A076610 -id:A076610 - cp's OEIS frontend

A379312 Positive integers whose prime indices include a unique 1 or prime number.

2, 3, 5, 11, 14, 17, 21, 26, 31, 35, 38, 39, 41, 46, 57, 58, 59, 65, 67, 69, 74, 77, 83, 86, 87, 94, 95, 98, 106, 109, 111, 115, 119, 122, 127, 129, 141, 142, 143, 145, 146, 147, 157, 158, 159, 178, 179, 182, 183, 185, 191, 194, 202, 206, 209, 211, 213, 214

Offset: 1

Gus Wiseman, Dec 28 2024

nonn

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 terms together with their prime indices begin:
    2: {1}
    3: {2}
    5: {3}
   11: {5}
   14: {1,4}
   17: {7}
   21: {2,4}
   26: {1,6}
   31: {11}
   35: {3,4}
   38: {1,8}
   39: {2,6}
   41: {13}
   46: {1,9}
   57: {2,8}
   58: {1,10}
   59: {17}
   65: {3,6}
   67: {19}
   69: {2,9}
   74: {1,12}
   77: {4,5}

These "old" primes are listed by A008578.
For no composite parts we have A302540, counted by A034891 (strict A036497).
For all composite parts we have A320629, counted by A023895 (strict A204389).
For a unique prime part we have A331915, counted by A379304 (strict A379305).
Positions of ones in A379311, see A379313.
Partitions of this type are counted by A379314, strict A379315.
A000040 lists the prime numbers, differences A001223.
A002808 lists the composite numbers, nonprimes A018252, differences A073783 or A065310.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798, counted by A001222.
A080339 is the characteristic function for the old prime numbers.
A376682 gives k-th differences of old prime numbers, see A030016, A075526.
Other counts of prime indices:
- A257991 odd, see A000041, A000070, A066207, A349158.
- A257992 even, see A000009, A038348, A066208, A379317.
- A257994 prime, see A000586, A000607, A076610, A331386.
- A330944 nonprime, see A002095, A096258, A320628, A330945.
- A379306 squarefree, see A302478, A379308, A379309, A379316.
- A379310 nonsquarefree, see A114374, A256012, A379307.
Cf. A038550, A073445, A087436, A173390, A379300, A379301, A379302.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[1000],Length[Select[prix[#],#==1||PrimeQ[#]&]]==1&]

A340019 MM-numbers of labeled graphs with half-loops, without isolated vertices.

Original entry on oeis.org

1, 3, 5, 11, 13, 15, 17, 29, 31, 33, 39, 41, 43, 47, 51, 55, 59, 65, 67, 73, 79, 83, 85, 87, 93, 101, 109, 123, 127, 129, 137, 139, 141, 143, 145, 149, 155, 157, 163, 165, 167, 177, 179, 187, 191, 195, 199, 201, 205, 211, 215, 219, 221, 233, 235, 237, 241, 249

Offset: 1

Gus Wiseman, Jan 02 2021

nonn

Here a half-loop is an edge with only one vertex, to be distinguished from a full loop, which has two equal vertices.
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}}.
Also products of distinct primes whose prime indices are either themselves prime or a squarefree semiprime (A006881).

			The sequence of terms together with their corresponding multisets of multisets (edge sets) begins:
     1: {}              55: {{2},{3}}      137: {{2,5}}
     3: {{1}}           59: {{7}}          139: {{1,7}}
     5: {{2}}           65: {{2},{1,2}}    141: {{1},{2,3}}
    11: {{3}}           67: {{8}}          143: {{3},{1,2}}
    13: {{1,2}}         73: {{2,4}}        145: {{2},{1,3}}
    15: {{1},{2}}       79: {{1,5}}        149: {{3,4}}
    17: {{4}}           83: {{9}}          155: {{2},{5}}
    29: {{1,3}}         85: {{2},{4}}      157: {{12}}
    31: {{5}}           87: {{1},{1,3}}    163: {{1,8}}
    33: {{1},{3}}       93: {{1},{5}}      165: {{1},{2},{3}}
    39: {{1},{1,2}}    101: {{1,6}}        167: {{2,6}}
    41: {{6}}          109: {{10}}         177: {{1},{7}}
    43: {{1,4}}        123: {{1},{6}}      179: {{13}}
    47: {{2,3}}        127: {{11}}         187: {{3},{4}}
    51: {{1},{4}}      129: {{1},{1,4}}    191: {{14}}

