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.

A327908 Nonprime squarefree numbers whose prime indices all have the same Omega (A001222) and the same sum of prime indices (A056239).

Original entry on oeis.org

1, 667, 2021, 4331, 5767, 7081, 7663, 11021, 11639, 12091, 13837, 15049, 20413, 23213, 26123, 31553, 34933, 35657, 35723, 37909, 37979, 38021, 41449, 42919, 44197, 45113, 51019, 53531, 58339, 63407, 66013, 70531, 72929, 73373, 73903, 75763, 83411, 87361, 90581
Offset: 1

Views

Author

Gus Wiseman, Sep 30 2019

Keywords

Examples

			The sequence of terms together with their prime indices begins:
      1: {}
    667: {9,10}
   2021: {14,15}
   4331: {18,20}
   5767: {21,22}
   7081: {21,25}
   7663: {22,25}
  11021: {27,28}
  11639: {27,30}
  12091: {28,30}
  13837: {26,33}
  15049: {26,35}
  20413: {33,35}
  23213: {34,39}
  26123: {36,40}
  31553: {34,49}
  34933: {42,44}
  35657: {42,45}
  35723: {34,55}
  37909: {39,49}

Links

Crossrefs

Equal omega: A327900
Equal sum of prime indices: A327901
Equal average of prime indices: A327902
Cf. A001222, A038041, A056239, A112798, A320324, A326574, A327899.

Programs

  • Mathematica
    primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[10000],!PrimeQ[#]&&SquareFreeQ[#]&&SameQ@@PrimeOmega/@primeMS[#]&&SameQ@@Total/@primeMS/@primeMS[#]&]

A327900 Nonprime squarefree numbers whose prime indices all have the same Omega (number of prime factors counted with multiplicity).

Original entry on oeis.org

1, 15, 33, 51, 55, 85, 91, 93, 123, 155, 161, 165, 177, 187, 201, 203, 205, 249, 255, 295, 299, 301, 327, 329, 335, 341, 377, 381, 415, 451, 465, 471, 511, 527, 537, 545, 553, 559, 561, 573, 611, 615, 633, 635, 649, 667, 679, 697, 703, 707, 723, 737, 785, 831
Offset: 1

Views

Author

Gus Wiseman, Sep 30 2019

Keywords

Comments

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

Examples

			The sequence of terms together with their prime indices begins:
    1: {}
   15: {2,3}
   33: {2,5}
   51: {2,7}
   55: {3,5}
   85: {3,7}
   91: {4,6}
   93: {2,11}
  123: {2,13}
  155: {3,11}
  161: {4,9}
  165: {2,3,5}
  177: {2,17}
  187: {5,7}
  201: {2,19}
  203: {4,10}
  205: {3,13}
  249: {2,23}
  255: {2,3,7}
  295: {3,17}

Links

Crossrefs

The case including primes and nonsquarefree numbers is A320324.
The version for sum of prime indices is A327901.
The version for mean of prime indices is A327902.
Cf. A001222, A038041, A056239, A078175, A112798, A306017, A306021, A316413, A317583, A322794, A327908.

Programs

  • Mathematica
    Select[Range[1000],!PrimeQ[#]&&SquareFreeQ[#]&&SameQ@@PrimeOmega/@PrimePi/@First/@FactorInteger[#]&]

A327902 Nonprime squarefree numbers whose prime indices all have the same average of prime indices (A326567/A326568).

Original entry on oeis.org

1, 21, 57, 115, 133, 145, 159, 371, 393, 399, 515, 535, 565, 667, 803, 869, 917, 933, 1007, 1067, 1113, 1963, 2021, 2095, 2157, 2165, 2177, 2249, 2285, 2315, 2363, 2369, 2461, 2489, 2599, 2705, 2751, 2839, 2987, 3021, 3103, 3277, 3335, 3707, 3859, 4331, 4367
Offset: 1

Views

Author

Gus Wiseman, Sep 30 2019

Keywords

Comments

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

Examples

			The sequence of terms together with their prime indices begins:
     1: {}
    21: {2,4}
    57: {2,8}
   115: {3,9}
   133: {4,8}
   145: {3,10}
   159: {2,16}
   371: {4,16}
   393: {2,32}
   399: {2,4,8}
   515: {3,27}
   535: {3,28}
   565: {3,30}
   667: {9,10}
   803: {5,21}
   869: {5,22}
   917: {4,32}
   933: {2,64}
  1007: {8,16}
  1067: {5,25}

Crossrefs

The version including primes and nonsquarefree numbers is A326536.
The version for number of prime indices is A327900.
The version for sum of prime indices is A327901.
Cf. A001222, A005117, A056239, A078175, A102627, A112798, A316413, A326567/A326568, A326621, A327908.

Programs

  • Mathematica
    primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[1000],!PrimeQ[#]&&SquareFreeQ[#]&&SameQ@@Mean/@primeMS/@primeMS[#]&];
Showing 1-3 of 3 results.

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

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