A132272 -id:A132272 - cp's OEIS frontend

A329624 Number of iterations of A329623 for starting value n before a repeated value appears, or -1 if this never happens.

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

Offset: 1

Scott R. Shannon, Nov 19 2019

This sequence gives the number of iterations of A329623 for start value n before a repeated value first appears. Unlike the "ghost iteration" of A329200 the only fixed points are the single digits 0 to 9. See A328865 for the first repeating value.
Due to A329623(n) being significantly larger than n for some values of n, the iterative sequence can grow to infinity for some n. The first value to do so is n = 1373. This appears due to the occurrence of the digit string '62637' at the end of the term of the third iteration. These five digits reappear every second iteration at the end of the term, but with more and more digits preceding it. A329917 lists other divergent n values.
The smallest value, for n >= 10, which acts as an end point before repeating is 9, which is the final value for many starting values.
The digit string '8091' forms the basis of a very long convergent series for some values of n. The digit string consisting of an arbitrary number of copies of '80' followed by '91' will eventually converge to 8091, then 891, then 91, which finally converges in ten more iterations. We can thus form a number of arbitrary length using this rule which is guaranteed to converge. This sequence appears naturally with the starting value n = 139100 which converges to 9 after 136 iterations after reaching a term with 72 digits after 20 iterations. See the linked file below.
From M. F. Hasler, Dec 03 2019: (Start)
It seems the a(n) = -1 are conjectural, i.e., we have no proof that the terms for which the trajectory seems to "explode" do not eventually end up in a cycle. For example, the 8th iterate of 1373 is 5218725017016262626273. If the 2nd digit is changed from 2 to 0, then the further iterates appear to explode up to a length of 157 bits, but finally end up in a 2-cycle of 41-digit numbers (26...26273, 62...62637).
The "repeating values" are members of cycles, listed in A328142. Only fixed points 1, ..., 9 and 4*(10^k-1)/9 + 11, k >=3, and 6 infinite families of 2-cycles are known.
(End)
The first escape value is a(1373) = -1 (without proof). - Georg Fischer, Jul 16 2020

			a(1) = 1 as A329623(1) = 1, so a repeating value occurs after 1 iteration.
a(10) = 2 as A329623(10) = 9 and A329623(9) = 9, so a repeating value occurs after 2 iterations.
a(128) = 3, as A329623(128) = 182, A329623(182) = 728, A329623(728) = 182, so a repeating value occurs after 3 iterations.

Scott R. Shannon, Table of n, a(n) for n = 1..10000
Scott R. Shannon, The iterative series for start value n = 139100.

Cf. A329623, A329197, A329200, A328865, A329917.

Sequences A324160, A226233, A179051, A140438, A132272 are unrelated; they begin with the same numbers as this sequence but differ after a(110) = 10, which ends the pattern of incrementing numbers, 2 through 11, repeated ten times.

A329624(n,L=n^10,U=[n])=-!for(i=1,oo,setsearch(U,n=A329623(n))&&return(i); nM. F. Hasler, Dec 02 2019

A132028 Product{0<=k<=floor(log_4(n)), floor(n/4^k)}, n>=1.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 16, 18, 20, 22, 36, 39, 42, 45, 64, 68, 72, 76, 100, 105, 110, 115, 144, 150, 156, 162, 196, 203, 210, 217, 512, 528, 544, 560, 648, 666, 684, 702, 800, 820, 840, 860, 968, 990, 1012, 1034, 1728, 1764, 1800, 1836, 2028, 2067, 2106, 2145, 2352

Offset: 1

Hieronymus Fischer, Aug 20 2007

nonn

If n is written in base-4 as n=d(m)d(m-1)d(m-2)...d(2)d(1)d(0) (where d(k) is the digit at position k) then a(n) is also the product d(m)d(m-1)d(m-2)...d(2)d(1)d(0)*d(m)d(m-1)d(m-2)...d(2)d(1)*d(m)d(m-1)d(m-2)...d(2)*...*d(m)d(m-1)d(m-2)*d(m)d(m-1)*d(m).

			a(26)=floor(26/4^0)*floor(26/4^1)*floor(26/4^2)=26*6*1=156; a(34)=544 since 34=202(base-4) and so
a(34)=202*20*2(base-4)=34*8*2=544.

