A007949 -id:A007949 - cp's OEIS frontend

A371638 a(n) = 2*n + valuation(n, 3) with valuation(n, 3) = A007949(n).

2, 4, 7, 8, 10, 13, 14, 16, 20, 20, 22, 25, 26, 28, 31, 32, 34, 38, 38, 40, 43, 44, 46, 49, 50, 52, 57, 56, 58, 61, 62, 64, 67, 68, 70, 74, 74, 76, 79, 80, 82, 85, 86, 88, 92, 92, 94, 97, 98, 100, 103, 104, 106, 111, 110, 112, 115, 116, 118, 121, 122, 124, 128, 128

Offset: 1

Peter Luschny, Mar 30 2024

See A371639 for the connection with Voronoi's congruence.

Cf. A007949, A371639, A292608 (c=2).

A371638 := n -> 2*n + padic:-ordp(n, 3): seq(A371638(n), n = 1..64);

Array[2 # + IntegerExponent[#, 3] &, 64] (* Michael De Vlieger, Mar 31 2024 *)

def A371638(n): return 2 * n + valuation(n, 3)
print([A371638(n) for n in range(1, 65)])

a(n) = valuation(denominator(Voronoi(3, n))) where Voronoi(c, n) = ((c^n - 1) * bernoulli(n)) / (n * c^(n - 1)).

A373593 Lexicographically earliest infinite sequence such that for all i, j >= 1, a(i) = a(j) => f(i) = f(j), where f(n<=3) = n, f(p) = 0 for primes p > 3, and for composite n, f(n) = [A007949(n), A046523(A248909(n)), A046523(A343430(n))].

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 5, 11, 7, 12, 6, 13, 5, 14, 7, 15, 16, 10, 5, 17, 4, 12, 18, 19, 5, 20, 7, 21, 6, 10, 12, 22, 7, 12, 16, 23, 5, 24, 7, 15, 14, 10, 5, 25, 26, 15, 6, 19, 5, 27, 10, 28, 16, 10, 5, 29, 7, 12, 30, 31, 12, 20, 7, 15, 6, 32, 5, 33, 7, 12, 11, 19, 12, 24, 7, 34, 35, 10, 5, 36, 10, 12, 6, 23, 5, 37, 38, 15, 16, 10, 12, 39, 7, 40, 14, 41

Offset: 1

Antti Karttunen, Jun 13 2024

nonn

Restricted growth sequence transform of the function f given in the definition.
For all i, j >= 1:
  a(i) = a(j) => A373595(i) = A373595(j),
  a(i) = a(j) => A353815(i) = A353815(j),
  a(i) = a(j) => A353816(i) = A353816(j).

Antti Karttunen, Table of n, a(n) for n = 1..100000

Cf. A007949, A046523, A248909, A343430.

Cf. A353815, A353816, A373595.

up_to = 100000;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
A007949(n) = valuation(n,3);
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };
A248909(n) = { my(f=factor(n)); for(k=1, #f~, if(1!=(f[k,1]%3), f[k,1]=1)); factorback(f); };
A343430(n) = { my(f=factor(n)); for(k=1, #f~, if(2!=(f[k,1]%3), f[k,1]=1)); factorback(f); };
Aux373593(n) = if(n<3, n, [A007949(n), A046523(A248909(n)), A046523(A343430(n))]);
v373593 = rgs_transform(vector(up_to, n, Aux373593(n)));
A373593(n) = v373593[n];

A373595 Lexicographically earliest infinite sequence such that for all i, j >= 1, a(i) = a(j) => f(i) = f(j), where f(n<=3) = n, f(p) = 0 for primes p > 3, and for composite n, f(n) = [A007949(n), A373591(n), A373592(n)].

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 5, 7, 8, 4, 5, 9, 5, 10, 6, 11, 5, 12, 5, 7, 13, 4, 5, 14, 4, 10, 15, 16, 5, 9, 5, 17, 6, 4, 10, 18, 5, 10, 13, 11, 5, 19, 5, 7, 12, 4, 5, 20, 21, 7, 6, 16, 5, 22, 4, 23, 13, 4, 5, 14, 5, 10, 24, 25, 10, 9, 5, 7, 6, 16, 5, 26, 5, 10, 9, 16, 10, 19, 5, 17, 27, 4, 5, 28, 4, 10, 6, 11, 5, 18, 21, 7, 13, 4, 10, 29, 5, 30, 12, 11, 5, 9, 5, 23, 19

Offset: 1

Antti Karttunen, Jun 13 2024

nonn

Restricted growth sequence transform of the function f given in the definition.
For all i, j > 1:
  A305900(i) = A305900(j) => A373594(i) = A373594(j) => a(i) = a(j),
  A373593(i) = A373593(j) => a(i) = a(j),
  a(i) = a(j) => b(i) = b(j), where b can be (but is not limited to) any of the sequences listed at the crossrefs-section, under "some of the matched sequences".

