A294674 -id:A294674 - cp's OEIS frontend

A356845 Odd numbers with gapless prime indices.

1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 23, 25, 27, 29, 31, 35, 37, 41, 43, 45, 47, 49, 53, 59, 61, 67, 71, 73, 75, 77, 79, 81, 83, 89, 97, 101, 103, 105, 107, 109, 113, 121, 125, 127, 131, 135, 137, 139, 143, 149, 151, 157, 163, 167, 169, 173, 175, 179, 181, 191

Offset: 1

Gus Wiseman, Sep 03 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.

A sequence is gapless if it covers an interval of positive integers.

			The terms together with their prime indices begin:
    1: {}
    3: {2}
    5: {3}
    7: {4}
    9: {2,2}
   11: {5}
   13: {6}
   15: {2,3}
   17: {7}
   19: {8}
   23: {9}
   25: {3,3}
   27: {2,2,2}
   29: {10}
   31: {11}
   35: {3,4}
   37: {12}
   41: {13}
   43: {14}

Consists of the odd terms of A073491.
These partitions are counted by A264396.
The strict case is A294674, counted by A136107.
The version for compositions is A356843, counted by A251729.
A001221 counts distinct prime factors, sum A001414.
A056239 adds up prime indices, row sums of A112798, lengths A001222.
A356069 counts gapless divisors, initial A356224 (complement A356225).
A356230 ranks gapless factorization lengths, firsts A356603.
A356233 counts factorizations into gapless numbers.
Cf. A003963, A034296, A055932, A073493, A107428, A287170, A289508, A325160, A356231, A356234, A356841.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
nogapQ[m_]:=Or[m=={},Union[m]==Range[Min[m],Max[m]]];
Select[Range[1,100,2],nogapQ[primeMS[#]]&]

A356014 Consider the exponents in the prime factorization of n, and replace each run of k consecutive e's by a unique e; the resulting list corresponds to the exponents in the prime factorization of a(n).

Original entry on oeis.org

1, 2, 3, 4, 3, 2, 3, 8, 9, 10, 3, 12, 3, 10, 3, 16, 3, 18, 3, 20, 21, 10, 3, 24, 9, 10, 27, 20, 3, 2, 3, 32, 21, 10, 3, 4, 3, 10, 21, 40, 3, 10, 3, 20, 45, 10, 3, 48, 9, 50, 21, 20, 3, 54, 21, 40, 21, 10, 3, 12, 3, 10, 63, 64, 21, 10, 3, 20, 21, 10, 3, 72, 3

Offset: 1

Rémy Sigrist, Jul 23 2022

nonn

We ignore the exponents (all 0's) for the prime numbers beyond the greatest prime factor of n.

This sequence operates on prime exponents as A090079 and A337864 operate on binary and decimal digits, respectively.

			For n = 99:
- 99 = 11^1 * 7^0 * 5^0 * 3^2 * 2^0,
- the list of exponents is: 1 0 0 2 0,
- compressing consecutive values, we obtain: 1 0 2 0,
- so a(99) = 7^1 * 5^0 * 3^2 * 2^0 = 63.

Cf. A007814, A061742, A067255, A090079, A115964, A294674, A319521, A337864, A356008, A356021 (fixed points).

a(n) = { my (v=1, e=-1, k=0); forprime (p=2, oo, if (n==1, return (v), if (e!=e=valuation(n,p), v*=prime(k++)^e); n/=p^e)) }

a(a(n)) = a(n).
a(n^k) = a(n)^k for any k >= 0.
a(n) = A319521(A356008(n)).
A007814(a(n)) = A007814(n).
a(n) = 3 iff n belongs to A294674 \ {1}.
a(n) = 4 iff n belongs to A061742 \ {1}.
a(n) = 8 iff n belongs to A115964.

A294472 Squarefree numbers whose odd prime factors are all consecutive primes.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17, 19, 22, 23, 26, 29, 30, 31, 34, 35, 37, 38, 41, 43, 46, 47, 53, 58, 59, 61, 62, 67, 70, 71, 73, 74, 77, 79, 82, 83, 86, 89, 94, 97, 101, 103, 105, 106, 107, 109, 113, 118, 122, 127, 131, 134, 137, 139, 142, 143, 146, 149, 151, 154, 157, 158, 163, 166

Offset: 1

Juri-Stepan Gerasimov, Oct 31 2017

nonn

The union of products of any number of consecutive odd primes and twice products of any number of consecutive odd primes.
A073485 lists the squarefree numbers with no gaps in their prime factors >= prime(1), and {a(n)} lists the squarefree numbers with no gaps in their prime factors >= prime(2). If we let {b(n)} be the squarefree numbers with no gaps in their prime factors >= prime(3), ..., and let {x(n)} be the squarefree numbers with no gaps in their prime factors >= prime(y), ..., then A073485(n) >= a(n) >= b(n) >= ... >= x(n) >= ... >= A005117(n). [edited by Jon E. Schoenfield, May 26 2018]
Conjecture: if z(n) is the smallest y such that n*k - k^2 is a squarefree number with no gaps in their prime factors >= prime(y) for some k < n, then z(n) >= 1 for all n > 1.
The terms a(n) for which a(n-1) + 1 = a(n) = a(n+1) - 1 begin 2, 6, 14, 30, 106, ... [corrected by Jon E. Schoenfield, May 26 2018]
Squarefree numbers for which any two neighboring odd prime factors in the ordered list of prime factors are consecutive primes. - Felix Fröhlich, Nov 01 2017

			70 is in this sequence because 2*5*7 = 70 is a squarefree number with two consecutive odd prime factors, 5 and 7.

Cf. A000040, A005117, A073485, A294674.

N:= 1000: # to get all terms <= N
R:= 1,2:
Oddprimes:= select(isprime, [seq(i,i=3..N,2)]):
for i from 1 to nops(Oddprimes) do
  p:= 1:
  for k from i to nops(Oddprimes) do
    p:= p*Oddprimes[k];
    if p > N then break fi;
    if 2*p <= N then R:= R, p, 2*p
    else R:= R,p
    fi
  od;
od:
R:= sort([R]); # Robert Israel, Nov 01 2017

Select[Range@ 166, And[Union@ #2 == {1}, Or[# == {1}, # == {}] &@ Union@ Differences@ PrimePi@ DeleteCases[#1, 2]] & @@ Transpose@ FactorInteger[#] &] (* Michael De Vlieger, Nov 01 2017 *)

Definition corrected by Michel Marcus, Nov 01 2017

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A356845 Odd numbers with gapless prime indices.

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A356014 Consider the exponents in the prime factorization of n, and replace each run of k consecutive e's by a unique e; the resulting list corresponds to the exponents in the prime factorization of a(n).

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A294472 Squarefree numbers whose odd prime factors are all consecutive primes.

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