Cf. A048651, A132020, A100221, A000217.
For formulas regarding a general parameter p (i.e. terms floor(n/p^k)) see A132264.
For the product of terms floor(n/p^k) for p=2 to p=12 see A098844(p=2), A132027(p=3)-A132033(p=9), A067080(p=10), A132263(p=11), A132264(p=12).
For the products of terms 1+floor(n/p^k) see A132269-A132272, A132327, A132328.

Recurrence: a(n)=n*a(floor(n/4)); a(n*4^m)=n^m*4^(m(m+1)/2)*a(n).

a(k*4^m)=k^(m+1)*4^(m(m+1)/2), for 0

Asymptotic behavior: a(n)=O(n^((1+log_4(n))/2)); this follows from the inequalities below.

a(n)<=b(n), where b(n)=n^(1+floor(log_4(n)))/4^((1+floor(log_4(n)))*floor(log_4(n))/2); equality holds for n=k*4^m, 0=0. b(n) can also be written n^(1+floor(log_4(n)))/4^A000217(floor(log_4(n))).

Also: a(n)<=2^(1/4)*n^((1+log_4(n))/2)=1.189207...*4^A000217(log_4(n)), equality holds for n=2*4^m, m>=0.

a(n)>c*b(n), where c=0.4194224417951075977... (see constant A132020).

Also: a(n)>c*2^(1/4)*n^((1+log_4(n))/2)=0.498780...*4^A000217(log_4(n)).

lim inf a(n)/b(n)=0.4194224417951075977..., for n-->oo.

lim sup a(n)/b(n)=1, for n-->oo.

lim inf a(n)/n^((1+log_4(n))/2)=0.4194224417951075977...*2^(1/4), for n-->oo.

lim sup a(n)/n^((1+log_4(n))/2)=2^(1/4), for n-->oo.

lim inf a(n)/a(n+1)=0.4194224417951075977... for n-->oo (see constant A132020).

A132325 Decimal expansion of Product_{k>=0} (1+1/10^k).

Original entry on oeis.org

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

Offset: 1

Hieronymus Fischer, Aug 20 2007

Twice the constant A132326.

			2.22446913827410126425215613418881160749501...

G. C. Greubel, Table of n, a(n) for n = 1..1200
Richard J. McIntosh, Some Asymptotic Formulae for q-Hypergeometric Series, Journal of the London Mathematical Society, Vol. 51, No. 1 (1995), pp. 120-136; alternative link.

Cf. A081845, A067080, A100220, A132019-A132026, A132034-A132038, A132323-A132326, A132271, A132272, A000593.

digits = 105; NProduct[1+1/10^k, {k, 1, Infinity}, NProductFactors -> 100, WorkingPrecision -> digits+3] // N[#, digits+3]& // RealDigits[#, 10, digits]& // First (* Jean-François Alcover, Feb 18 2014 *)
2*N[QPochhammer[-1/10,1/10]] (* G. C. Greubel, Dec 02 2015 *)

prodinf(x=0, 1+(1/10)^x) \\ Altug Alkan, Dec 03 2015

Equals lim sup_{n->oo} Product_{0<=k<=floor(log_10(n))} (1+1/floor(n/10^k)).
Equals lim sup_{n->oo} A132271(n)/n^((1+log_10(n))/2).
Equals lim sup_{n->oo} A132272(n)/n^((log_10(n)-1)/2).
Equals 2*exp(Sum_{n>0} 10^(-n)*Sum_{k|n} -(-1)^k/k) = 2*exp(Sum_{n>0} A000593(n)/(n*10^n)).
Equals lim sup_{n->oo} A132271(n+1)/A132271(n).
Equals 2*(-1/10; 1/10){infinity}, where (a;q){infinity} is the q-Pochhammer symbol. - G. C. Greubel, Dec 02 2015
Equals sqrt(2) * exp(log(10)/24 + Pi^2/(12*log(10))) * Product_{k>=1} (1 - exp(-2*(2*k-1)*Pi^2/log(10))) (McIntosh, 1995). - Amiram Eldar, May 20 2023

