A276086 -id:A276086 - cp's OEIS frontend

A328571 Primorial base expansion of n converted into its prime product form, but with all nonzero digits replaced by 1's: a(n) = A007947(A276086(n)).

1, 2, 3, 6, 3, 6, 5, 10, 15, 30, 15, 30, 5, 10, 15, 30, 15, 30, 5, 10, 15, 30, 15, 30, 5, 10, 15, 30, 15, 30, 7, 14, 21, 42, 21, 42, 35, 70, 105, 210, 105, 210, 35, 70, 105, 210, 105, 210, 35, 70, 105, 210, 105, 210, 35, 70, 105, 210, 105, 210, 7, 14, 21, 42, 21, 42, 35, 70, 105, 210, 105, 210, 35, 70, 105, 210, 105, 210, 35, 70

Offset: 0

Antti Karttunen, Oct 20 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 0..32768
Index entries for sequences related to primorial base

Cf. A001221, A007947, A083346, A267263, A276086, A328572.

Cf. A276156 (gives the indices where this coincides with A276086).

rad[n_] := Times @@ FactorInteger[n][[All, 1]];
A276086[n0_] := Module[{m = 1, i = 1, n = n0, p}, While[n > 0, p = Prime[i]; m *= p^Mod[n, p]; n = Quotient[n, p]; i++]; m];
a[n_] := rad[A276086[n]];
Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Dec 01 2021, after Antti Karttunen in A276086 *)

A328571(n) = { my(m=1, p=2); while(n, m *= (p^!!(n%p)); n = n\p; p = nextprime(1+p)); (m); };

a(n) = A007947(A276086(n)).
a(n) = A276086(n) / A328572(n).
a(A276156(n)) = A276086(A276156(n)). [And at no other points the equality holds]
A001221(a(n)) = A267263(n).
a(n) = A083346(A276086(n)). - Antti Karttunen, Feb 28 2021

A329041 Square array read by antidiagonals: A(n, k) = A327936(A276086(n) * A276086(k)).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 2, 3, 2, 1, 1, 1, 3, 3, 3, 3, 1, 1, 1, 2, 1, 6, 3, 6, 1, 2, 1, 1, 1, 1, 1, 3, 3, 1, 1, 1, 1, 1, 2, 1, 2, 1, 6, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 2, 3, 2, 1, 2, 3, 2, 3, 2, 1, 1, 1, 3, 3, 3, 3, 1, 1, 3, 3, 3, 3, 1, 1

Offset: 0

Antti Karttunen, Nov 03 2019

Array A(n, k) is symmetric, and is read as (n,k) = (0, 0), (1, 0), (0, 1), (2, 0), (1, 1), (0, 2), (3, 0), (2, 1), (1, 2), (0, 3), ...

If A(n, k) is 1, it tells that adding of n and k do not generate any carries, when done in primorial base (A049345). If A(n, k) is larger than one, then its prime factors indicate in which specific moduli (digit positions) the sum was larger than allowed for that position.

			The top left corner of the array:
        0  1  2  3  4  5  6  7  8  9 10 11 12
      +--------------------------------------
   0: | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
   1: | 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, ...
   2: | 1, 1, 1, 1, 3, 3, 1, 1, 1, 1, 3, 3, 1, ...
   3: | 1, 2, 1, 2, 3, 6, 1, 2, 1, 2, 3, 6, 1, ...
   4: | 1, 1, 3, 3, 3, 3, 1, 1, 3, 3, 3, 3, 1, ...
   5: | 1, 2, 3, 6, 3, 6, 1, 2, 3, 6, 3, 6, 1, ...
   6: | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
   7: | 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, ...
   8: | 1, 1, 1, 1, 3, 3, 1, 1, 1, 1, 3, 3, 1, ...
   9: | 1, 2, 1, 2, 3, 6, 1, 2, 1, 2, 3, 6, 1, ...
  10: | 1, 1, 3, 3, 3, 3, 1, 1, 3, 3, 3, 3, 1, ...
  11: | 1, 2, 3, 6, 3, 6, 1, 2, 3, 6, 3, 6, 1, ...
  12: | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
...
A(11,25) = A(25,11) = 10 because 11 is written in primorial base representation (A049345) as "121" and 25 as "401", and when these are added together digit by digit, we see that the maximal allowed digits "421" for the rightmost three positions are exceeded in positions 1 and 3, with the 1st and 3rd primes 2 and 5 as their moduli, thus A(11,25) = 2*5 = 10.

Cf. A049345, A276086, A327936.

