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.

Previous Showing 11-19 of 19 results.

A339003 Numbers of the form prime(x) * prime(y) where x and y are distinct and both odd.

Original entry on oeis.org

10, 22, 34, 46, 55, 62, 82, 85, 94, 115, 118, 134, 146, 155, 166, 187, 194, 205, 206, 218, 235, 253, 254, 274, 295, 298, 314, 334, 335, 341, 358, 365, 382, 391, 394, 415, 422, 451, 454, 466, 482, 485, 514, 515, 517, 527, 538, 545, 554, 566, 614, 626, 635, 649
Offset: 1

Views

Author

Gus Wiseman, Nov 21 2020

Keywords

Comments

The squarefree semiprimes in A332822. - Peter Munn, Dec 25 2020

Examples

			The sequence of terms together with their prime indices begins:
     10: {1,3}     187: {5,7}     358: {1,41}    527: {7,11}
     22: {1,5}     194: {1,25}    365: {3,21}    538: {1,57}
     34: {1,7}     205: {3,13}    382: {1,43}    545: {3,29}
     46: {1,9}     206: {1,27}    391: {7,9}     554: {1,59}
     55: {3,5}     218: {1,29}    394: {1,45}    566: {1,61}
     62: {1,11}    235: {3,15}    415: {3,23}    614: {1,63}
     82: {1,13}    253: {5,9}     422: {1,47}    626: {1,65}
     85: {3,7}     254: {1,31}    451: {5,13}    635: {3,31}
     94: {1,15}    274: {1,33}    454: {1,49}    649: {5,17}
    115: {3,9}     295: {3,17}    466: {1,51}    662: {1,67}
    118: {1,17}    298: {1,35}    482: {1,53}    685: {3,33}
    134: {1,19}    314: {1,37}    485: {3,25}    694: {1,69}
    146: {1,21}    334: {1,39}    514: {1,55}    697: {7,13}
    155: {3,11}    335: {3,19}    515: {3,27}    706: {1,71}
    166: {1,23}    341: {5,11}    517: {5,15}    713: {9,11}

Crossrefs

A338910 is the not necessarily squarefree version.
A339004 is the even instead of odd version.
A001358 lists semiprimes, with odd and even terms A046315 and A100484.
A005117 lists squarefree numbers.
A006881 lists squarefree semiprimes, with odd and even terms A046388 and A100484.
A289182/A115392 list the positions of odd/even terms of A001358.
A300912 lists products of two primes of relatively prime index.
A320656 counts factorizations into squarefree semiprimes.
A338898, A338912, and A338913 give the prime indices of semiprimes, with product A087794, sum A176504, and difference A176506.
A338899, A270650, and A270652 give the prime indices of squarefree semiprimes, with difference A338900.
A338904 groups semiprimes by weight.
A338906/A338907 list semiprimes of even/odd weight.
A339002 lists products of two distinct primes of non-relatively prime index.
A339005 lists products of two distinct primes of divisible index.
Cf. A001221, A001222, A056239, A112798, A166237, A195017, A318990, A320911, A338901, A338903, A338911.
Subsequence of A332822.

Programs

  • Mathematica
    Select[Range[100],SquareFreeQ[#]&&PrimeOmega[#]==2&&OddQ[Times@@PrimePi/@First/@FactorInteger[#]]&]
  • Python
    from math import isqrt
from sympy import primepi, primerange
def A339003(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        kmin = kmax >> 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x): return n+x-sum(primepi(x//p)-a>>1 for a,p in enumerate(primerange(isqrt(x)+1),1) if a&1)
    return bisection(f,n,n) # Chai Wah Wu, Apr 03 2025

Formula

Numbers m such that A001221(m) = A001222(m) = A195017(m) = 2. - Peter Munn, Dec 31 2020

A339004 Numbers of the form prime(x) * prime(y) where x and y are distinct and both even.

