A112765 -id:A112765 - cp's OEIS frontend

A379005 Lexicographically earliest infinite sequence such that a(i) = a(j) => v_2(i) = v_2(j), v_3(i) = v_3(j) and v_5(i) = v_5(j), for all i, j, where v_2 (A007814), v_3 (A007949) and v_5 (A112765) give the 2-, 3- and 5-adic valuations of n respectively.

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

Offset: 1

Antti Karttunen, Dec 15 2024

nonn

Restricted growth sequence transform of A355582.
For all i, j:
  A379001(i) = A379001(j) => a(i) = a(j),
  a(i) = a(j) => A322026(i) = A322026(j),
  a(i) = a(j) => A379004(i) = A379004(j).

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

Cf. A007814, A007949, A112765, A355582, A379006 (ordinal transform).

Cf. A322026, A379001, A379004.

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; };
v379005 = rgs_transform(vector(up_to, n, [valuation(n,2), valuation(n,3), valuation(n,5)]));
A379005(n) = v379005[n];

A379001 Lexicographically earliest infinite sequence such that a(i) = a(j) => A046523(i) = A046523(j), v_2(i) = v_2(j), v_3(i) = v_3(j) and v_5(i) = v_5(j), for all i, j, where v_2 (A007814), v_3 (A007949) and v_5 (A112765) give the 2-, 3- and 5-adic valuations of n respectively.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 7, 11, 7, 12, 13, 14, 7, 15, 7, 16, 17, 12, 7, 18, 19, 12, 20, 21, 7, 22, 7, 23, 17, 12, 24, 25, 7, 12, 17, 26, 7, 27, 7, 21, 28, 12, 7, 29, 30, 31, 17, 21, 7, 32, 24, 33, 17, 12, 7, 34, 7, 12, 35, 36, 24, 27, 7, 21, 17, 37, 7, 38, 7, 12, 39, 21, 40, 27, 7, 41, 42, 12, 7, 43, 24, 12, 17, 33, 7, 44, 40, 21, 17, 12, 24, 45, 7, 46, 35

Offset: 1

Antti Karttunen, Dec 15 2024

nonn

Restricted growth sequence transform of ordered 4-tuple [A046523(n), A007814(n), A007949(n), A112765(n)].
For all i, j:
  A379000(i) = A379000(j) => a(i) = a(j),
  a(i) = a(j) => A358230(i) = A358230(j),
  a(i) = a(j) => A379002(i) = A379002(j),
  a(i) = a(j) => A379005(i) = A379005(j).

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

Cf. A046523, A007814, A007949, A112765.

Cf. A358230, A379000, A379002, A379005.

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; };
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };
v379001 = rgs_transform(vector(up_to, n, [A046523(n), valuation(n,2), valuation(n,3), valuation(n,5)]));
A379001(n) = v379001[n];

A379004 Lexicographically earliest infinite sequence such that a(i) = a(j) => v_2(i) = v_2(j) and v_5(i) = v_5(j), for all i, j, where v_2 (A007814) and v_5 (A112765) give the 2- and 5-adic valuations of n respectively.

Original entry on oeis.org

1, 2, 1, 3, 4, 2, 1, 5, 1, 6, 1, 3, 1, 2, 4, 7, 1, 2, 1, 8, 1, 2, 1, 5, 9, 2, 1, 3, 1, 6, 1, 10, 1, 2, 4, 3, 1, 2, 1, 11, 1, 2, 1, 3, 4, 2, 1, 7, 1, 12, 1, 3, 1, 2, 4, 5, 1, 2, 1, 8, 1, 2, 1, 13, 4, 2, 1, 3, 1, 6, 1, 5, 1, 2, 9, 3, 1, 2, 1, 14, 1, 2, 1, 3, 4, 2, 1, 5, 1, 6, 1, 3, 1, 2, 4, 10, 1, 2, 1, 15, 1, 2, 1, 5, 4

Offset: 1

Antti Karttunen, Dec 15 2024

Restricted growth sequence transform of A132741, or equally, of the ordered pair [A007814(n), A112765(n)].
For all i, j:
  A379005(i) = A379005(j) => a(i) = a(j).
A379003 (after its initial 0) and this sequence are ordinal transforms of each other.