The version with full loops covering an initial interval is A320461.
The case covering an initial interval is A340018.
The version with full loops is A340020.
A006450 lists primes of prime index.
A106349 lists primes of semiprime index.
A257994 counts prime prime indices.
A302242 is the weight of the multiset of multisets with MM-number n.
A302494 lists MM-numbers of sets of sets, with connected case A328514.
A309356 lists MM-numbers of simple graphs.
A322551 lists primes of squarefree semiprime index.
A330944 counts nonprime prime indices.
A339112 lists MM-numbers of multigraphs with loops.
A339113 lists MM-numbers of multigraphs.
Cf. A000040, A000720, A001222, A005117, A056239, A076610, A112798, A289509, A302590, A305079, A326788.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[1000],And[SquareFreeQ[#],And@@(PrimeQ[#]||(SquareFreeQ[#]&&PrimeOmega[#]==2)&/@primeMS[#])]&]

A355535 Odd numbers of which it is not possible to choose a different prime factor of each prime index.

Original entry on oeis.org

9, 21, 25, 27, 45, 49, 57, 63, 75, 81, 99, 105, 115, 117, 121, 125, 133, 135, 147, 153, 159, 171, 175, 189, 195, 207, 225, 231, 243, 245, 261, 273, 275, 279, 285, 289, 297, 315, 325, 333, 343, 345, 351, 357, 361, 363, 369, 371, 375, 387, 393, 399, 405, 423

Offset: 1

Gus Wiseman, Jul 22 2022

nonn

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 terms together with their prime indices begin:
    9: {2,2}
   21: {2,4}
   25: {3,3}
   27: {2,2,2}
   45: {2,2,3}
   49: {4,4}
   57: {2,8}
   63: {2,2,4}
   75: {2,3,3}
   81: {2,2,2,2}
   99: {2,2,5}
  105: {2,3,4}
For example, the prime indices of 897 are {2,6,9}, of which we can choose prime factors in two ways: (2,2,3) or (2,3,3); but neither of these has all distinct elements, so 897 is in the sequence.

Including evens gives A355529.
The version for all divisors including evens is A355740, zeros of A355739.
Choices of a prime factor of each prime index: A355741, unordered A355744.
A001221 counts distinct prime factors, with sum A001414.
A001222 counts prime factors with multiplicity.
A003963 multiplies together the prime indices of n.
A056239 adds up prime indices, row sums of A112798.
A120383 lists numbers divisible by all of their prime indices.
Cf. A000720, A076610, A289509, A302796, A327486, A355731, A355733, A355742.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],OddQ[#]&&Select[Tuples[primeMS/@primeMS[#]],UnsameQ@@#&]=={}&]

A357852 Replace prime(k) with prime(k+2) in the prime factorization of n.

Original entry on oeis.org

1, 5, 7, 25, 11, 35, 13, 125, 49, 55, 17, 175, 19, 65, 77, 625, 23, 245, 29, 275, 91, 85, 31, 875, 121, 95, 343, 325, 37, 385, 41, 3125, 119, 115, 143, 1225, 43, 145, 133, 1375, 47, 455, 53, 425, 539, 155, 59, 4375, 169, 605, 161, 475, 61, 1715, 187, 1625, 203

Offset: 1

Gus Wiseman, Oct 28 2022

This is the same as A045966 except the first term is 1 instead of 3.

			The terms together with their prime indices begin:
    1: {}
    5: {3}
    7: {4}
   25: {3,3}
   11: {5}
   35: {3,4}
   13: {6}
  125: {3,3,3}
   49: {4,4}
   55: {3,5}
   17: {7}
  175: {3,3,4}
   19: {8}
   65: {3,6}
   77: {4,5}
  625: {3,3,3,3}

Applying the transformation only once gives A003961.
A permutation of A007310.
Other multiplicative sequences: A064988, A064989, A357977, A357980, A357983.
A000040 lists the primes.
A056239 adds up prime indices, row-sums of A112798.
Cf. A000720, A003964, A066207, A076610, A215366, A296150, A299201, A357979.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Product[Prime[i+2],{i,primeMS[n]}],{n,30}]

a(n) = my(f=factor(n)); for (k=1, #f~, f[k,1] = nextprime(nextprime(f[k,1]+1)+1)); factorback(f); \\ Michel Marcus, Oct 28 2022

from math import prod
from sympy import nextprime, factorint
def A357852(n): return prod(nextprime(p,ith=2)**e for p, e in factorint(n).items()) # Chai Wah Wu, Oct 29 2022

a(n) = A003961(A003961(n)).