Cf. also A317836, A324351, A328770, A328841, A328842.

up_to = 105;
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A327936(n) = { my(f = factor(n)); for(k=1, #f~, f[k,2] = (f[k,2]>=f[k,1])); factorback(f); };
A329041sq(row,col) = A327936(A276086(row)*A276086(col));
A329041list(up_to) = { my(v = vector(up_to), i=0); for(a=0,oo, for(col=0,a, if(i++ > up_to, return(v)); v[i] = A329041sq(a-col,col))); (v); };
v329041 = A329041list(up_to);
A329041(n) = v329041[1+n];

A(n, k) = A327936(A276086(n) * A276086(k)).

For all n, A(A328841(n), A328842(n)) = 1 and A(A328770(n), A328770(n)) = 1.

A351255 Numbers whose k-th arithmetic derivative is zero for some k>0, ordered by their position in A276086.

Original entry on oeis.org

1, 2, 3, 6, 9, 18, 5, 10, 30, 25, 150, 375, 750, 5625, 7, 14, 21, 42, 126, 70, 105, 315, 350, 1575, 3150, 1750, 2625, 49, 98, 882, 490, 735, 4410, 2450, 3675, 11025, 12250, 30625, 61250, 183750, 686, 3430, 5145, 25725, 77175, 385875, 1929375, 3858750, 4802, 72030, 120050, 180075, 33614, 100842, 117649, 705894, 26471025

Offset: 1

Antti Karttunen, Feb 10 2022

Equal to nonzero terms of A099308 when sorted into ascending order. In this order, which is dictated by the primorial base expansion of n (A049345) and mapped to products of prime powers by A276086, all terms of A099308 that are prime(k)-smooth appear before the terms that are not prime(k)-smooth.
Number of terms whose greatest prime factor (A006530) is prime(n) [in other words, that are prime(n)-smooth but not prime(n-1)-smooth] is given by A351071(n): 1, 4, 8, 44, 216, 1474, 11130, ...
For all n > 1, A003415(a(n)) is also a term of the sequence.
Note that only 451 of the first 105367 terms (all 19-smooth terms) are such that there occurs a 19-smooth number (A080682) larger than 1 on the path before 1 is encountered, when starting from x = a(n) and iterating with map x -> A003415(x).

Antti Karttunen, Table of n, a(n) for n = 1..12878 (all 17-smooth terms of this sequence)
Antti Karttunen, 105368 initial terms, without indices (all 19-smooth terms of this sequence, and also A276086(9699690) = 23, the first 23-smooth term)
Index entries for sequences related to primorial base

Cf. A003415, A049345, A099307, A099308, A276086, A328116, A351071, A351072 (number of prime(n)-smooth terms).

Cf. A351256 [= A051903(a(n))], A351257 [= A099307(a(n))], A351258, A351259 [= A351078(a(n))], A351261 [= A351079(a(n))].

A003415checked(n) = if(n<=1, 0, my(f=factor(n), s=0); for(i=1, #f~, if(f[i,2]>=f[i,1],return(0), s += f[i, 2]/f[i, 1])); (n*s));
A099307(n) = { my(s=1); while(n>1, n = A003415checked(n); s++); if(n,s,0); };
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
for(n=0, 2^9, u=A276086(n); c = A099307(u); if(c>0,print1(u, ", ")));

a(n) = A276086(A328116(n)).

A351458 Numbers k for which k * gcd(sigma(k), A276086(k)) is equal to sigma(k) * gcd(k, A276086(k)), where A276086 is the primorial base exp-function, and sigma gives the sum of divisors of its argument.

Original entry on oeis.org

1, 10, 56, 9196, 9504, 56160, 121176, 239096, 354892, 411264, 555520, 716040, 804384, 904704, 1063348, 1387386, 1444352, 1454112, 1884800, 2708640, 3317248, 3548920, 4009824, 4634784, 6179712, 6795360, 7285248, 14511744, 16328466, 28377216, 29855232, 31940280, 37444736, 42711552, 49762944, 52815744