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

Cf. A007814, A112765, A132741, A379003 (ordinal transform), A379005.

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; };
v379004 = rgs_transform(vector(up_to, n, [valuation(n,2), valuation(n,5)]));
A379004(n) = v379004[n];

A379002 Lexicographically earliest infinite sequence such that a(i) = a(j) => A046523(i) = A046523(j) and A112765(i) = A112765(j), for all i, j, where A046523 gives the least representative of the prime signature of n and A112765 gives the 5-adic valuation of n.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Dec 15 2024

nonn

Restricted growth sequence transform of ordered pair [A046523(n), A112765(n)].
For all i, j:
  A379001(i) = A379001(j) => a(i) = a(j).

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

Cf. A046523, A112765, A379001.

Cf. also A305891, A305893.

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; };
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };
v379002 = rgs_transform(vector(up_to, n, [A046523(n), valuation(n,5)]));
A379002(n) = v379002[n];

A381836 k/25 is in this list if A053824(k) < A112765(k), i.e. if digitsum(k, 5) < valuation(k, 5).

Original entry on oeis.org

1, 5, 10, 25, 30, 50, 75, 125, 130, 150, 175, 250, 275, 375, 500, 625, 630, 650, 675, 750, 775, 875, 1000, 1250, 1275, 1375, 1500, 1875, 2000, 2500, 3125, 3130, 3150, 3175, 3250, 3275, 3375, 3500, 3750, 3775, 3875, 4000, 4375, 4500, 5000, 5625, 6250, 6275, 6375

Offset: 1

Peter Luschny, Mar 08 2025

Cf. A053824, A112765.

Cf. A371176 (base 2), A381838 (base 3), A381837 (base 4).

aList := upto -> local k; [seq(k/25, k in select(n -> add(convert(n, base, 5)) < padic[ordp](n, 5), [seq(25..upto,25)]))]: aList(160000);

Select[Range[160000],DigitSum[#,5]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(160000, 5))

A127428 v_5( (10n)! ) - v_5( (5n)! ), where v_p(N) is the p-adic valuation defined in A112765.

Original entry on oeis.org

0, 1, 2, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, 17, 18, 19, 20, 21, 23, 24, 25, 26, 27, 29, 30, 31, 32, 33, 35, 36, 37, 38, 39, 41, 42, 43, 44, 45, 48, 49, 50, 51, 52, 54, 55, 56, 57, 58, 60, 61, 62, 63, 64, 66, 67, 68, 69, 70, 72, 73, 74, 75, 76, 80, 81, 82, 83, 84, 86, 87, 88, 89, 90, 92, 93

Offset: 0

N. J. A. Sloane, Apr 02 2007

nonn

v_5(n) is given in A112765. Cf. A001511, A127427.

A127428 := proc(n)
    padic[ordp]((10*n)!,5)-padic[ordp]((5*n)!,5) ;
end proc:
seq(A127428(n),n=0..100) ; # R. J. Mathar, Mar 29 2017

a(n) = valuation((10*n)!, 5) - valuation((5*n)!, 5); \\ Michel Marcus, Jul 29 2017

a(n) - n = a( [(n+2)/5] ).

Typo in name corrected by Michel Marcus, Jul 29 2017

A195760 G.f.: A(x) = exp( Sum_{n>=1} 55^A112765(n) x^n/n ), where A112765 is the exponent of the highest power of 5 dividing n.