A379316 Positive integers whose prime indices include a unique squarefree number.

Original entry on oeis.org

2, 3, 5, 11, 13, 14, 17, 21, 29, 31, 35, 38, 41, 43, 46, 47, 57, 59, 67, 69, 73, 74, 77, 79, 83, 91, 95, 98, 101, 106, 109, 111, 113, 115, 119, 122, 127, 137, 139, 142, 147, 149, 157, 159, 163, 167, 178, 179, 181, 183, 185, 191, 194, 199, 203, 206, 209, 211

Offset: 1

Gus Wiseman, Dec 29 2024

nonn

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 terms together with their prime indices begin:
    2: {1}
    3: {2}
    5: {3}
   11: {5}
   13: {6}
   14: {1,4}
   17: {7}
   21: {2,4}
   29: {10}
   31: {11}
   35: {3,4}
   38: {1,8}
   41: {13}
   43: {14}
   46: {1,9}

For all squarefree parts we have A302478, zeros of A379310.
Positions of 1 in A379306.
For no squarefree parts we have A379307, counted by A114374, strict A256012.
Partitions of this type are counted by A379308, strict A379309.
A000040 lists the primes, differences A001223.
A005117 lists the squarefree numbers, differences A076259.
A008966 is the characteristic function for the squarefree numbers.
A013929 lists the nonsquarefree numbers, differences A078147.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798, counted by A001222.
Other counts of prime indices:
- A257991 odd, see A000041, A000070, A066207, A349158.
- A257992 even, see A000009, A038348, A066208, A379317.
- A257994 prime, see A002095, A096258, A320628, A331386, A331915, A379304, A379305.
- A330944 nonprime, see A000586, A000607, A076610, A330945.
- A379300 composite, see A023895, A034891, A036497, A302540, A379301.
- A379311 prime or 1, see A204389, A320629, A379312-A379315.
Cf. A000720, A013928, A038550, A057627, A068361, A070321, A072284, A087436, A112929, A377430.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[1000],Length[Select[prix[#],SquareFreeQ]]==1&]

A355747 Number of multisets that can be obtained by choosing a divisor of each positive integer from 1 to n.

Original entry on oeis.org

1, 1, 2, 4, 10, 20, 58, 116, 320, 772, 2170, 4340, 14112, 28224, 78120, 212004, 612232, 1224464, 3873760, 7747520, 24224608, 64595088, 175452168, 350904336

Offset: 0

Gus Wiseman, Jul 20 2022

			The a(0) = 1 through a(4) = 10 multisets:
  {}  {1}  {1,1}  {1,1,1}  {1,1,1,1}
           {1,2}  {1,1,2}  {1,1,1,2}
                  {1,1,3}  {1,1,1,3}
                  {1,2,3}  {1,1,1,4}
                           {1,1,2,2}
                           {1,1,2,3}
                           {1,1,2,4}
                           {1,1,3,4}
                           {1,2,2,3}
                           {1,2,3,4}

The sum of the same integers is A000096.
The product of the same integers is A000142, Heinz number A070826.
Counting sequences instead of multisets gives A066843.
The integers themselves are the rows of A131818 (shifted).
For prime indices we have A355733, only prime factors A355744.
For prime factors instead of divisors we have A355746, factors A355537.
A000005 counts divisors.
A000040 lists the prime numbers.
A001221 counts distinct prime factors, with sum A001414.
A001222 counts prime factors with multiplicity.
Cf. A000720, A002110, A006218, A076610, A327486, A355538, A355731, A355737, A355741, A355745.

Table[Length[Union[Sort/@Tuples[Divisors/@Range[n]]]],{n,0,10}]

from sympy import divisors
from itertools import count, islice
def agen():
    s = {tuple()}
    for n in count(1):
        yield len(s)
        s = set(tuple(sorted(t+(d,))) for t in s for d in divisors(n))
print(list(islice(agen(), 16))) # Michael S. Branicky, Aug 03 2022

a(n) = A355733(A070826(n)).

a(p) = 2*a(p-1) for p prime. - Michael S. Branicky, Aug 03 2022

a(15)-a(21) from Michael S. Branicky, Aug 03 2022

a(22)-a(23) from Michael S. Branicky, Aug 08 2022

A340020 MM-numbers of labeled graphs with loops, without isolated vertices.