Offset: 1

Antti Karttunen, Feb 13 2022

nonn

Numbers k such that k * A324644(k) = A000203(k) * A324198(k).
Numbers k such that gcd(A064987(k), A324580(k)) = gcd(A064987(k), A351252(k)).
Numbers k such that their abundancy index [sigma(k)/k] is equal to A324644(k)/A324198(k). See A364286.
A324644 gives odd values for even numbers and for the odd squares. A324198 is odd on all arguments, therefore on odd squares the above equation reduces to odd * odd = odd * odd, and on odd nonsquares as odd * even = even * odd. It is an open question whether there are any odd terms after the initial a(1)=1.
If k is even, but not a multiple of 3, then A276086(k) is a multiple of 3, but not even (i.e., is an odd multiple of 3). If for such k also sigma(k) = 3*k, then A007949(A324644(k)) = min(A007949(sigma(k)), A007949(A276086(k))) = 1, while A007949(A324198(k)) = min(A007949(k), A007949(A276086(k))) = 0, therefore all such k's do occur in this sequence, for example, the two known terms of A005820 (3-perfect numbers) that are not multiples of three: 459818240, 51001180160, but also any hypothetical term of A005820 of the form 4u+2, where 2u+1 is not multiple of 3, and which by necessity is then also an odd perfect number.
Similarly, of the 65 known 5-multiperfect numbers (A046060), those 20 that are not multiples of five are included in this sequence. Note that all 65 are multiples of six.
It is conjectured that the intersection of this sequence with the multiperfect numbers (A007691) gives A323653, see comments in the latter.
For all even terms k of this sequence, A007814(A000203(k)) = A007814(k), sigma preserves the 2-adic valuation, and A007949(A000203(k)) >= A007949(k), i.e., does not decrease the 3-adic valuation. The condition is equivalence (=) when k is a multiple of 6. With odd terms, any hypothetical odd perfect number x would yield a one greater 2-adic valuation for sigma(x) than for x, but would satisfy the main condition of this sequence. - Corrected Feb 17 2022
If k is a nonsquare positive odd number (in A088828), then it must be a term of A191218. - Antti Karttunen, Mar 10 2024

Cf. A000203, A005820, A007691, A007814, A007949, A046060, A088828, A191218, A276086, A323653, A324198, A324644, A327938, A364286, A351459 (intersection with a similar sequence, A349745).

Cf. also A351549.

A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
isA351458(n) = { my(s=sigma(n), z=A276086(n)); (n*gcd(s,z))==(s*gcd(n,z)); };

A006530(n) = if(1==n, n, my(f=factor(n)); f[#f~, 1]); \\ Works OK with rationals also!
isA351458(n) = { my(orgn=n, s=sigma(n), abi=s/n, p=2, q=A006530(abi), d, e1, e2); while((1!=abi)&&(p<=q), d = n%p; e1 = min(d, valuation(s, p)); e2 = min(d, valuation(orgn, p)); d = e1-e2; if(valuation(abi,p)!=d, return(0), abi /= (p^d)); n = n\p; p = nextprime(1+p)); (abi==1); }; \\ (This implementation does not require the construction of largish intermediate numbers, A276086, but might still be slower and return a few false positives on the long run, so please check the results with the above program). - Antti Karttunen, Feb 19 2022

A354347 Dirichlet inverse of A345000, where A345000(n) = gcd(A003415(n), A003415(A276086(n))), with A003415 the arithmetic derivative, and A276086 the primorial base exp-function.

Original entry on oeis.org

1, -1, -1, -1, -1, 1, -1, -1, 0, 1, -1, 1, -1, 1, 1, -9, -1, -2, -1, 1, -3, 1, -1, 1, -4, -3, 0, 1, -1, -1, -1, 21, 1, 1, -1, -6, -1, 1, 1, 3, -1, 7, -1, -1, 0, -3, -1, 23, 0, 4, -3, 7, -1, 2, 1, 3, 1, 1, -1, -1, -1, 1, 8, 15, -1, -1, -1, 1, 1, 3, -1, 14, -1, 1, -46, -7, -1, 7, -1, 5, 0, 1, -1, 3, 1, -3, 1, -131

Offset: 1

Antti Karttunen, Jun 07 2022

sign

Antti Karttunen, Table of n, a(n) for n = 1..30030
Index entries for sequences related to primorial base

Cf. A003415, A276086, A345000.
Cf. A038838 (positions of even terms), A122132 (of odd terms), A353627 (parity of terms), A354815 (positions of 0's), A354816 (of -1's).
Cf. also A346242, A354348, A354823, A354824.

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A345000(n) = gcd(A003415(n),A003415(A276086(n)));
memoA354347 = Map();
A354347(n) = if(1==n,1,my(v); if(mapisdefined(memoA354347,n,&v), v, v = -sumdiv(n,d,if(dA345000(n/d)*A354347(d),0)); mapput(memoA354347,n,v); (v)));

a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, dA345000(n/d) * a(d).