Original entry on oeis.org

1, 5, 15, 35, 70, 130, 230, 390, 635, 995, 1515, 2255, 3290, 4710, 6620, 9160, 12505, 16865, 22485, 29645, 38695, 50055, 64215, 81735, 103245, 129505, 161405, 199965, 246335, 301795, 367855, 446255, 538965, 648185, 776345, 926265, 1101155, 1304615, 1540635

Offset: 0

Paul D. Hanna, Sep 23 2011

nonn

			G.f.: A(x) = 1 + 5*x + 15*x^2 + 35*x^3 + 70*x^4 + 130*x^5 + 230*x^6 +...
log(A(x)) = 5*x + 5*x^2/2 + 5*x^3/3 + 5*x^4/4 + 25*x^5/5 + 5*x^6/6 + 5*x^7/7 + 5*x^8/8 + 5*x^9/9 + 25*x^10/10 +...
The coefficients in the QUINTISECTIONS of g.f. A(x) begin:
Q0: [1, 130, 1515, 9160, 38695, 129505, 367855, 926265, 2128510, ...];
Q1: [5, 230, 2255, 12505, 50055, 161405, 446255, 1101155, 2491030, ...];
Q2: [15, 390, 3290, 16865, 64215, 199965, 538965, 1304615, 2907440, ...];
Q3: [35, 635, 4710, 22485, 81735, 246335, 648185, 1540635, 3384660, ...];
Q4: [70, 995, 6620, 29645, 103245, 301795, 776345, 1813595, 3930245, ...].
The coefficients in the products Q2*Q3 and Q1*Q4 begin:
Q2(x)*Q3(x): [525, 23175, 433450, 4853600, 38447875, 236756775, ...];
Q1(x)*Q4(x): [350, 21075, 419800, 4789900, 38209000, 235990975, ...];
where Q2(x)*Q3(x) - Q1(x)*Q4(x) = 175*(1-x)^6/R(x)^2, and
R(x) = 1 - 9*x + 36*x^2 - 84*x^3 + 126*x^4 - 130*x^5 + 120*x^6 +...

Paul D. Hanna, Table of n, a(n) for n = 0..1000

Cf. A195761, A161809, A112765.

{a(n)=local(N=ceil(log(n+6)/log(5)));polcoeff(1/(1-x+x*O(x^n))^5/prod(k=1,N,(1-x^(5^k) +x*O(x^n))^4),n)}

{a(n)=local(L=sum(m=1, n, 5*5^valuation(m, 5)*x^m/m)+x*O(x^n)); polcoeff(exp(L), n)}

G.f.: A(x) = 1/(1-x)^5 * Product_{n>=1} 1/(1 - x^(5^n))^4.
G.f. satisfies: A(x) = A(x^5)*(1-x^5)/(1-x)^5.
Let the QUINTISECTIONS of g.f. A(x) be defined by:
A(x) = Q0(x^5) + x*Q1(x^5) + x^2*Q3(x^5) + x^3*Q3(x^5) + x^4*Q4(x^5),
then:
_ Q0(x) = (1 + 121*x + 381*x^2 + 121*x^3 + x^4)/R(x)
_ Q1(x) = 5*(1 + 37*x + 73*x^2 + 14*x^3)/R(x)
_ Q2(x) = 5*(3 + 51*x + 64*x^2 + 7*x^3)/R(x)
_ Q3(x) = 5*(7 + 64*x + 51*x^2 + 3*x^3)/R(x)
_ Q4(x) = 5*(14 + 73*x + 37*x^2 + 1*x^3)/R(x)
where R(x) = (1-x)^5 * Product_{n>=0} (1 - x^(5^n))^4.
Further, the quintisections are related by:
_ Q2(x)*Q3(x) - Q1(x)*Q4(x) = 175*(1-x)^6/R(x)^2.

A229291 n is in the sequence if n is prime, (n-1)/5^A112765(n-1) is a squarefree number, A112765(n-1) < 3 and every prime divisor of n-1 is in the sequence.

Original entry on oeis.org

2, 3, 5, 7, 11, 23, 31, 43, 47, 67, 71, 139, 151, 211, 283, 311, 331, 431, 463, 659, 683, 691, 863, 907, 947, 967, 1051, 1151, 1291, 1303, 1319, 1367, 1427, 1511, 1699, 1867, 1979, 1987, 2011, 2111, 2131, 2311, 2351, 2531, 3011, 3023, 3083, 3323, 3851, 4099

Offset: 1

José María Grau Ribas, Oct 06 2013

nonn

If n is in A226963 then n is some product of elements of this sequence.

