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.

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A368728 Numbers whose prime indices are 1, prime, or semiprime.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 54, 55, 56, 58, 59, 60, 62, 63, 64, 65, 66, 67, 68, 69, 70, 72, 73, 75
Offset: 1

Views

Author

Gus Wiseman, Jan 07 2024

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.

Crossrefs

These are products of primes indexed by elements of A037143.
For just primes we have A076610, strict A302590.
For just semiprimes we have A339112, strict A340020.
For squarefree semiprimes we have A339113, strict A309356.
The odd case is A368729, strict A340019.
The complement is A368833.
A000607 counts partitions into primes, A034891 with ones allowed.
A001358 lists semiprimes, squarefree A006881.
A006450, A106349, A322551, A368732 list selected primes.
A056239 adds up prime indices, row sums of A112798.
A101048 counts partitions into semiprimes.
Cf. A000040, A000720, A001222, A003963, A005117, A302242, A320628, A320911, A320912.

Programs

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

Formula

Closed under multiplication.

A368729 Numbers whose prime indices are prime or semiprime. MM-numbers of labeled multigraphs with loops and half-loops without isolated (uncovered) nodes.

Original entry on oeis.org

1, 3, 5, 7, 9, 11, 13, 15, 17, 21, 23, 25, 27, 29, 31, 33, 35, 39, 41, 43, 45, 47, 49, 51, 55, 59, 63, 65, 67, 69, 73, 75, 77, 79, 81, 83, 85, 87, 91, 93, 97, 99, 101, 105, 109, 115, 117, 119, 121, 123, 125, 127, 129, 135, 137, 139, 141, 143, 145, 147, 149
Offset: 1

Views

Author

Gus Wiseman, Jan 07 2024

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. The multiset of multisets with MM-number n is formed by taking the multiset of prime indices of each part of the multiset of prime indices of n. For example, the prime indices of 78 are {1,2,6}, so the multiset of multisets with MM-number 78 is {{},{1},{1,2}}.

Examples

			The terms together with the corresponding multigraphs begin:
   1: {}
   3: {{1}}
   5: {{2}}
   7: {{1,1}}
   9: {{1},{1}}
  11: {{3}}
  13: {{1,2}}
  15: {{1},{2}}
  17: {{4}}
  21: {{1},{1,1}}
  23: {{2,2}}
  25: {{2},{2}}
  27: {{1},{1},{1}}
  29: {{1,3}}
  31: {{5}}
  33: {{1},{3}}
  35: {{2},{1,1}}
  39: {{1},{1,2}}
  41: {{6}}
  43: {{1,4}}
  45: {{1},{1},{2}}
  47: {{2,3}}
  49: {{1,1},{1,1}}

Crossrefs

In the unlabeled case these multigraphs are counted by A320663.
These are products of primes indexed by elements of A037143 greater than 1.
For just primes we have A076610, squarefree A302590.
For just semiprimes we have A339112, squarefree A340020.
For just half-loops we have A340019.
This is the odd case of A368728, complement A368833.
A000607 counts partitions into primes, with ones allowed A034891.
A001358 lists semiprimes, squarefree A006881.
A006450, A106349, A322551, A368732 list selected primes.
A056239 adds up prime indices, row sums of A112798.
A101048 counts partitions into semiprimes.
Cf. A000040, A000720, A001222, A003963, A005117, A302242, A309356, A320628, A320912, A339113.