Original entry on oeis.org

21, 39, 57, 87, 91, 111, 129, 133, 159, 183, 203, 213, 237, 247, 259, 267, 301, 303, 321, 339, 371, 377, 393, 417, 427, 453, 481, 489, 497, 519, 543, 551, 553, 559, 579, 597, 623, 669, 687, 689, 703, 707, 717, 749, 753, 789, 791, 793, 813, 817, 843, 879, 917
Offset: 1

Views

Author

Gus Wiseman, Nov 22 2020

Keywords

Comments

The squarefree semiprimes in A332821. - Peter Munn, Dec 25 2020

Examples

			The sequence of terms together with their prime indices begins:
     21: {2,4}     267: {2,24}    543: {2,42}
     39: {2,6}     301: {4,14}    551: {8,10}
     57: {2,8}     303: {2,26}    553: {4,22}
     87: {2,10}    321: {2,28}    559: {6,14}
     91: {4,6}     339: {2,30}    579: {2,44}
    111: {2,12}    371: {4,16}    597: {2,46}
    129: {2,14}    377: {6,10}    623: {4,24}
    133: {4,8}     393: {2,32}    669: {2,48}
    159: {2,16}    417: {2,34}    687: {2,50}
    183: {2,18}    427: {4,18}    689: {6,16}
    203: {4,10}    453: {2,36}    703: {8,12}
    213: {2,20}    481: {6,12}    707: {4,26}
    237: {2,22}    489: {2,38}    717: {2,52}
    247: {6,8}     497: {4,20}    749: {4,28}
    259: {4,12}    519: {2,40}    753: {2,54}

Crossrefs

A338911 is the not necessarily squarefree version.
A339003 is the odd instead of even version, with not necessarily squarefree version A338910.
A001358 lists semiprimes, with odd/even terms A046315/A100484.
A005117 lists squarefree numbers.
A006881 lists squarefree semiprimes, with odd/even terms A046388/A100484.
A289182/A115392 list the positions of odd/even terms in A001358.
A300912 lists products of pairs of primes with relatively prime indices.
A318990 lists products of pairs of primes with divisible indices.
A320656 counts factorizations into squarefree semiprimes.
A338898, A338912, and A338913 give the prime indices of semiprimes, with product A087794, sum A176504, and difference A176506.
A338899, A270650, and A270652 give the prime indices of squarefree semiprimes, with difference A338900.
A338904 groups semiprimes by weight.
A338906/A338907 list semiprimes of even/odd weight.
Cf. A000040, A001221, A001222, A056239, A112798, A166237, A195017, A320911, A338901, A338903, A339002.
Subsequence of A332821.

Programs

  • Mathematica
    Select[Range[100],SquareFreeQ[#]&&PrimeOmega[#]==2&&OddQ[Times@@(1+ PrimePi/@First/@FactorInteger[#])]&]
  • Python
    from math import isqrt
from sympy import primepi, primerange
def A339004(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        kmin = kmax >> 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x): return n+x-sum(primepi(x//p)-a>>1 for a,p in enumerate(primerange(isqrt(x)+1),1) if a&1^1)
    return bisection(f,n,n) # Chai Wah Wu, Apr 03 2025

Formula

Numbers m such that A001221(m) = A001222(m) = 2 and A195017(m) = -2. - Peter Munn, Dec 31 2020

A339005 Numbers of the form prime(x) * prime(y) where x properly divides y. Squarefree semiprimes with divisible prime indices.

Original entry on oeis.org

6, 10, 14, 21, 22, 26, 34, 38, 39, 46, 57, 58, 62, 65, 74, 82, 86, 87, 94, 106, 111, 115, 118, 122, 129, 133, 134, 142, 146, 158, 159, 166, 178, 183, 185, 194, 202, 206, 213, 214, 218, 226, 235, 237, 254, 259, 262, 267, 274, 278, 298, 302, 303, 305, 314, 319
Offset: 1

Views

Author

Gus Wiseman, Dec 05 2020

Keywords

Comments

A squarefree semiprime (A006881) is a product of any two distinct prime numbers. A prime index of n is a number m such that the m-th prime number 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:
    6: {1,2}    82: {1,13}  159: {2,16}  259: {4,12}
   10: {1,3}    86: {1,14}  166: {1,23}  262: {1,32}
   14: {1,4}    87: {2,10}  178: {1,24}  267: {2,24}
   21: {2,4}    94: {1,15}  183: {2,18}  274: {1,33}
   22: {1,5}   106: {1,16}  185: {3,12}  278: {1,34}
   26: {1,6}   111: {2,12}  194: {1,25}  298: {1,35}
   34: {1,7}   115: {3,9}   202: {1,26}  302: {1,36}
   38: {1,8}   118: {1,17}  206: {1,27}  303: {2,26}
   39: {2,6}   122: {1,18}  213: {2,20}  305: {3,18}
   46: {1,9}   129: {2,14}  214: {1,28}  314: {1,37}
   57: {2,8}   133: {4,8}   218: {1,29}  319: {5,10}
   58: {1,10}  134: {1,19}  226: {1,30}  321: {2,28}
   62: {1,11}  142: {1,20}  235: {3,15}  326: {1,38}
   65: {3,6}   146: {1,21}  237: {2,22}  334: {1,39}
   74: {1,12}  158: {1,22}  254: {1,31}  339: {2,30}