Cf. A112765, A226963, A229289, A229290.

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[5]

A289772 a(n) is the numerator of b(n) where b(n) = 1/(3(1+2A112765(n) - b(n-1))) and b(0) = 0, where A112765(n) is the 5-adic valuation of n.

Original entry on oeis.org

0, 1, 1, 2, 1, 1, 2, 5, 3, 4, 1, 5, 4, 7, 3, 2, 3, 7, 4, 5, 1, 4, 3, 5, 2, 1, 3, 8, 5, 7, 2, 11, 9, 16, 7, 5, 8, 19, 11, 14, 3, 13, 10, 17, 7, 4, 5, 11, 6, 7, 1, 8, 7, 13, 6, 5, 9, 22, 13, 17, 4, 19, 15, 26, 11, 7, 10, 23, 13, 16, 3, 11, 8, 13, 5, 2, 5, 13, 8, 11, 3

Offset: 0

Michel Marcus, Jul 12 2017

For n>0, a(n)/A289773(n) lists the rationals of a quinary analog of the Calkin-Wilf tree. See the Ponton link.

			Tree of rationals begin:
0;
1/3;
1/2, 2/3, 1, 1/6, 2/5;
5/9, 3/4, 4/3, 1/5, 5/12, 4/7, 7/9, 3/2, 2/9, 3/7, 7/12, 4/5, 5/3, 1/4, 4/9, 3/5, 5/6, 2, 1/9, 3/8, 8/15, 5/7, 7/6, 2/11, 11/27;
...

Robert Israel, Table of n, a(n) for n = 0..10000
Lionel Ponton, Two trees enumerating the positive rationals, arXiv:1707.02366 [math.NT], 2017. See p. 7.
Lionel Ponton, Two trees enumerating the positive rationals, Integers, Electronic Journal of Combinatorial Number Theory 18A (2018), #A17.

Cf. A002487, A277749, A277750, A289773.

b:= proc(n) option remember; 1/(3*(1+2*padic:-ordp(n,5)-procname(n-1))) end proc:
b(0):= 0:
map(numer@b, [$0..100]); # Robert Israel, Jul 12 2017

a[0] = 0; a[n_] := a[n] = 1/(3 (1 + 2 IntegerExponent[n, 5] - a[n - 1])); Table[Numerator@ a@ n, {n, 0, 80}] (* Michael De Vlieger, Jul 12 2017 *)

b(n) = if (n==0, 0, 1/(3*(1+2*valuation(n, 5) - b(n-1))));
lista(nn) = for (n=0, nn, print1(numerator(b(n)), ", "));

A289773 a(n) is the denominator of b(n) where b(n) = 1/(3(1+2A112765(n) - b(n-1))) and b(0) = 0, with A112765(n) being the 5-adic valuation of n.

Original entry on oeis.org

1, 3, 2, 3, 1, 6, 5, 9, 4, 3, 5, 12, 7, 9, 2, 9, 7, 12, 5, 3, 4, 9, 5, 6, 1, 9, 8, 15, 7, 6, 11, 27, 16, 21, 5, 24, 19, 33, 14, 9, 13, 30, 17, 21, 4, 15, 11, 18, 7, 3, 8, 21, 13, 18, 5, 27, 22, 39, 17, 12, 19, 45, 26, 33, 7, 30, 23, 39, 16, 9, 11, 24, 13, 15, 2, 15, 13

Offset: 0

Michel Marcus, Jul 12 2017

For n>0, A289772(n)/a(n) lists the rationals of a quinary analog of the Calkin-Wilf tree. See the Ponton link.

			Tree of rationals begin:
0;
1/3;
1/2, 2/3, 1, 1/6, 2/5;
5/9, 3/4, 4/3, 1/5, 5/12, 4/7, 7/9, 3/2, 2/9, 3/7, 7/12, 4/5, 5/3, 1/4, 4/9, 3/5, 5/6, 2, 1/9, 3/8, 8/15, 5/7, 7/6, 2/11, 11/27;
...