Antti Karttunen, Table of n, a(n) for n = 1..100000

Cf. A007949, A373591, A373592.
Cf. A305900, A373593, A373594.
Some of the matched sequences (see comments): A001222, A359430, A369643, A369658, A373371, A373383, A373474, A373491, A373493, A373585, A373588, A373596.

up_to = 100000;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
A007949(n) = valuation(n,3);
A373591(n) = sum(i=1, #n=factor(n)~, (1==n[1, i]%3)*n[2, i]);
A373592(n) = sum(i=1, #n=factor(n)~, (2==n[1, i]%3)*n[2, i]);
Aux373595(n) = if(n<=3, n, if(isprime(n), 0, [A007949(n), A373591(n), A373592(n)]));
v373595 = rgs_transform(vector(up_to, n, Aux373595(n)));
A373595(n) = v373595[n];

A381838 k/9 is in this list if A053735(k) < A007949(k), i.e. if digitsum(k, 3) < valuation(k, 3).

Original entry on oeis.org

1, 3, 6, 9, 12, 18, 27, 30, 36, 45, 54, 63, 81, 84, 90, 99, 108, 117, 135, 162, 171, 189, 216, 243, 246, 252, 261, 270, 279, 297, 324, 333, 351, 378, 405, 432, 486, 495, 513, 540, 567, 594, 648, 729, 732, 738, 747, 756, 765, 783, 810, 819, 837, 864, 891, 918, 972, 981, 999

Offset: 1

Peter Luschny, Mar 08 2025

Cf. A007949, A053735.

Cf. A371176 (base 2), A381837 (base 4), A381836 (base 5).

aList := upto -> local k; [seq(k/9, k in select(n -> add(convert(n, base, 3)) < padic[ordp](n, 3), [seq(9..upto,9)]))]: aList(9000);

Select[Range[9000],DigitSum[#,3]Stefano Spezia, Mar 08 2025 *)

def aList(upto, b): return [n/b^2 for n in srange(b^2, upto, b^2) if sum(n.digits(b)) < valuation(n, b)]
print(aList(9000, 3))

A127427 a(n) = v_3( (6n)! ) - v_3( (3n)! ), where v_3(N) is the 3-adic valuation A007949(N).

Original entry on oeis.org

0, 1, 3, 4, 5, 8, 9, 10, 12, 13, 14, 16, 17, 18, 22, 23, 24, 26, 27, 28, 30, 31, 32, 35, 36, 37, 39, 40, 41, 43, 44, 45, 48, 49, 50, 52, 53, 54, 56, 57, 58, 63, 64, 65, 67, 68, 69, 71, 72, 73, 76, 77, 78, 80, 81, 82, 84, 85, 86, 89, 90, 91, 93, 94, 95, 97, 98, 99, 103, 104, 105, 107

Offset: 0

N. J. A. Sloane, Apr 02 2007

nonn

Sung-Hyuk Cha, On Integer Sequences Derived from Balanced k-ary Trees, Applied Mathematics in Electrical and Computer Engineering, 2012. - From _N. J. A. Sloane_, Jun 12 2012
Sung-Hyuk Cha, On Complete and Size Balanced k-ary Tree Integer Sequences, International Journal of Applied Mathematics and Informatics, Issue 2, Volume 6, 2012, pp. 67-75. - From _N. J. A. Sloane_, Dec 24 2012

Cf. A007949, A053735, A054861, A004128.

Essentially partial sums of A127427.

s[n_] := Plus @@ IntegerDigits[n, 3]; a[n_] := (3*n + s[3*n] - s[6*n])/2; Array[a, 100, 0] (* Amiram Eldar, Feb 21 2021 *)

a(n) = valuation((6*n)!, 3) - valuation((3*n)!, 3); \\ Michel Marcus, Jul 29 2017

a(n) - n = a( [(n+1)/3] ).

a(n) = (3*n + A053735(n) - A053735(6*n))/2. - Amiram Eldar, Feb 21 2021

A229290 n is in the sequence if n is prime, (n-1)/3^A007949(n-1) is a squarefree number, A007949(n-1) < 3 and every prime divisor of n-1 is in the sequence.