Programs

  • Mathematica
    prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],OddQ[#]&&Max@@Length/@prix/@prix[#]<=2&]

A368732 Primes whose index is one, another prime number, or a semiprime.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 17, 23, 29, 31, 41, 43, 47, 59, 67, 73, 79, 83, 97, 101, 109, 127, 137, 139, 149, 157, 163, 167, 179, 191, 199, 211, 227, 233, 241, 257, 269, 271, 277, 283, 293, 313, 331, 347, 353, 367, 373, 389, 401, 421, 431, 439, 443, 449, 461, 467, 487
Offset: 1

Views

Author

Gus Wiseman, Jan 08 2024

Keywords

Crossrefs

For just primes we have A006450, products A076610, strict A302590.
These indices are A037143.
For just semiprimes we have A106349, products A339112, strict A340020.
Products of these primes are A368728, odd A368729, odd strict A340019.
Products of the complementary primes are A368833.
A000607 counts partitions into primes, with ones allowed A034891.
A001358 lists semiprimes, squarefree A006881.
A056239 adds up prime indices, row sums of A112798.
A101048 counts partitions into semiprimes.
A322551 lists primes of squarefree semiprime index.
Cf. A000040, A000720, A001222, A003963, A005117, A302242, A309356, A320628, A320911, A320912, A339113.

Programs

  • Mathematica
    Prime/@Select[Range[100],PrimeOmega[#]<=2&]

A368833 Numbers whose prime indices are not 1, prime, or semiprime.

Original entry on oeis.org

19, 37, 38, 53, 57, 61, 71, 74, 76, 89, 95, 103, 106, 107, 111, 113, 114, 122, 131, 133, 142, 148, 151, 152, 159, 171, 173, 178, 181, 183, 185, 190, 193, 197, 206, 209, 212, 213, 214, 222, 223, 226, 228, 229, 239, 244, 247, 251, 259, 262, 263, 265, 266, 267
Offset: 1

Views

Author

Gus Wiseman, Jan 08 2024

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 terms together with their prime indices begin:
   19: {8}
   37: {12}
   38: {1,8}
   53: {16}
   57: {2,8}
   61: {18}
   71: {20}
   74: {1,12}
   76: {1,1,8}
   89: {24}
   95: {3,8}
  103: {27}
  106: {1,16}
  107: {28}
  111: {2,12}
  113: {30}
  114: {1,2,8}
  122: {1,18}
  131: {32}
  133: {4,8}
  142: {1,20}
  148: {1,1,12}

Crossrefs

These are non-products of primes indexed by elements of A037143.
The complement for just primes is A076610, strict A302590.
The complement for just semiprimes is A339112, strict A340020.
The complement for just squarefree semiprimes is A339113, strict A309356.
The complement is A368728.
The complement for just primes and semiprimes is A368729, strict A340019.
A000607 counts partitions into primes, with ones allowed A034891.
A001358 lists semiprimes, squarefree A006881.
A006450, A106349, A322551, A368732 list selected primes.
A056239 adds up prime indices, row sums of A112798.
A101048 counts partitions into semiprimes.
Cf. A000040, A000720, A001222, A003963, A005117, A302242, A320628.

Programs

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

A371454 Numbers whose binary indices are all semiprimes.

Original entry on oeis.org

8, 32, 40, 256, 264, 288, 296, 512, 520, 544, 552, 768, 776, 800, 808, 8192, 8200, 8224, 8232, 8448, 8456, 8480, 8488, 8704, 8712, 8736, 8744, 8960, 8968, 8992, 9000, 16384, 16392, 16416, 16424, 16640, 16648, 16672, 16680, 16896, 16904, 16928, 16936, 17152
Offset: 1

Views

Author

Gus Wiseman, Apr 02 2024

Keywords

Comments

A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793.

Examples

			The terms together with their binary expansions and binary indices begin:
     8:           1000 ~ {4}
    32:         100000 ~ {6}
    40:         101000 ~ {4,6}
   256:      100000000 ~ {9}
   264:      100001000 ~ {4,9}
   288:      100100000 ~ {6,9}
   296:      100101000 ~ {4,6,9}
   512:     1000000000 ~ {10}
   520:     1000001000 ~ {4,10}
   544:     1000100000 ~ {6,10}
   552:     1000101000 ~ {4,6,10}
   768:     1100000000 ~ {9,10}
   776:     1100001000 ~ {4,9,10}
   800:     1100100000 ~ {6,9,10}
   808:     1100101000 ~ {4,6,9,10}

Crossrefs

Partitions of this type are counted by A101048, squarefree case A002100.
For primes instead of semiprimes we get A326782.
For prime indices instead of binary indices we have A339112, A339113.
The squarefree case is A371453.
A001358 lists semiprimes, squarefree A006881.
A005117 lists squarefree numbers.
A048793 lists binary indices, reverse A272020, length A000120, sum A029931.
A070939 gives length of binary expansion.
A096111 gives product of binary indices.
Cf. A087086, A296119, A302478, A326031, A367905, A368109, A368533, A371450.

