A099800 -id:A099800 - cp's OEIS frontend

A342050 Numbers k which have an odd number of trailing zeros in their primorial base representation A049345(k).

2, 4, 8, 10, 14, 16, 20, 22, 26, 28, 30, 32, 34, 38, 40, 44, 46, 50, 52, 56, 58, 60, 62, 64, 68, 70, 74, 76, 80, 82, 86, 88, 90, 92, 94, 98, 100, 104, 106, 110, 112, 116, 118, 120, 122, 124, 128, 130, 134, 136, 140, 142, 146, 148, 150, 152, 154, 158, 160, 164, 166, 170, 172, 176, 178, 180, 182, 184, 188, 190, 194, 196, 200, 202, 206, 208, 212

Offset: 1

Amiram Eldar, Feb 26 2021

Numbers k such that A276084(k) is odd.
All the terms are even since odd numbers have 0 trailing zeros, and 0 is not odd.
The number of terms not exceeding A002110(m) for m>=1 is A002110(m) * Sum_{k=1..m}(-1)^k/A002110(k) = 1, 2, 11, 76, 837, 10880, 184961, ...
The asymptotic density of this sequence is Sum_{k>=1} (-1)^(k+1)/A002110(k) = 0.362306... (A132120).
Also Heinz numbers of partitions with even least gap. The least gap (mex or minimal excludant) of a partition is the least positive integer that is not a part. The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), giving a bijective correspondence between positive integers and integer partitions. - Gus Wiseman, Apr 23 2021
Numbers k such that A000720(A053669(k)) is even. Differences from the related A353531 seem to be terms that are multiples of 210, but not all of them, for example primorial 30030 (= 143*210) is in neither sequence. Consider also A038698. - Antti Karttunen, Apr 25 2022

			2 is a term since A049345(2) = 10 has 1 trailing zero.
4 is a term since A049345(2) = 20 has 1 trailing zero.
30 is a term since A049345(2) = 1000 has 3 trailing zeros.
From _Gus Wiseman_, Apr 23 2021: (Start)
The sequence of terms together with their prime indices begins:
      2: {1}             46: {1,9}             90: {1,2,2,3}
      4: {1,1}           50: {1,3,3}           92: {1,1,9}
      8: {1,1,1}         52: {1,1,6}           94: {1,15}
     10: {1,3}           56: {1,1,1,4}         98: {1,4,4}
     14: {1,4}           58: {1,10}           100: {1,1,3,3}
     16: {1,1,1,1}       60: {1,1,2,3}        104: {1,1,1,6}
     20: {1,1,3}         62: {1,11}           106: {1,16}
     22: {1,5}           64: {1,1,1,1,1,1}    110: {1,3,5}
     26: {1,6}           68: {1,1,7}          112: {1,1,1,1,4}
     28: {1,1,4}         70: {1,3,4}          116: {1,1,10}
     30: {1,2,3}         74: {1,12}           118: {1,17}
     32: {1,1,1,1,1}     76: {1,1,8}          120: {1,1,1,2,3}
     34: {1,7}           80: {1,1,1,1,3}      122: {1,18}
     38: {1,8}           82: {1,13}           124: {1,1,11}
     40: {1,1,1,3}       86: {1,14}           128: {1,1,1,1,1,1,1}
     44: {1,1,5}         88: {1,1,1,5}        130: {1,3,6}
(End)

Amiram Eldar, Table of n, a(n) for n = 1..10000
George E. Andrews and David Newman, Partitions and the Minimal Excludant, Annals of Combinatorics, Volume 23, May 2019, Pages 249-254.
Brian Hopkins, James A. Sellers, and Dennis Stanton, Dyson's Crank and the Mex of Integer Partitions, arXiv:2009.10873 [math.CO], 2020.
Wikipedia, Mex (mathematics)
Index entries for sequences computed from indices in prime factorization
Index entries for sequences related to primorial base

Cf. A002110, A049345, A053669, A132120, A276084.
Complement of A342051.
A099800 is subsequence.
Analogous sequences: A001950 (Zeckendorf representation), A036554 (binary), A145204 (ternary), A217319 (base 4), A232745 (factorial base).
The version for reversed binary expansion is A079523.
Positions of even terms in A257993.
A000070 counts partitions with a selected part.
A056239 adds up prime indices, row sums of A112798.
A073491 lists numbers with gap-free prime indices.
A079067 counts gaps in prime indices.
A238709 counts partitions by sum and least difference.
A333214 lists positions of adjacent unequal prime gaps.
A339662 gives greatest gap in prime indices.
Cf. A000720, A001223, A005408, A026794, A029707, A038698, A047235, A079068, A121539, A286469, A286470, A325351, A353525 (characteristic function).
Differs from A353531 for the first time at n=77, where a(77) = 212, as this sequence misses A353531(77) = 210.

seq[max_] := Module[{bases = Prime@Range[max, 1, -1], nmax}, nmax = Times @@ bases - 1; Select[Range[nmax], OddQ @ LengthWhile[Reverse @ IntegerDigits[#, MixedRadix[bases]], #1 == 0 &] &]]; seq[4]
Select[Range[100],EvenQ[Min@@Complement[Range[PrimeNu[#]+1],PrimePi/@First/@FactorInteger[#]]]&] (* Gus Wiseman, Apr 23 2021 *)

A353525(n) = { for(i=1,oo,if(n%prime(i),return((i+1)%2))); }
isA342050(n) = A353525(n);
k=0; n=0; while(k<77, n++; if(isA342050(n), k++; print1(n,", "))); \\ Antti Karttunen, Apr 25 2022

More terms added (to differentiate from A353531) by Antti Karttunen, Apr 25 2022

A332821 One part of a 3-way classification of the positive integers. Numbers n for which A048675(n) == 1 (mod 3).