Crossrefs

A300912 is the version for relative primality.
A318990 is the not necessarily squarefree version.
A339002 is the version for non-relative primality.
A339003 is the version for odd indices.
A339004 is the version for even indices
A001358 lists semiprimes.
A005117 lists squarefree numbers.
A006881 lists squarefree semiprimes, with odd/even terms A046388/A100484.
A320655 counts factorizations into semiprimes.
A320656 counts factorizations into squarefree semiprimes.
A338898/A338912/A338913 give the prime indices of semiprimes, with product A087794, sum A176504, and difference A176506.
A338899/A270650/A270652 give the prime indices of squarefree semiprimes, with difference A338900.
Cf. A001221, A112798, A166237, A320911, A338901, A338904, A338909.

Programs

  • Mathematica
    Select[Range[100],SquareFreeQ[#]&&PrimeOmega[#]==2&& Divisible@@Reverse[PrimePi/@First/@FactorInteger[#]]&]

Formula

Equals A318990 \ A000290.

A338909 Numbers of the form prime(x) * prime(y) where x and y have a common divisor > 1.

Original entry on oeis.org

9, 21, 25, 39, 49, 57, 65, 87, 91, 111, 115, 121, 129, 133, 159, 169, 183, 185, 203, 213, 235, 237, 247, 259, 267, 289, 299, 301, 303, 305, 319, 321, 339, 361, 365, 371, 377, 393, 417, 427, 445, 453, 481, 489, 497, 515, 517, 519, 529, 543, 551, 553, 559, 565
Offset: 1

Views

Author

Gus Wiseman, Nov 20 2020

Keywords

Examples

			The sequence of terms together with their prime indices begins:
      9: {2,2}     169: {6,6}     319: {5,10}
     21: {2,4}     183: {2,18}    321: {2,28}
     25: {3,3}     185: {3,12}    339: {2,30}
     39: {2,6}     203: {4,10}    361: {8,8}
     49: {4,4}     213: {2,20}    365: {3,21}
     57: {2,8}     235: {3,15}    371: {4,16}
     65: {3,6}     237: {2,22}    377: {6,10}
     87: {2,10}    247: {6,8}     393: {2,32}
     91: {4,6}     259: {4,12}    417: {2,34}
    111: {2,12}    267: {2,24}    427: {4,18}
    115: {3,9}     289: {7,7}     445: {3,24}
    121: {5,5}     299: {6,9}     453: {2,36}
    129: {2,14}    301: {4,14}    481: {6,12}
    133: {4,8}     303: {2,26}    489: {2,38}
    159: {2,16}    305: {3,18}    497: {4,20}

Crossrefs

A082023 counts partitions with these as Heinz numbers, complement A023022.
A300912 is the complement in A001358.
A339002 is the squarefree case.
A001221 counts distinct prime indices.
A001222 counts prime indices.
A001358 lists semiprimes, with odds A046315 and evens A100484.
A004526 counts 2-part partitions, with strict case A140106 (shifted left).
A006881 lists squarefree semiprimes, with odds A046388 and evens A100484.
A176504/A176506/A087794 give sum/difference/product of semiprime indices.
A318990 lists semiprimes with divisible indices.
A320655 counts factorizations into semiprimes.
A338898, A338912, and A338913 give semiprime indices.
A338899, A270650, and A270652 give squarefree semiprime indices.
A338910 lists semiprimes with odd indices.
A338911 lists semiprimes with even indices.
Cf. A005117, A037143, A055684, A056239, A065516, A112798, A115392, A128301, A289182, A338900, A338904.

Programs

  • Mathematica
    Select[Range[100],PrimeOmega[#]==2&&GCD@@PrimePi/@First/@FactorInteger[#]>1&]

Formula

Equals A001358 \ A300912.
Equals A339002 \/ (A001248 \ {4}).

A339002 Numbers of the form prime(x) * prime(y) where x and y are distinct and have a common divisor > 1.

