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 1-3 of 3 results.

A249242 Squarefree primitive abundant numbers (using the second definition: having no abundant proper divisors, cf. A091191).

Original entry on oeis.org

30, 42, 66, 70, 78, 102, 114, 138, 174, 186, 222, 246, 258, 282, 318, 354, 366, 402, 426, 438, 474, 498, 534, 582, 606, 618, 642, 654, 678, 762, 786, 822, 834, 894, 906, 942, 978, 1002, 1038, 1074, 1086, 1146, 1158, 1182, 1194, 1266, 1338, 1362, 1374, 1398, 1430, 1434, 1446, 1506
Offset: 1

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Author

Derek Orr, Oct 23 2014

Keywords

Comments

Primitive numbers in A087248.
Squarefree numbers in A091191.
According to the definition of A091191, all terms of the form 6*p, p > 3, are in this sequence (and similarly for other perfect numbers). Primitive abundant can also be defined as "having only deficient proper divisors", cf. A071395. The corresponding squarefree terms are listed in A298973, and those with n prime factors are counted in A295369. (The preceding remark shows that this count would be infinite for n = 3, using the definition of A091191.) - M. F. Hasler, Feb 16 2018

Links

Crossrefs

Intersection of A087248 and A091191.
Cf. A071395, A295369, A298973.

Programs

  • Mathematica
    Select[Range@1506, SquareFreeQ[#] && DivisorSigma[1, #] > 2 #  && Times @@ Boole@ Map[DivisorSigma[1, #] <= 2 # &, Most@ Divisors@ #] == 1 &] (* Amiram Eldar, Jun 26 2019 after Michael De Vlieger at A091191 *)
  • PARI
    v=[];for(n=1,10^5,d=0;for(j=2,ceil(sqrt(n)),if(n%(j^2),d++));if(d==ceil(sqrt(n))-1,if(sigma(n)>2*n,c=0;for(i=1,#v,if(n%v[i],c++));if(c==#v,print1(n,", ");v=concat(v,n)))))

Extensions

Definition edited by M. F. Hasler, Feb 16 2018

A295369 Number of squarefree primitive abundant numbers (A071395) with n prime factors.

Original entry on oeis.org

0, 0, 1, 18, 610, 216054, 12566567699
Offset: 1

Views

Author

Gianluca Amato, Feb 12 2018

Keywords

Comments

Here primitive abundant number means an abundant number all of whose proper divisors are deficient numbers (A071395). The alternative definition (an abundant number having no abundant proper divisor, see A091191) would yield an infinite count for a(3): since 2*3 = 6 is perfect, all numbers of the kind 2*3*p with p > 3 would be primitive abundant.
See A287590 for the number of squarefree ODD primitive abundant numbers with n prime factors.
The actual numbers are listed in A298973. - M. F. Hasler, Feb 16 2018

Examples

			For n=3, the only squarefree primitive abundant number (SFPAN) is 2*5*7 = 70, which is also a primitive weird number, see A002975.
For n=4, the 18 SFPAN range from 2*5*11*13 = 1430 to 2*5*11*53 = 5830.
For n=5, the 610 SFPAN range from 3*5*7*11*13 = 15015 to 2*5*11*59*647 = 4199030.

Links

Crossrefs

Cf. A071395 (primitive abundant numbers), A287590 (counts of odd SFPAN), A298973, A249242 (using A091191).

Programs

  • PARI
    A295369(n, p=1, m=1, sigmam=1) = {
  my(centerm = sigmam/(2*m-sigmam), s=0);
  if (n==1,
    if (centerm > p, primepi(ceil(centerm)-1) - primepi(p), 0),
    p = max(floor(centerm),p); while (0A295369(n-1, p=nextprime(p+1), m*p, sigmam*(p+1)), s+=c); s
  )
}
  • SageMath
    def A295369(n, p=1, m=1, sigmam=1):
  centerm = sigmam/(2*m-sigmam)
  if n==1:
    return prime_pi(ceil(centerm)-1) - prime_pi(p) if centerm > p else 0
  else:
    p = max(floor(centerm), p)
    s = 0
    while True:
       p = next_prime(p)
       c = A295369(n-1, p, m*p, sigmam*(p+1))
       if c <= 0: return s
       s+=c

A372298 Primitive infinitary abundant numbers (definition 1): infinitary abundant numbers (A129656) whose all proper infinitary divisors are infinitary deficient numbers.

Original entry on oeis.org

40, 56, 70, 72, 88, 104, 756, 924, 945, 1092, 1188, 1344, 1386, 1428, 1430, 1596, 1638, 1760, 1870, 2002, 2016, 2080, 2090, 2142, 2176, 2210, 2394, 2432, 2470, 2530, 2584, 2720, 2750, 2944, 2990, 3040, 3128, 3190, 3200, 3230, 3250, 3400, 3410, 3496, 3712, 3770
Offset: 1

Views

Author

Amiram Eldar, Apr 25 2024

Keywords

Examples

			40 is a term since it is an infinitary abundant number and all its proper infinitary divisors, {1, 2, 4, 5, 8, 10, 20}, are infinitary deficient numbers.
24 and 30, which are infinitary abundant numbers, are not primitive, because they are divisible by 6 which is an infinitary perfect number.

Links

Crossrefs

Subsequence of A129656 and A372299.
A372300 is a subsequence.
Cf. A007357, A077609, A129657.
Similar sequences: A071395, A298973, A302573, A307112, A307114, A307115.

Programs

  • Mathematica
    f[p_, e_] := Module[{b = IntegerDigits[e, 2]}, m = Length[b]; Product[If[b[[j]] > 0, 1 + p^(2^(m - j)), 1], {j, 1, m}]];
isigma[1] = 1; isigma[n_] := Times @@ f @@@ FactorInteger[n]; idefQ[n_] := isigma[n] < 2*n; idivs[1] = {1};
idivs[n_] := Sort@ Flatten@ Outer[Times, Sequence @@ (FactorInteger[n] /. {p_, e_Integer} :> p^Select[Range[0, e], BitOr[e, #] == e &])];
q[n_] := Module[{d = idivs[n]}, Total[d] > 2*n && AllTrue[Most[d], idefQ]]; Select[Range[4000], q]
  • PARI
    isidiv(d, f) = {if (d==1, return (1)); for (k=1, #f~, bne = binary(f[k, 2]); bde = binary(valuation(d, f[k, 1])); if (#bde < #bne, bde = concat(vector(#bne-#bde), bde)); for (j=1, #bne, if (! bne[j] && bde[j], return (0)); ); ); return (1); }
idivs(n) = {my(f = factor(n), d = divisors(f), idiv = []); for (k=1, #d, if (isidiv(d[k], f), idiv = concat(idiv, d[k])); ); idiv; } \\ Michel Marcus at A077609
isigma(n) = {my(f = factor(n), b); prod(i=1, #f~, b = binary(f[i, 2]); prod(k=1, #b, if(b[k], 1+f[i, 1]^(2^(#b-k)), 1)))} ;
is(n) = isigma(n) > 2*n && select(x -> x < n && isigma(x) >= 2*x, idivs(n)) == [];
Showing 1-3 of 3 results.

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