Original entry on oeis.org

2, 3, 7, 19, 43, 127, 2287, 4903, 5419, 13723, 14479, 82339, 98299, 101347, 304039, 617767, 688087, 1676827, 3735583, 3736087, 4130323, 4324363, 4693267, 4951819, 10621603, 11032999, 11208259, 11554243, 11737783, 12198859, 26152603, 26563939, 28159603

Offset: 1

José María Grau Ribas, Oct 05 2013

nonn

If n is in A226961 then n is some product of elements of this sequence.

Cf. A007949, A226961, A229289, A229291.

fa = FactorInteger; free[n_] := n == Product[fa[n][[i, 1]], {i,
  Length[fa[ n]]}]; Os[b_, 1] = True; Os[b_, 2] = True; Os[b_, b_] = True; Os[b_, n_] := Os[b, n] = PrimeQ[n] && free[(n-1)/ b^IntegerExponent[n - 1,  b]] && IntegerExponent[n - 1, b] < 3 && Union@Table[Os[b, fa[n - 1][[i,1]]], {i, Length[fa[n - 1]]}] == {True}; G[b_] := Select[Prime[Range[2000]], Os[b, #] &]; G[3]

A336069 Decimal expansion of the asymptotic density of numbers k divisible by A007949(k) (A336068).

Original entry on oeis.org

2, 8, 7, 1, 0, 6, 1, 3, 1, 8, 6, 6, 3, 4, 7, 1, 7, 3, 2, 2, 2, 8, 6, 1, 0, 3, 8, 3, 0, 0, 4, 9, 5, 5, 1, 0, 5, 9, 1, 4, 4, 3, 8, 1, 2, 3, 0, 9, 3, 8, 8, 9, 0, 6, 1, 0, 5, 5, 9, 2, 6, 9, 5, 7, 1, 0, 3, 8, 1, 4, 2, 1, 3, 2, 1, 0, 9, 1, 2, 0, 9, 0, 1, 9, 0, 3, 4

Offset: 0

Amiram Eldar, Jul 07 2020

			0.287106131866347173222861038300495510591443812309388...

Tibor Šalát, On the function a_p, p^a_p(n) || n (n > 1), Mathematica Slovaca, Vol. 44, No. 2 (1994), pp. 143-151.

Cf. A007949, A336067, A336068.

RealDigits[2 * Sum[1/k/3^(k + 1 - IntegerExponent[k, 3]), {k, 1, 1000}], 10, 100][[1]]

Equals 2 * Sum_{k>=1} 1/(k * 3^(k - A007949(k) + 1)).

A340683 a(n) = A007949((A003961(A003961(n))+1)/2), where A003961 shifts the prime factorization of n one step towards larger primes, and A007949(x) gives the exponent of largest power of 3 dividing x.

Original entry on oeis.org

0, 1, 0, 0, 1, 2, 0, 2, 0, 0, 2, 0, 0, 1, 1, 0, 1, 1, 1, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 1, 1, 0, 2, 0, 0, 0, 0, 0, 1, 1, 3, 1, 3, 1, 1, 0, 0, 1, 4, 0, 0, 1, 0, 1, 1, 1, 0, 2, 2, 0, 0, 0, 1, 0, 0, 2, 0, 0, 0, 1, 1, 3, 0, 1, 1, 2, 2, 2, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 1, 0, 2, 1, 0, 3, 0, 0, 3, 1

Offset: 1

Antti Karttunen, Feb 14 2021

nonn

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences computed from indices in prime factorization

Cf. A003961, A007949, A048673, A292251, A341347.

A003961(n) = { my(f=factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); }; \\ From A003961
A340683(n) = valuation((A003961(A003961(n))+1)/2, 3);

a(n) = A292251(A003961(n)) = A007949(A048673(A003961(n))).

A358747 Lexicographically earliest infinite sequence such that for all i, j, a(i) = a(j) => f(i) = f(j), where f(n) = [A007814(n), A007949(n), A324198(n)] when n > 1, with f(1) = 1.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 5, 7, 8, 9, 5, 10, 5, 2, 11, 12, 5, 13, 5, 14, 3, 2, 5, 15, 16, 2, 17, 4, 5, 6, 5, 18, 3, 2, 19, 20, 5, 2, 3, 21, 5, 22, 5, 4, 23, 2, 5, 24, 19, 25, 3, 4, 5, 26, 27, 28, 3, 2, 5, 10, 5, 2, 29, 30, 5, 6, 5, 4, 3, 31, 5, 32, 5, 2, 33, 4, 19, 6, 5, 34, 35, 2, 5, 36, 27, 2, 3, 7, 5, 13, 19, 4, 3, 2, 5, 37, 5, 38, 8, 14, 5, 6, 5, 7, 39

Offset: 1

Antti Karttunen, Dec 01 2022

For all i, j: A305900(i) = A305900(j) => a(i) = a(j).