Original entry on oeis.org

1, 7, 13, 23, 29, 43, 47, 73, 79, 91, 97, 101, 137, 139, 149, 161, 163, 167, 199, 203, 227, 233, 257, 269, 271, 293, 299, 301, 313, 329, 347, 373, 377, 389, 421, 439, 443, 449, 467, 487, 491, 499, 511, 553, 559, 577, 607, 611, 631, 647, 653, 661, 667, 673, 677

Offset: 1

Gus Wiseman, Jan 02 2021

nonn

Here a loop is an edge with two equal vertices, distinguished from a half-loop, which has only one vertex.
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}}.
Also products of distinct primes whose prime indices are semiprimes, where a semiprime (A001358) is a product of any two prime numbers.

			The sequence of terms together with their corresponding multisets of multisets (edge sets) begins:
      1: {}              161: {{1,1},{2,2}}    347: {{2,9}}
      7: {{1,1}}         163: {{1,8}}          373: {{1,12}}
     13: {{1,2}}         167: {{2,6}}          377: {{1,2},{1,3}}
     23: {{2,2}}         199: {{1,9}}          389: {{4,5}}
     29: {{1,3}}         203: {{1,1},{1,3}}    421: {{1,13}}
     43: {{1,4}}         227: {{4,4}}          439: {{3,7}}
     47: {{2,3}}         233: {{2,7}}          443: {{1,14}}
     73: {{2,4}}         257: {{3,5}}          449: {{2,10}}
     79: {{1,5}}         269: {{2,8}}          467: {{4,6}}
     91: {{1,1},{1,2}}   271: {{1,10}}         487: {{2,11}}
     97: {{3,3}}         293: {{1,11}}         491: {{1,15}}
    101: {{1,6}}         299: {{1,2},{2,2}}    499: {{3,8}}
    137: {{2,5}}         301: {{1,1},{1,4}}    511: {{1,1},{2,4}}
    139: {{1,7}}         313: {{3,6}}          553: {{1,1},{1,5}}
    149: {{3,4}}         329: {{1,1},{2,3}}    559: {{1,2},{1,4}}

The case with only one edge is A106349.
The case covering an initial interval is A320461.
The version allowing multiple edges is A339112.
The half-loop version covering an initial interval is A340018.
The half-loop version is A340019.
A006450 lists primes of prime index.
A302242 is the weight of the multiset of multisets with MM-number n.
A302494 lists MM-numbers of sets of sets, with connected case A328514.
A309356 lists MM-numbers of simple graphs.
A339113 lists MM-numbers of multigraphs.
Cf. A000040, A000720, A001222, A005117, A056239, A076610, A112798, A289509, A302590, A305079, A326754, A326788.

Select[Range[100],SquareFreeQ[#]&&FreeQ[If[#==1,{},FactorInteger[#]],{p_,k_}/;PrimeOmega[PrimePi[p]]!=2]&]

A356065 Squarefree numbers whose prime indices are all prime-powers.

Original entry on oeis.org

1, 3, 5, 7, 11, 15, 17, 19, 21, 23, 31, 33, 35, 41, 51, 53, 55, 57, 59, 67, 69, 77, 83, 85, 93, 95, 97, 103, 105, 109, 115, 119, 123, 127, 131, 133, 155, 157, 159, 161, 165, 177, 179, 187, 191, 201, 205, 209, 211, 217, 227, 231, 241, 249, 253, 255, 265, 277

Offset: 1

Gus Wiseman, Jul 25 2022

nonn

			105 has prime indices {2,3,4}, all three of which are prime-powers, so 105 is in the sequence.

The multiplicative version (factorizations) is A050361, non-strict A000688.
Heinz numbers of the partitions counted by A054685, with 1's A106244, non-strict A023894, non-strict with 1's A023893.
Counting twice-partitions of this type gives A279786, non-strict A279784.
Counting twice-factorizations gives A295935, non-strict A296131.
These are the odd products of distinct elements of A302493.
Allowing prime index 1 gives A302496, non-strict A302492.
The case of primes (instead of prime-powers) is A302590, non-strict A076610.
These are the squarefree positions of 1's in A355741.
This is the squarefree case of A355743, complement A356066.
A001222 counts prime-power divisors.
A005117 lists the squarefree numbers.
A034699 gives maximal prime-power divisor.
A246655 lists the prime-powers (A000961 includes 1), towers A164336.
A355742 chooses a prime-power divisor of each prime index.
Cf. A001970, A055887, A063834, A302601, A355731, A355744, A356064.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],SquareFreeQ[#]&&And@@PrimePowerQ/@primeMS[#]&]

Intersection of A005117 and A355743.

