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.

A367589 Numbers with exactly two distinct prime factors, both appearing with different exponents.

Original entry on oeis.org

12, 18, 20, 24, 28, 40, 44, 45, 48, 50, 52, 54, 56, 63, 68, 72, 75, 76, 80, 88, 92, 96, 98, 99, 104, 108, 112, 116, 117, 124, 135, 136, 144, 147, 148, 152, 153, 160, 162, 164, 171, 172, 175, 176, 184, 188, 189, 192, 200, 207, 208, 212, 224, 232, 236, 242, 244
Offset: 1

Views

Author

Gus Wiseman, Dec 01 2023

Keywords

Comments

First differs from A177425 in lacking 360.
First differs from A182854 in lacking 360.
These are the Heinz numbers of the partitions counted by A182473.

Examples

			The terms together with their prime indices begin:
  12: {1,1,2}
  18: {1,2,2}
  20: {1,1,3}
  24: {1,1,1,2}
  28: {1,1,4}
  40: {1,1,1,3}
  44: {1,1,5}
  45: {2,2,3}
  48: {1,1,1,1,2}
  50: {1,3,3}
  52: {1,1,6}
  54: {1,2,2,2}
  56: {1,1,1,4}
  63: {2,2,4}
  68: {1,1,7}
  72: {1,1,1,2,2}

Crossrefs

The case of any multiplicities is A007774, counts A002133.
These partitions are counted by A182473.
The case of equal exponents is A367590, counts A367588.
A000041 counts integer partitions, strict A000009.
A091602 counts partitions by greatest multiplicity, least A243978.
A098859 counts partitions with distinct multiplicities, ranks A130091.
A116608 counts partitions by number of distinct parts.
Cf. A071625, A072233, A072774, A109297, A367580.

Programs

  • Mathematica
    Select[Range[100], PrimeNu[#]==2&&UnsameQ@@Last/@FactorInteger[#]&]

A367590 Numbers with exactly two distinct prime factors, both appearing with the same exponent.

Original entry on oeis.org

6, 10, 14, 15, 21, 22, 26, 33, 34, 35, 36, 38, 39, 46, 51, 55, 57, 58, 62, 65, 69, 74, 77, 82, 85, 86, 87, 91, 93, 94, 95, 100, 106, 111, 115, 118, 119, 122, 123, 129, 133, 134, 141, 142, 143, 145, 146, 155, 158, 159, 161, 166, 177, 178, 183, 185, 187, 194
Offset: 1

Views

Author

Gus Wiseman, Dec 01 2023

Keywords

Comments

First differs from A268390 in lacking 210.
First differs from A238748 in lacking 210.
These are the Heinz numbers of the partitions counted by A367588.

Examples

			The terms together with their prime indices begin:
     6: {1,2}         57: {2,8}        106: {1,16}
    10: {1,3}         58: {1,10}       111: {2,12}
    14: {1,4}         62: {1,11}       115: {3,9}
    15: {2,3}         65: {3,6}        118: {1,17}
    21: {2,4}         69: {2,9}        119: {4,7}
    22: {1,5}         74: {1,12}       122: {1,18}
    26: {1,6}         77: {4,5}        123: {2,13}
    33: {2,5}         82: {1,13}       129: {2,14}
    34: {1,7}         85: {3,7}        133: {4,8}
    35: {3,4}         86: {1,14}       134: {1,19}
    36: {1,1,2,2}     87: {2,10}       141: {2,15}
    38: {1,8}         91: {4,6}        142: {1,20}
    39: {2,6}         93: {2,11}       143: {5,6}
    46: {1,9}         94: {1,15}       145: {3,10}
    51: {2,7}         95: {3,8}        146: {1,21}
    55: {3,5}        100: {1,1,3,3}    155: {3,11}

Links

Crossrefs

The case of any multiplicities is A007774, counts A002133.
Partitions of this type are counted by A367588.
The case of distinct exponents is A367589, counts A182473.
A000041 counts integer partitions, strict A000009.
A091602 counts partitions by greatest multiplicity, least A243978.
A116608 counts partitions by number of distinct parts.
Cf. A006881, A039956, A071625, A072233, A072774, A109297, A303661, A367580.

Programs

  • Mathematica
    Select[Range[100], SameQ@@Last/@If[#==1, {}, FactorInteger[#]]&&PrimeNu[#]==2&]
Select[Range[200],PrimeNu[#]==2&&Length[Union[FactorInteger[#][[;;,2]]]]==1&] (* Harvey P. Dale, Aug 04 2025 *)

Formula

Union of A006881 and A303661. - Michael De Vlieger, Dec 01 2023

A367449 Numbers k for which there are exactly k pairs (i, j), 1 <= i < j < k, such that i + j is a divisor of k.

Original entry on oeis.org

30, 42, 54, 66, 78, 102, 114, 138, 174, 186, 208, 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, 1312, 1338, 1362, 1374
Offset: 1

Views

Author

Marius A. Burtea, Dec 10 2023

Keywords

Comments

Numbers k >= 1 for which A367588(k) = Sum_{d|k} floor((d-1)/2) = k;
Numbers k >= 1 for which A000203(k) - A000005(k) - A183063(k) = 2*k.
The sequence is infinite because all numbers of the form m = 6*p, p >= 5 prime (A138636), are terms.
Indeed: sigma(6*p) - tau(6*p) - A183063(6*p) = 3*4*(p + 1) - 8 - 4 = 12*p = 2*m.
If m = 2^k*p, p = 2^(k + 1) - 4*k - 3 prime number, then m is a term. Indeed: sigma(m) - tau(m) - A183063(m) = (2^(k + 1) - 1)*(p + 1) - 2*(k + 1) - 2*k = 2*m.

Examples

			30 is a term since it has exactly 30 pairs (i,j): (1, 2), (2, 3), (1, 4), (2, 4), (1, 5), (4, 6), (3, 7), (2, 8), (7, 8), (1,9), (6, 9), (5, 10), (4, 11), (3, 12), (2, 13), (1, 14), (14, 16), (13, 17),(12, 18), (11, 19), (10, 20), (9, 21), (8, 22), (7, 23), (6, 24), (5, 25), (4,26), (3, 27), (2, 28), (1, 29).

Crossrefs

Fixed points of A367588.
Cf. A000005, A000203, A138636, A183063.

Programs

  • Magma
    [k:k in [1..1000]|(DivisorSigma(1,k)-#Divisors(k)-#[d:d in Divisors(k)| IsEven(d)]) eq 2*k ];
  • Maple
    filter:= proc(n) uses numtheory;
  sigma(n) - tau(n) - `if`(n::even, tau(n/2),0) = 2*n
end proc:
select(filter, [$1..10000]); # Robert Israel, Dec 12 2023
  • Mathematica
    f1[p_, e_] := e+1; f1[2, e_] := 2*e+1; f2[p_, e_] := (p^(e+1)-1)/(p-1); s[1] = 0; s[n_] := Module[{fct = FactorInteger[n]}, Times @@ f2 @@@ fct - Times @@ f1 @@@ fct]; Select[Range[1400], s[#] == 2*# &] (* Amiram Eldar, Dec 16 2023 *)
  • PARI
    isok(k) = sumdiv(k, d, (d-1)\2) == k; \\ Michel Marcus, Dec 19 2023
Showing 1-3 of 3 results.

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