Original entry on oeis.org

21, 39, 57, 65, 87, 91, 111, 115, 129, 133, 159, 183, 185, 203, 213, 235, 237, 247, 259, 267, 299, 301, 303, 305, 319, 321, 339, 365, 371, 377, 393, 417, 427, 445, 453, 481, 489, 497, 515, 517, 519, 543, 551, 553, 559, 565, 579, 597, 611, 623, 669, 685, 687
Offset: 1

Views

Author

Gus Wiseman, Nov 22 2020

Keywords

Examples

			The sequence of terms together with their prime indices begins:
     21: {2,4}     235: {3,15}    393: {2,32}
     39: {2,6}     237: {2,22}    417: {2,34}
     57: {2,8}     247: {6,8}     427: {4,18}
     65: {3,6}     259: {4,12}    445: {3,24}
     87: {2,10}    267: {2,24}    453: {2,36}
     91: {4,6}     299: {6,9}     481: {6,12}
    111: {2,12}    301: {4,14}    489: {2,38}
    115: {3,9}     303: {2,26}    497: {4,20}
    129: {2,14}    305: {3,18}    515: {3,27}
    133: {4,8}     319: {5,10}    517: {5,15}
    159: {2,16}    321: {2,28}    519: {2,40}
    183: {2,18}    339: {2,30}    543: {2,42}
    185: {3,12}    365: {3,21}    551: {8,10}
    203: {4,10}    371: {4,16}    553: {4,22}
    213: {2,20}    377: {6,10}    559: {6,14}

Crossrefs

A300912 is the complement in A001358.
A338909 is the not necessarily squarefree version.
A001358 lists semiprimes, with odd and even terms A046315 and A100484.
A005117 lists squarefree numbers.
A006881 lists squarefree semiprimes, with odd/even terms A046388/A100484.
A339005 lists products of pairs of distinct primes of divisible index.
A320656 counts factorizations into squarefree semiprimes.
A338898, A338912, and A338913 give the prime indices of semiprimes, with product A087794, sum A176504, and difference A176506.
A338899, A270650, and A270652 give the prime indices of squarefree semiprimes, with difference A338900.
A338910/A338911 list products of pairs of primes both of odd/even index.
A339003/A339004 list squarefree semiprimes of odd/even index.
Cf. A001221, A001222, A055684, A056239, A112798, A115392, A166237, A167171, A318990, A320892, A320911, A338901.

Programs

  • Mathematica
    Select[Range[100],SquareFreeQ[#]&&PrimeOmega[#]==2&&GCD@@PrimePi/@First/@FactorInteger[#]>1&]

A358106 Quotient of the n-th divisible pair, where pairs are ordered first by sum and then by denominator.

Original entry on oeis.org

1, 2, 3, 1, 4, 5, 2, 1, 6, 7, 3, 1, 8, 2, 9, 4, 1, 10, 11, 5, 3, 2, 1, 12, 13, 6, 1, 14, 4, 2, 15, 7, 3, 1, 16, 17, 8, 5, 2, 1, 18, 19, 9, 4, 3, 1, 20, 6, 2, 21, 10, 1, 22, 23, 11, 7, 5, 3, 2, 1, 24, 4, 25, 12, 1, 26, 8, 2, 27, 13, 6, 3, 1, 28, 29, 14, 9, 5, 4, 2, 1
Offset: 2

Views

Author

Gus Wiseman, Nov 03 2022

Keywords

Examples

			Grouping by sum gives:
   2:  1
   3:  2
   4:  3 1
   5:  4
   6:  5 2 1
   7:  6
   8:  7 3 1
   9:  8 2
  10:  9 4 1
  11: 10
  12: 11 5 3 2 1
  13: 12
  14: 13 6 1
  15: 14 4 2
  16: 15 7 3 1
  17: 16
  18: 17 8 5 2 1

Crossrefs

Row-lengths are A032741.
This is A208460/A027751.
A ranking of divisible pairs is A318990, proper A339005.
A different ordering is A358103 = A358104 / A358105.
A000041 counts partitions, strict A000009.
A001358 lists semiprimes, squarefree A006881.
A318991 ranks divisor-chains.
A358192/A358193 gives quotients of semiprime indices.
Cf. A000837, A003238, A122934.

Programs

  • Mathematica
    Table[Divide@@@Select[IntegerPartitions[n,{2}],Divisible@@#&],{n,2,30}]

Formula

a(n) = A208460(n)/A027751(n).