Programs

  • Mathematica
    bix[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
semi[n_]:=PrimeOmega[n]==2;
Select[Range[10000],And@@semi/@bix[#]&]
  • Python
    from math import isqrt
from sympy import primepi, primerange
def A371454(n):
    def f(x,n): return int(n+x+((t:=primepi(s:=isqrt(x)))*(t-1)>>1)-sum(primepi(x//k) for k in primerange(1, s+1)))
    def A001358(n):
        m, k = n, f(n,n)
        while m != k:
            m, k = k, f(k,n)
        return m
    return sum(1<<A001358(i)-1 for i, j in enumerate(bin(n)[:1:-1],1) if j=='1') # Chai Wah Wu, Aug 16 2024

A281617 Expansion of Sum_{i = p*q, p prime, q prime} x^i/(1 - x^i) / Product_{j = p*q, p prime, q prime} (1 - x^j).

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 1, 0, 2, 1, 3, 0, 5, 2, 6, 3, 9, 3, 14, 7, 16, 10, 23, 12, 32, 20, 37, 28, 52, 35, 69, 49, 80, 68, 110, 83, 137, 112, 166, 150, 215, 178, 268, 239, 324, 303, 406, 365, 504, 472, 604, 584, 747, 708, 917, 888, 1089, 1085, 1337, 1311, 1618, 1606, 1916, 1954, 2332, 2334, 2782, 2829, 3300, 3407, 3963
Offset: 0

Views

Author

Ilya Gutkovskiy, Jan 25 2017

Keywords

Comments

Total number of parts in all partitions of n into semiprimes (A001358).
Convolution of A086971 and A101048.

Examples

			a(12) = 5 because we have [6, 6], [4, 4, 4] and 2 + 3 = 5.

Links

Crossrefs

Cf. A001358, A084993, A086971, A101048, A344447.

Programs

  • Maple
    h:= proc(n) option remember; `if`(n=0, 0,
     `if`(numtheory[bigomega](n)=2, n, h(n-1)))
    end:
b:= proc(n, i) option remember; `if`(n=0, [1, 0], `if`(i<1, [0$2],
     `if`(i>n, 0, (p-> p+[0, p[1]])(b(n-i, h(min(n-i, i)))))+b(n, h(i-1))))
    end:
a:= n-> b(n, h(n))[2]:
seq(a(n), n=0..70);  # Alois P. Heinz, May 19 2021
  • Mathematica
    nmax = 70; Rest[CoefficientList[Series[Sum[Floor[PrimeOmega[i]/2] Floor[2/PrimeOmega[i]] x^i/(1 - x^i), {i, 2, nmax}]/Product[1 - Floor[PrimeOmega[j]/2] Floor[2/PrimeOmega[j]] x^j, {j, 2, nmax}], {x, 0, nmax}], x]]

Formula

G.f.: Sum_{i = p*q, p prime, q prime} x^i/(1 - x^i) / Product_{j = p*q, p prime, q prime} (1 - x^j).
a(n) = Sum_{k>0} k * A344447(n,k). - Alois P. Heinz, May 19 2021

A280238 Expansion of 1/(1 - Sum_{k>=2} floor(bigomega(k)/2)*floor(2/bigomega(k))*x^k), where bigomega(k) is the number of prime divisors of k counted with multiplicity (A001222).