Antti Karttunen, Table of n, a(n) for n = 1..100000

Cf. A007814, A007949, A324198.

Cf. also A305900, A358230.

up_to = 100000;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
A007814(n) = valuation(n,2);
A007949(n) = valuation(n,3);
A324198(n) = { my(m=1, p=2, orgn=n); while(n, m *= (p^min(n%p, valuation(orgn, p))); n = n\p; p = nextprime(1+p)); (m); };
Aux358747(n) = if(1==n,n,[A007814(n), A007949(n), A324198(n)]);
v358747 = rgs_transform(vector(up_to, n,Aux358747(n)));
A358747(n) = v358747[n];

A374040 Lexicographically earliest infinite sequence such that a(i) = a(j) => f(i) = f(j), where f(n) = [A003415(n), A085731(n), A007814(n), A007949(n)], for all i, j >= 1.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 5, 7, 8, 9, 5, 10, 5, 11, 12, 13, 5, 14, 5, 15, 16, 17, 5, 18, 19, 20, 21, 22, 5, 23, 5, 24, 25, 26, 27, 28, 5, 29, 30, 31, 5, 32, 5, 33, 34, 35, 5, 36, 37, 38, 39, 40, 5, 41, 42, 43, 44, 45, 5, 46, 5, 47, 48, 49, 50, 51, 5, 52, 53, 54, 5, 55, 5, 56, 57, 58, 50, 59, 5, 60, 61, 62, 5, 63, 64, 65, 66, 67, 5, 68, 69, 70, 71, 72, 73, 74, 5, 75

Offset: 1

Antti Karttunen, Jul 01 2024

nonn

Restricted growth sequence transform of the quadruple [A003415(n), A085731(n), A007814(n), A007949(n)].
For all i, j >= 1:
  A305900(i) = A305900(j) => a(i) = a(j),
  a(i) = a(j) => A322026(i) = A322026(j),
  a(i) = a(j) => A369051(i) = A369051(j) => A083345(i) = A083345(j),
  a(i) = a(j) => b(i) = b(j), where b can be any of the sequences listed at the crossrefs-section, under "some of the other matched sequences".

Antti Karttunen, Table of n, a(n) for n = 1..100000

Cf. A003415, A007814, A007949, A085731.
Some of the other matched sequences (see comments): A083345, A359430, A369001, A369004, A369643, A369658, A373143, A373474, A373483.
Cf. also A322026, A353521, A369051, A373268, A372573, A374131 for similar and related constructions.
Differs from A305900 first at n=77, where a(77) = 50, while A305900(77) = 59.

up_to = 100000;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
Aux374040(n) = { my(d=A003415(n)); [d, gcd(n,d), valuation(n,2), valuation(n,3)]; };
v374040 = rgs_transform(vector(up_to, n, Aux374040(n)));
A374040(n) = v374040[n];

cp's OEIS Frontend

A371638 a(n) = 2*n + valuation(n, 3) with valuation(n, 3) = A007949(n).

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A373593 Lexicographically earliest infinite sequence such that for all i, j >= 1, a(i) = a(j) => f(i) = f(j), where f(n<=3) = n, f(p) = 0 for primes p > 3, and for composite n, f(n) = [A007949(n), A046523(A248909(n)), A046523(A343430(n))].

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A373595 Lexicographically earliest infinite sequence such that for all i, j >= 1, a(i) = a(j) => f(i) = f(j), where f(n<=3) = n, f(p) = 0 for primes p > 3, and for composite n, f(n) = [A007949(n), A373591(n), A373592(n)].

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A381838 k/9 is in this list if A053735(k) < A007949(k), i.e. if digitsum(k, 3) < valuation(k, 3).

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A127427 a(n) = v_3( (6n)! ) - v_3( (3n)! ), where v_3(N) is the 3-adic valuation A007949(N).

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A229290 n is in the sequence if n is prime, (n-1)/3^A007949(n-1) is a squarefree number, A007949(n-1) < 3 and every prime divisor of n-1 is in the sequence.

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A336069 Decimal expansion of the asymptotic density of numbers k divisible by A007949(k) (A336068).

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A340683 a(n) = A007949((A003961(A003961(n))+1)/2), where A003961 shifts the prime factorization of n one step towards larger primes, and A007949(x) gives the exponent of largest power of 3 dividing x.

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A358747 Lexicographically earliest infinite sequence such that for all i, j, a(i) = a(j) => f(i) = f(j), where f(n) = [A007814(n), A007949(n), A324198(n)] when n > 1, with f(1) = 1.

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