For all n >= 1, A000035(a(n)) = A353627(n).

A324653 a(n) = A000203(A276086(n)).

Original entry on oeis.org

1, 3, 4, 12, 13, 39, 6, 18, 24, 72, 78, 234, 31, 93, 124, 372, 403, 1209, 156, 468, 624, 1872, 2028, 6084, 781, 2343, 3124, 9372, 10153, 30459, 8, 24, 32, 96, 104, 312, 48, 144, 192, 576, 624, 1872, 248, 744, 992, 2976, 3224, 9672, 1248, 3744, 4992, 14976, 16224, 48672, 6248, 18744, 24992, 74976, 81224, 243672, 57, 171, 228

Offset: 0

Antti Karttunen, Mar 10 2019

nonn

Cf. A000203, A002110, A276086.
Cf. A267263, A276150, A324650, A324655 for omega, bigomega, phi and tau analogs, and also A324654.
Cf. also A324054.

A276086(n) = { my(i=0,m=1,pr=1,nextpr); while((n>0),i=i+1; nextpr = prime(i)*pr; if((n%nextpr),m*=(prime(i)^((n%nextpr)/pr));n-=(n%nextpr));pr=nextpr); m; };
A324653(n) = sigma(A276086(n));

a(n) = A000203(A276086(n)).

For n >= 1, a(A002110(n-1)) = 1+A000040(n).

A328389 Maximal digit value in primorial base expansion of A276086(n): a(n) = A328114(A276086(n)).

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 2, 2, 2, 1, 2, 3, 4, 3, 2, 5, 2, 2, 4, 2, 5, 4, 5, 10, 6, 6, 8, 6, 5, 9, 1, 2, 3, 2, 2, 4, 2, 2, 3, 1, 3, 3, 5, 4, 3, 5, 7, 4, 4, 8, 3, 3, 4, 9, 9, 8, 7, 11, 4, 8, 3, 3, 4, 4, 3, 4, 2, 2, 3, 7, 10, 10, 5, 4, 6, 3, 8, 9, 7, 5, 10, 10, 10, 8, 5, 5, 8, 6, 9, 7, 4, 4, 6, 9, 4, 7, 8, 5, 3, 5, 7, 4, 7, 7, 11, 9

Offset: 0

Antti Karttunen, Oct 15 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 0..32589
Index entries for sequences related to primorial base

Cf. A002110, A051903, A143293, A276086, A276087, A328114, A328322, A328390, A328391, A328392, A328395.

A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A328114(n) = { my(s=0, p=2); while(n, s = max(s,(n%p)); n = n\p; p = nextprime(1+p)); (s); };
A328389(n) = A328114(A276086(n));

a(n) = A328114(A276086(n)).

a(n) = A051903(A276087(n)).

A324650 a(n) = A000010(A276086(n)).

Original entry on oeis.org

1, 1, 2, 2, 6, 6, 4, 4, 8, 8, 24, 24, 20, 20, 40, 40, 120, 120, 100, 100, 200, 200, 600, 600, 500, 500, 1000, 1000, 3000, 3000, 6, 6, 12, 12, 36, 36, 24, 24, 48, 48, 144, 144, 120, 120, 240, 240, 720, 720, 600, 600, 1200, 1200, 3600, 3600, 3000, 3000, 6000, 6000, 18000, 18000, 42, 42, 84, 84, 252, 252, 168, 168, 336, 336, 1008, 1008

Offset: 0

Antti Karttunen, Mar 10 2019

nonn

Terms are duplicated because phi(2*(2n+1)) = phi(2n+1) for all n >= 0.

Cf. A000010, A002110, A276086, A324651 (bisection).
Cf. also A267263, A276150, A324653, A324655 for omega, bigomega, sigma and tau analogs.
Cf. also A290077.

A324650(n) = { my(i=0,m=1,pr=1,nextpr); while((n>0),i=i+1; nextpr = prime(i)*pr; if((n%nextpr), m *= (prime(i)-1)*(prime(i)^(((n%nextpr)/pr)-1)); n-=(n%nextpr));pr=nextpr); (m); };

A276086(n) = { my(i=0,m=1,pr=1,nextpr); while((n>0),i=i+1; nextpr = prime(i)*pr; if((n%nextpr),m*=(prime(i)^((n%nextpr)/pr));n-=(n%nextpr));pr=nextpr); m; };
A324650(n) = eulerphi(A276086(n));

a(n) = A000010(A276086(n)).
a(2n+1) = a(2n) for all n >= 0.
For n >= 1, a(A002110(n-1)) = A000040(n)-1.