Original entry on oeis.org

1, 0, 0, 0, 1, 0, 1, 0, 1, 1, 3, 0, 2, 2, 6, 3, 6, 3, 11, 10, 16, 10, 23, 23, 40, 34, 52, 52, 93, 94, 130, 133, 209, 234, 330, 352, 488, 570, 804, 909, 1198, 1405, 1918, 2283, 2980, 3512, 4622, 5636, 7340, 8811, 11321, 13864, 17937, 21957, 27936, 34262, 43857, 54290, 68915, 84940, 107685, 133811, 169615, 210375, 265305
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 29 2016

Keywords

Comments

Number of compositions (ordered partitions) into semiprimes (A001358).

Examples

			a(10) = 3 because we have [4, 6], [6, 4] and [10].

Links

Crossrefs

Cf. A001222, A001358, A023360, A101048.

Programs

  • Mathematica
    nmax = 44; CoefficientList[Series[1/(1 - Sum[Floor[PrimeOmega[k]/2] Floor[2/PrimeOmega[k]] x^k, {k, 2, nmax}]), {x, 0, nmax}], x]

Formula

G.f.: 1/(1 - Sum_{k>=2} floor(bigomega(k)/2)*floor(2/bigomega(k))*x^k).

A280912 Number of partitions of n into odd semiprimes (A046315).

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 1, 0, 0, 2, 0, 0, 2, 1, 1, 2, 0, 0, 3, 1, 0, 3, 1, 1, 3, 1, 0, 4, 2, 2, 5, 1, 1, 5, 3, 1, 6, 3, 2, 8, 2, 1, 7, 5, 4, 9, 4, 3, 11, 6, 3, 11, 6, 6, 14, 7, 5, 15, 9, 7, 16, 9, 8, 20, 14, 9, 21, 13, 11, 26, 16, 12, 28, 19, 17, 29, 19, 17, 37, 27
Offset: 0

Views

Author

Ilya Gutkovskiy, Jan 10 2017

Keywords

Examples

			a(39) = 3 because we have [39], [21, 9, 9] and [15, 15, 9].

Links

Crossrefs

Cf. A001222, A001358, A046315, A099773, A101048, A112020.

Programs

Formula

G.f.: Product_{k>=1} 1/(1 - floor(bigomega(2*k+1)/2)*floor(2/bigomega(2*k+ 1))*x^(2*k+1)), where bigomega(k) is the number of prime divisors of k counted with multiplicity (A001222).

A108793 Semiprimes that can be partitioned into a sum of semiprimes in more than one way.

Original entry on oeis.org

10, 14, 15, 21, 22, 25, 26, 33, 34, 35, 38, 39, 46, 49, 51, 55, 57, 58, 62, 65, 69, 74, 77, 82, 85, 86, 87, 91, 93, 94, 95, 106, 111, 115, 118, 119, 121, 122, 123, 129, 133, 134, 141, 142, 143, 145, 146, 155, 158, 159, 161, 166, 169, 177, 178, 183, 185, 187, 194
Offset: 1

Views

Author

Giovanni Teofilatto, Jul 09 2005

Keywords

Comments

Essentially semiprimes greater than 9.

Crossrefs

Cf. A101048.

Extensions

Corrected and extended by Ray Chandler, Jul 11 2005

A114325 Number of partitions of n into brilliant numbers (A078972).

Original entry on oeis.org

0, 0, 0, 1, 0, 1, 0, 1, 1, 2, 0, 2, 1, 3, 2, 3, 1, 5, 3, 5, 4, 6, 4, 9, 7, 8, 8, 12, 10, 15, 12, 15, 16, 21, 19, 24, 22, 27, 30, 34, 31, 40, 40, 46, 49, 54, 52, 65, 68, 74
Offset: 1

Views

Author

Giovanni Resta, Feb 06 2006

Keywords

Examples

			a(18)=5 since 18 has the following 5 "brilliant" partitions: {14,4}, {10,4,4}, {9,9}, {6,6,6}, {6,4,4,4}.

Crossrefs

Cf. A101048, A078972.
Previous Showing 21-30 of 31 results. Next

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