Original entry on oeis.org

2, 5, 9, 11, 12, 16, 17, 21, 23, 28, 30, 31, 39, 40, 41, 47, 49, 52, 54, 57, 59, 66, 67, 70, 72, 73, 75, 76, 83, 87, 88, 91, 96, 97, 100, 102, 103, 109, 111, 116, 126, 127, 128, 129, 130, 133, 135, 136, 137, 138, 148, 149, 154, 157, 159, 165, 167, 168, 169, 172, 175, 179, 180, 183, 184, 186, 190, 191, 197, 203, 211, 212

Offset: 1

Antti Karttunen and Peter Munn, Feb 25 2020

nonn

The positive integers are partitioned between A332820, this sequence and A332822.
For each prime p, the terms include exactly one of p and p^2. The primes alternate between this sequence and A332822. This sequence has the primes with odd indexes, those in A031368.
The terms are the even numbers in A332822 halved. The terms are also the numbers m such that 5m is in A332822, and so on for alternate primes: 11, 17, 23 etc. Likewise, the terms are the numbers m such that 3m is in A332820, and so on for alternate primes: 7, 13, 19 etc.
The numbers that are half of the even terms of this sequence are in A332820, which consists exactly of those numbers. The numbers that are one third of the terms that are multiples of 3 are in A332822, which consists exactly of those numbers. For larger primes, an alternating pattern applies as described in the previous paragraph.
If we take each odd term of this sequence and replace each prime in its factorization by the next smaller prime, the resulting number is in A332822, which consists entirely of those numbers.
The product of any 2 terms of this sequence is in A332822, the product of any 3 terms is in A332820, and the product of a term of A332820 and a term of this sequence is in this sequence. So if a number k is present, k^2 is in A332822, k^3 is in A332820, and k^4 is in this sequence.
If k is an even number, exactly one of {k/2, k, 2k} is in the sequence (cf. A191257 / A067368 / A213258); and generally if k is a multiple of a prime p, exactly one of {k/p, k, k*p} is in the sequence.

Positions of ones in A332823; equivalently, numbers in row 3k+1 of A277905 for some k >= 0.
Cf. A048675, A332820, A332822.
Comparable 2 or 3-way classifications: A000379/A000028, A001969/A000069, A003159/A036554, A005843/A005408, A028260/A026424, A191257/A067368/A213258, A325431/A325432, A329609/A329604/A332812.
Subsequences: intersection of A026478 and A066208, A031368 (prime terms), A033431\{0}, A052934\{1}, A069486, A099800, A167747\{1}, A244725\{0}, A244728\{0}, A338911 (semiprime terms).

Select[Range@ 212, Mod[Total@ #, 3] == 1 &@ Map[#[[-1]]*2^(PrimePi@ #[[1]] - 1) &, FactorInteger[#]] &] (* Michael De Vlieger, Mar 15 2020 *)

isA332821(n) =  { my(f = factor(n)); (1==((sum(k=1, #f~, f[k, 2]*2^primepi(f[k, 1]))/2)%3)); };

{a(n) : n >= 1} = {2 * A332820(k) : k >= 1} U {A003961(A332822(k)) : k >= 1}.

{a(n) : n >= 1} = {A332822(k)^2 : k >= 1} U {A331590(2, A332820(k)) : k >= 1}.

A099788 a(n) = Product_{i=1..2n} prime(i).

Original entry on oeis.org

1, 6, 210, 30030, 9699690, 6469693230, 7420738134810, 13082761331670030, 32589158477190044730, 117288381359406970983270, 557940830126698960967415390, 3217644767340672907899084554130, 23768741896345550770650537601358310, 232862364358497360900063316880507363070

Offset: 0

N. J. A. Sloane, Nov 19 2004

Vincenzo Librandi, Table of n, a(n) for n = 0..100

Cf. A000040, A099800.
Bisection of A002110.
Subset of A030229.

[1] cat [&*[NthPrime(j): j in [1..2*n]]: n in [1..20]]; // G. C. Greubel, Sep 04 2019

a:=n-> mul(ithprime(j),j=1..2*n): seq(a(n),n=0..13); # Emeric Deutsch

Table[Product[Prime[i], {i, 2n}], {n, 0, 20}] (* Wesley Ivan Hurt, Mar 13 2014 *)

a(n) = prod(i=1, 2*n, prime(i)); \\ Michel Marcus, Mar 15 2014

[1]+[product(nth_prime(j) for j in (1..2*n)) for n in (1..20)] # G. C. Greubel, Sep 04 2019

More terms from Emeric Deutsch, Feb 23 2005

Name revised by Wesley Ivan Hurt, Mar 13 2014

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A342050 Numbers k which have an odd number of trailing zeros in their primorial base representation A049345(k).

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A332821 One part of a 3-way classification of the positive integers. Numbers n for which A048675(n) == 1 (mod 3).

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A099788 a(n) = Product_{i=1..2n} prime(i).

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