on oeis.org

2, 3, 15, 5, 35, 7, 39, 65, 51, 11, 95, 13, 115, 161, 87, 17, 155, 19, 111, 185, 123, 23, 215, 141, 235, 329, 159, 29, 371, 31, 183, 305, 427, 201, 335, 37, 219, 365, 511, 41, 395, 43, 415, 581, 267, 47, 623, 1501, 291, 485, 303, 53, 515, 321, 327, 545, 339, 59
Offset: 2

Views

Author

David Wasserman, May 06 2003

Keywords

Comments

Are there any terms with more than 3 prime factors?

Examples

			a(6) = 35 because the prime factors of 35 are {5, 7}, which have mean 6.

Links

Crossrefs

Cf. A070009, A078177, A200612.

A266618 Least number whose arithmetic mean of all prime factors, counted with multiplicity, is equal to n.

Original entry on oeis.org

2, 3, 15, 5, 35, 7, 39, 65, 51, 11, 95, 13, 115, 161, 87, 17, 155, 19, 111, 185, 123, 23, 215, 141, 235, 329, 159, 29, 371, 31, 183, 305, 427, 201, 335, 37, 219, 365, 511, 41, 395, 43, 415, 524, 267, 47, 623, 1501, 291, 485, 303, 53, 515, 321, 327, 545, 339, 59
Offset: 2

Views

Author

Paolo P. Lava, Feb 22 2016

Keywords

Comments

Obviously a(p) = p if p is prime.
Similar to A082572 but here the prime factors are not necessarily distinct. First difference for a(45) = 524 while A082572(45) = 581.

Examples

			Prime factor of 15 are 3 and 5: (3 + 5) / 2 = 4 and no other number less than 15 has arithmetic mean of all its prime factors, counted with multiplicity, equal to 4.

Links

Crossrefs

Cf. A078177, A082572, A200612.

Programs

  • Maple
    with(numtheory): P:= proc(q) local a,b,i,k,n; for i from 2 to q do
for n from 2 to q do a:=ifactors(n)[2]; b:=add(a[k][1]*a[k][2],k=1..nops(a))/add(a[k][2],k=1..nops(a));
if type(b,integer) then if i=b then lprint(b,n); break; fi; fi; od; od; end: P(10^9);
  • PARI
    ampf(n) = my(f = factor(n)); (sum(k=1, #f~, f[k,1]*f[k,2]) / vecsum(f[,2]));
a(n) = {m = 2; while (ampf(m) != n, m++); m;} \\ Michel Marcus, Feb 22 2016

A275384 Composite squarefree numbers such that the arithmetic mean of its prime factors is an integer.

Original entry on oeis.org

15, 21, 33, 35, 39, 42, 51, 55, 57, 65, 69, 77, 78, 85, 87, 91, 93, 95, 105, 110, 111, 114, 115, 119, 123, 129, 133, 141, 143, 145, 155, 159, 161, 170, 177, 183, 185, 186, 187, 195, 201, 203, 205, 209, 213, 215, 217, 219, 221, 222, 230, 231, 235, 237, 247, 249, 253, 258, 259, 265, 267
Offset: 1

Views

Author

Antonio Roldán, Jul 25 2016

Keywords

Comments

Sopf(a(n)) is multiple of omega(a(n)) (sopf(n) is the sum of the distinct prime factors of n, and omega(n) is the number of distinct primes dividing n).
This sequence is subsequence of A078177 and supersequence of A187073.

Examples

			170 is in the sequence because 170 = 17*2*5 (squarefree number) and (17+2+5)/3 = 8 is an integer.

Links

Crossrefs

Cf. A005117, A078177, A134344, A187073, A229094.

Programs

  • Mathematica
    Select[Range@ 270, And[CompositeQ@ #, SquareFreeQ@ #, IntegerQ@ Mean@ FactorInteger[#][[All, 1]]] &] (* Michael De Vlieger, Jul 25 2016 *)
  • PARI
    sopf(n)= my(f, s=0); f=factor(n); for(i=1, matsize(f)[1], s+=f[i, 1]); s
for(i=2,500,if(issquarefree(i)&&!isprime(i),m=sopf(i)/omega(i);if(m==truncate(m),print1(i,", "))))
Showing 1-4 of 4 results.

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