A358192 Numerator of the quotient of the prime indices of the n-th semiprime.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 3, 1, 1, 1, 1, 2, 3, 1, 1, 1, 1, 2, 1, 4, 1, 3, 1, 1, 2, 2, 1, 3, 1, 1, 1, 1, 4, 1, 1, 2, 1, 1, 1, 2, 1, 5, 3, 1, 3, 1, 1, 4, 1, 1, 2, 1, 1, 1, 5, 1, 2, 1, 2, 3, 1, 5, 1, 1, 3, 4, 1, 2, 6, 1, 1, 1, 3, 2, 5, 1, 1, 1, 3, 1, 1
Offset: 1

Views

Author

Gus Wiseman, Nov 03 2022

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 31st semiprime has prime indices (4,6), so the quotient is 4/6 = 2/3; hence a(31) = 2.

Crossrefs

The divisible pairs are ranked by A318990, proper A339005.
The unreduced pair is (A338912, A338913).
The quotients of divisible pairs are A358103.
The restriction to divisible pairs is A358104, denominator A358105.
The denominator is A358193.
A000040 lists the primes.
A001222 counts prime indices, distinct A001221.
A001358 lists the semiprimes, squarefree A006881.
A003963 multiplies together prime indices.
A056239 adds up prime indices.
Cf. A000720, A027751, A128301, A215366, A289508, A289509, A296150, A300912, A318991, A358106.

Programs

  • Mathematica
    primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Numerator/@Divide@@@primeMS/@Select[Range[100],PrimeOmega[#]==2&]

A358193 Denominator of the quotient of the prime indices of the n-th semiprime.

Original entry on oeis.org

1, 2, 1, 3, 4, 3, 2, 5, 1, 6, 5, 7, 4, 8, 3, 9, 1, 7, 5, 4, 10, 11, 2, 9, 12, 5, 13, 7, 14, 5, 3, 11, 15, 8, 16, 6, 3, 17, 7, 1, 18, 13, 7, 2, 19, 15, 20, 6, 10, 21, 11, 22, 8, 9, 23, 1, 17, 24, 9, 4, 7, 25, 19, 26, 5, 13, 27, 8, 10, 28, 14, 11, 29, 21, 7, 30
Offset: 1

Views

Author

Gus Wiseman, Nov 03 2022

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 31-st semiprime has prime indices (4,6), so the quotient is 4/6 = 2/3; hence a(31) = 3.

Crossrefs

The divisible pairs are ranked by A318990, proper A339005.
The unreduced pair is (A338912, A338913).
The quotients of divisible pairs are A358103.
The restriction to divisible pairs is A358105, numerator A358104.
The numerator is A358192.
A000040 lists the primes.
A001222 counts prime indices, distinct A001221.
A001358 lists the semiprimes, squarefree A006881.
A003963 multiplies together prime indices.
A056239 adds up prime indices.
Cf. A000720, A027751, A032741, A128301, A215366, A289508, A289509, A296150, A300912, A318991, A358106.

Programs

  • Mathematica
    primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Denominator/@Divide@@@primeMS/@Select[Range[100],PrimeOmega[#]==2&]

A318993 Matula-Goebel number of the planted achiral tree determined by the n-th number whose consecutive prime indices are divisible.

Original entry on oeis.org

1, 2, 4, 3, 8, 7, 16, 5, 9, 19, 32, 17, 64, 53, 11, 128, 23, 256, 67, 49, 131, 512, 59, 27, 311, 25, 241, 1024, 2048, 31, 719, 83, 4096, 1619, 361, 331, 8192, 227, 16384, 739, 3671, 32768, 277, 81, 103, 2063, 65536, 97, 1523, 2809, 8161, 131072, 262144, 17863
Offset: 1

Views

Author

Gus Wiseman, Sep 06 2018

Keywords

Examples

			The sequence of all planted achiral trees begins: o, (o), (oo), ((o)), (ooo), ((oo)), (oooo), (((o))), ((o)(o)), ((ooo)), (ooooo), (((oo))), (oooooo), ((oooo)), ((((o)))), (ooooooo), (((o)(o))), (oooooooo), (((ooo))), ((oo)(oo)).

Links

Crossrefs

Cf. A000040, A001221, A003238, A008480, A061775, A214577, A318990, A318991, A318992.

Programs

  • Mathematica
    primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
ptnToAch[y_]:=Fold[Table[#1,{#2}]&,{},Divide@@@Partition[Append[y,1],2,1]];
MGNumber[[]]:=1;MGNumber[x:[__]]:=If[Length[x]==1,Prime[MGNumber[x[[1]]]],Times@@Prime/@MGNumber/@x];
MGNumber/@ptnToAch/@Reverse/@primeMS/@Select[Range[100],Or[#==1,PrimeQ[#],Divisible@@Reverse[primeMS[#]]]&]
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