Robert Israel, Table of n, a(n) for n = 0..10000
Lionel Ponton, Two trees enumerating the positive rationals, arXiv:1707.02366 [math.NT], 2017. See p. 7.
Lionel Ponton, Two trees enumerating the positive rationals, Integers, Electronic Journal of Combinatorial Number Theory 18A (2018), #A17.

Cf. A002487, A277749, A277750, A289772.

b:= proc(n) option remember; 1/(3*(1+2*padic:-ordp(n,5)-procname(n-1))) end proc:
b(0):= 0:
map(denom@b, [$0..100]); # Robert Israel, Jul 12 2017

a[0] = 0; a[n_] := a[n] = 1/(3 (1 + 2 IntegerExponent[n, 5] - a[n - 1])); Table[Denominator@ a@ n, {n, 0, 76}] (* Michael De Vlieger, Jul 12 2017 *)

b(n) = if (n==0, 0, 1/(3*(1+2*valuation(n, 5) - b(n-1))));
lista(nn) = for (n=0, nn, print1(denominator(b(n)), ", "));

cp's OEIS Frontend

A379005 Lexicographically earliest infinite sequence such that a(i) = a(j) => v_2(i) = v_2(j), v_3(i) = v_3(j) and v_5(i) = v_5(j), for all i, j, where v_2 (A007814), v_3 (A007949) and v_5 (A112765) give the 2-, 3- and 5-adic valuations of n respectively.

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A379001 Lexicographically earliest infinite sequence such that a(i) = a(j) => A046523(i) = A046523(j), v_2(i) = v_2(j), v_3(i) = v_3(j) and v_5(i) = v_5(j), for all i, j, where v_2 (A007814), v_3 (A007949) and v_5 (A112765) give the 2-, 3- and 5-adic valuations of n respectively.

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A379004 Lexicographically earliest infinite sequence such that a(i) = a(j) => v_2(i) = v_2(j) and v_5(i) = v_5(j), for all i, j, where v_2 (A007814) and v_5 (A112765) give the 2- and 5-adic valuations of n respectively.

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A379002 Lexicographically earliest infinite sequence such that a(i) = a(j) => A046523(i) = A046523(j) and A112765(i) = A112765(j), for all i, j, where A046523 gives the least representative of the prime signature of n and A112765 gives the 5-adic valuation of n.

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A381836 k/25 is in this list if A053824(k) < A112765(k), i.e. if digitsum(k, 5) < valuation(k, 5).

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Maple

Mathematica

SageMath

A127428 v_5( (10n)! ) - v_5( (5n)! ), where v_p(N) is the p-adic valuation defined in A112765.

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Formula

Extensions

A195760 G.f.: A(x) = exp( Sum_{n>=1} 5*5^A112765(n) * x^n/n ), where A112765 is the exponent of the highest power of 5 dividing n.

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Formula

A229291 n is in the sequence if n is prime, (n-1)/5^A112765(n-1) is a squarefree number, A112765(n-1) < 3 and every prime divisor of n-1 is in the sequence.

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Mathematica

A289772 a(n) is the numerator of b(n) where b(n) = 1/(3*(1+2*A112765(n) - b(n-1))) and b(0) = 0, where A112765(n) is the 5-adic valuation of n.

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A195760 G.f.: A(x) = exp( Sum_{n>=1} 55^A112765(n) x^n/n ), where A112765 is the exponent of the highest power of 5 dividing n.

A289772 a(n) is the numerator of b(n) where b(n) = 1/(3(1+2A112765(n) - b(n-1))) and b(0) = 0, where A112765(n) is the 5-adic valuation of n.

A289773 a(n) is the denominator of b(n) where b(n) = 1/(3(1+2A112765(n) - b(n-1))) and b(0) = 0, with A112765(n) being the 5-adic valuation of n.