A328116 Numbers n such that the k-th arithmetic derivative of A276086(n) is zero for some k.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 9, 12, 15, 20, 21, 28, 30, 31, 32, 33, 35, 37, 38, 40, 43, 46, 47, 49, 50, 60, 61, 65, 67, 68, 71, 73, 74, 76, 79, 84, 85, 87, 91, 97, 98, 104, 106, 112, 118, 119, 121, 129, 133, 134, 151, 153, 180, 183, 196, 207, 210, 211, 212, 213, 218, 220, 221, 223, 225, 226, 227, 228, 229, 231, 235, 239, 240

Offset: 1

Antti Karttunen, Oct 08 2019

nonn

Numbers x such that A276086(x) [which is A351255(a(x))] is in A099308.

Antti Karttunen, Table of n, a(n) for n = 1..10000
Antti Karttunen, First 105368 terms (from 0 to 9699690)

Cf. A002110 (subsequence), A003415, A099308, A276086, A327969, A328306 (characteristic function), A328307 (its partial sums).

Cf. A351255 [= A276086(a(n))], A351256 [= A328114(a(n))].

A003415checked(n) = if(n<=1, 0, my(f=factor(n), s=0); for(i=1, #f~, if(f[i,2]>=f[i,1],return(0), s += f[i, 2]/f[i, 1])); (n*s));
A276086(n) = { my(i=0,m=1,pr=1,nextpr); while((n>0),i=i+1; nextpr = prime(i)*pr; if((n%nextpr),m*=(prime(i)^((n%nextpr)/pr));n-=(n%nextpr));pr=nextpr); m; };
isA099308(n) = { while(n>1, n = A003415checked(n)); (n); };
isA328116(n) = isA099308(A276086(n));

For all n >= 1, A328307(a(n)) = n.

A355442 a(n) = gcd(A003961(n), A276086(n)), where A003961 is fully multiplicative with a(p) = nextprime(p), and A276086 is primorial base exp-function.

Original entry on oeis.org

1, 3, 1, 9, 1, 5, 1, 3, 5, 3, 1, 5, 1, 3, 5, 9, 1, 25, 1, 3, 5, 3, 1, 5, 1, 3, 125, 9, 1, 7, 1, 3, 1, 3, 7, 5, 1, 3, 5, 63, 1, 5, 1, 3, 175, 3, 1, 5, 1, 21, 5, 9, 1, 125, 7, 3, 5, 3, 1, 7, 1, 3, 1, 9, 7, 5, 1, 3, 5, 21, 1, 25, 1, 3, 245, 9, 1, 5, 1, 21, 125, 3, 1, 5, 7, 3, 5, 9, 1, 7, 1, 3, 1, 3, 7, 5, 1, 3, 5, 441

Offset: 1

Antti Karttunen, Jul 13 2022

nonn

Antti Karttunen, Table of n, a(n) for n = 1..11550
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537
Index entries for sequences computed from indices in prime factorization
Index entries for sequences related to primorial base

Cf. A003961, A020639, A276086, A355001 [smallest prime factor of a(n)], A355456 [= gcd(sigma(n), a(n))], A355692 (Dirichlet inverse), A355820, A355821 (positions of 1's).

Cf. also A322361, A324198, A351459.

A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A355442(n) = gcd(A003961(n), A276086(n));

a(n) = gcd(A003961(n), A276086(n)).

cp's OEIS Frontend

A328571 Primorial base expansion of n converted into its prime product form, but with all nonzero digits replaced by 1's: a(n) = A007947(A276086(n)).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A329041 Square array read by antidiagonals: A(n, k) = A327936(A276086(n) * A276086(k)).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

A351255 Numbers whose k-th arithmetic derivative is zero for some k>0, ordered by their position in A276086.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Formula

A351458 Numbers k for which k * gcd(sigma(k), A276086(k)) is equal to sigma(k) * gcd(k, A276086(k)), where A276086 is the primorial base exp-function, and sigma gives the sum of divisors of its argument.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

PARI

A354347 Dirichlet inverse of A345000, where A345000(n) = gcd(A003415(n), A003415(A276086(n))), with A003415 the arithmetic derivative, and A276086 the primorial base exp-function.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

A324653 a(n) = A000203(A276086(n)).

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

A328389 Maximal digit value in primorial base expansion of A276086(n): a(n) = A328114(A276086(n)).

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

A324650 a(n) = A000010(A276086(n)).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

PARI

Formula