A132029 Product{0<=k<=floor(log_5(n)), floor(n/5^k)}, n>=1.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 20, 22, 24, 26, 28, 45, 48, 51, 54, 57, 80, 84, 88, 92, 96, 125, 130, 135, 140, 145, 180, 186, 192, 198, 204, 245, 252, 259, 266, 273, 320, 328, 336, 344, 352, 405, 414, 423, 432, 441, 1000, 1020, 1040, 1060, 1080, 1210, 1232, 1254, 1276

Offset: 1

Hieronymus Fischer, Aug 20 2007

nonn

If n is written in base-5 as n=d(m)d(m-1)d(m-2)...d(2)d(1)d(0) (where d(k) is the digit at position k) then a(n) is also the product d(m)d(m-1)d(m-2)...d(2)d(1)d(0)*d(m)d(m-1)d(m-2)...d(2)d(1)*d(m)d(m-1)d(m-2)...d(2)*...*d(m)d(m-1)d(m-2)*d(m)d(m-1)*d(m).

			a(26)=floor(26/5^0)*floor(26/5^1)*floor(26/5^2)=26*5*1=130; a(34)=204 since 34=114(base-5) and so a(34)=114*11*1(base-5)=34*6*1=204.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A048651, A132021, A100222, A000217.
For formulas regarding a general parameter p (i.e. terms floor(n/p^k)) see A132264.
For the product of terms floor(n/p^k) for p=2 to p=12 see A098844(p=2), A132027(p=3)-A132033(p=9), A067080(p=10), A132263(p=11), A132264(p=12).
For the products of terms 1+floor(n/p^k) see A132269-A132272, A132327, A132328.

Table[Product[Floor[n/5^k],{k,0,Floor[Log[5,n]]}],{n,60}] (* Harvey P. Dale, Oct 16 2019 *)

Recurrence: a(n)=n*a(floor(n/5)); a(n*5^m)=n^m*5^(m(m+1)/2)*a(n).

a(k*5^m)=k^(m+1)*5^(m(m+1)/2), for 0

Asymptotic behavior: a(n)=O(n^((1+log_5(n))/2)); this follows from the inequalities below.

a(n)<=b(n), where b(n)=n^(1+floor(log_5(n)))/5^((1+floor(log_5(n)))*floor(log_5(n))/2); equality holds for n=k*5^m, 0=0. b(n) can also be written n^(1+floor(log_5(n)))/5^A000217(floor(log_5(n))).

Also: a(n)<=2^((1-log_5(2))/2)*n^((1+log_5(n))/2)=1.2181246...*5^A000217(log_5(n)), equality holds for n=2*5^m, m>=0.

a(n)>c*b(n), where c=0.438796837203638531... (see constant A132021).

Also: a(n)>c*(sqrt(2)/2^log_5(sqrt(2)))*n^((1+log_5(n))/2)=0.534509224...*5^A000217(log_5(n)).

lim inf a(n)/b(n)=0.438796837203638531..., for n-->oo.

lim sup a(n)/b(n)=1, for n-->oo.

lim inf a(n)/n^((1+log_5(n))/2)=0.438796837203638531...*sqrt(2)/2^log_5(sqrt(2)), for n-->oo.

lim sup a(n)/n^((1+log_5(n))/2)=sqrt(2)/2^log_5(sqrt(2))=1.2181246..., for n-->oo.

lim inf a(n)/a(n+1)=0.438796837203638531... for n-->oo (see constant A132021).