A320633 Composite numbers whose prime indices are also composite.

Original entry on oeis.org

49, 91, 133, 161, 169, 203, 247, 259, 299, 301, 329, 343, 361, 371, 377, 427, 437, 481, 497, 511, 529, 551, 553, 559, 611, 623, 637, 667, 679, 689, 703, 707, 721, 749, 791, 793, 817, 841, 851, 893, 917, 923, 931, 949, 959, 973, 989, 1007, 1027, 1043, 1057

Offset: 1

Gus Wiseman, Oct 18 2018

nonn

A prime index of n is a number m such that prime(m) divides n.

			The sequence of terms begins:
   49 = prime(4)^2
   91 = prime(4)*prime(6)
  133 = prime(4)*prime(8)
  161 = prime(4)*prime(9)
  169 = prime(6)^2
  203 = prime(4)*prime(10)
  247 = prime(6)*prime(8)
  259 = prime(4)*prime(12)
  299 = prime(6)*prime(9)
  301 = prime(4)*prime(14)
  329 = prime(4)*prime(15)
  343 = prime(4)^3
  361 = prime(8)^2
  371 = prime(4)*prime(16)
  377 = prime(6)*prime(10)
  427 = prime(4)*prime(18)
  437 = prime(8)*prime(9)
  481 = prime(6)*prime(12)
  497 = prime(4)*prime(20)
  511 = prime(4)*prime(21)
  529 = prime(9)^2
  551 = prime(8)*prime(10)
  553 = prime(4)*prime(22)
  559 = prime(6)*prime(14)
  611 = prime(6)*prime(15)
  623 = prime(4)*prime(24)
  637 = prime(4)^2*prime(6)

Cf. A000040, A006450, A007821, A018252, A050370, A056239, A076610, A112798, A302242, A302478, A320533, A320628, A320629.

Select[Range[2,1000],And[OddQ[#],!PrimeQ[#],And@@Not/@PrimeQ/@PrimePi/@First/@FactorInteger[#]]&]

A346068 Numbers that are the product of distinct primes with prime subscripts raised to prime powers.

Original entry on oeis.org

1, 9, 25, 27, 121, 125, 225, 243, 289, 675, 961, 1089, 1125, 1331, 1681, 2187, 2601, 3025, 3125, 3267, 3375, 3481, 4489, 4913, 6075, 6889, 7225, 7803, 8649, 11881, 11979, 15125, 15129, 16129, 24025, 24649, 25947, 27225, 28125, 29403, 29791, 30375, 31329, 32041, 33275, 34969

Offset: 1

Ilya Gutkovskiy, Jul 30 2021

nonn

			675 = 3^3 * 5^2 = prime(prime(1))^prime(2) * prime(prime(2))^prime(1), therefore 675 is a term.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Intersection of A056166 and A076610.

Cf. A006450, A302590, A302596, A321874.

Join[{1}, Select[Range[35000], AllTrue[Join[PrimePi[(t = Transpose @ FactorInteger[#])[[1]]], t[[2]]], PrimeQ] &]] (* Amiram Eldar, Jul 30 2021 *)

from sympy import factorint, isprime, primepi
def ok(n):
    f = factorint(n)
    if not all(isprime(e) for e in f.values()): return False
    return all(isprime(primepi(p)) for p in f)
print(list(filter(ok, range(35000)))) # Michael S. Branicky, Jul 30 2021

Sum_{n>=1} 1/a(n) = Product_{p in A006450} (1 + Sum_{q prime} 1/p^q) = 1.2271874... - Amiram Eldar, Jul 31 2021

cp's OEIS Frontend

A379312 Positive integers whose prime indices include a unique 1 or prime number.

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A340019 MM-numbers of labeled graphs with half-loops, without isolated vertices.

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A355535 Odd numbers of which it is not possible to choose a different prime factor of each prime index.

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A357852 Replace prime(k) with prime(k+2) in the prime factorization of n.

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A379316 Positive integers whose prime indices include a unique squarefree number.

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A355747 Number of multisets that can be obtained by choosing a divisor of each positive integer from 1 to n.

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A340020 MM-numbers of labeled graphs with loops, without isolated vertices.

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A356065 Squarefree numbers whose prime indices are all prime-powers.

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A320633 Composite numbers whose prime indices are also composite.

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