A132030 a(n) = Product_{k=0..floor(log_6(n))} floor(n/6^k), n>=1.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 24, 26, 28, 30, 32, 34, 54, 57, 60, 63, 66, 69, 96, 100, 104, 108, 112, 116, 150, 155, 160, 165, 170, 175, 216, 222, 228, 234, 240, 246, 294, 301, 308, 315, 322, 329, 384, 392, 400, 408, 416, 424, 486, 495, 504, 513, 522, 531, 600

Offset: 1

Hieronymus Fischer, Aug 20 2007

nonn

If n is written in base 6 as n = d(m)d(m-1)d(m-2)...d(2)d(1)d(0) (where d(k) is the digit at position k) then a(n) is also the product d(m)d(m-1)d(m-2)...d(2)d(1)d(0)*d(m)d(m-1)d(m-2)...d(2)d(1)*d(m)d(m-1)d(m-2)...d(2)*...*d(m)d(m-1)d(m-2)*d(m)d(m-1)*d(m).

			a(52) = floor(52/6^0)*floor(52/6^1)*floor(52/6^2) = 52*8*1 = 416;
a(58) = 522 since 58 = 134_6 and so a(58) = 134_6 * 13_6 * 1_6 = 58*9*1 = 522.

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A048651, A132022, A132034, A000217.
For formulas regarding a general parameter p (i.e., terms floor(n/p^k)) see A132264.
For the product of terms floor(n/p^k) for p=2 to p=12 see A098844(p=2), A132027(p=3)-A132033(p=9), A067080(p=10), A132263(p=11), A132264(p=12).
For the products of terms 1+floor(n/p^k) see A132269-A132272, A132327, A132328.

f:= proc(n) option remember; n*procname(floor(n/6)) end proc:
f(0):= 1:
seq(f(i),i=1..100); # Robert Israel, Dec 20 2015

Table[Product[Floor[n/6^k], {k, 0, Floor[Log[6, n]]}], {n, 1, 100}] (* G. C. Greubel, Dec 20 2015 *)

Recurrence: a(n)=n*a(floor(n/6)); a(n*6^m)=n^m*6^(m(m+1)/2)*a(n).

a(k*6^m) = k^(m+1)*6^(m(m+1)/2), for 0

Asymptotic behavior: a(n) = O(n^((1+log_6(n))/2)); this follows from the inequalities below.

a(n) <= b(n), where b(n) = n^(1+floor(log_6(n)))/6^((1+floor(log_6(n)))*floor(log_6(n))/2); equality holds for n=k*6^m, 0=0. b(n) can also be written n^(1+floor(log_6(n)))/6^A000217(floor(log_6(n))).

Also: a(n) <= 2^((1-log_6(2))/2)*n^((1+log_6(n))/2) = 1.236766885...*6^A000217(log_6(n)), equality holds for n=2*6^m and for n=3*6^m, m>=0 (consider 2^((1-log_6(2))/2)=3^((1-log_6(3))/2) since 6=2*3).

a(n) > c*b(n), where c = 0.45071262522603913... (see constant A132022).

Also: a(n) > c*(sqrt(2)/2^log_6(sqrt(2)))*n^((1+log_6(n))/2) = 0.557426449...*6^A000217(log_6(n)).

lim inf a(n)/b(n) = 0.45071262522603913..., for n-->oo.

lim sup a(n)/b(n) = 1, for n-->oo.

lim inf a(n)/n^((1+log_6(n))/2) = 0.45071262522603913...*sqrt(2)/2^log_6(sqrt(2)), for n-->oo.

lim sup a(n)/n^((1+log_6(n))/2) = sqrt(3)/3^log_6(sqrt(3))=1.236766885..., for n-->oo.

lim inf a(n)/a(n+1) = 0.45071262522603913... for n-->oo (see constant A132022).

G.f. g(x) satisfies g(x) = (x+2x^2+3x^3+4x^4+5x^5)*(1 + g(x^6)) + 6*(x^6+x^7+x^8+x^9+x^10+x^11)*g'(x^6). - Robert Israel, Dec 20 2015

A132032 Product{0<=k<=floor(log_8(n)), floor(n/8^k)}, n>=1.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 32, 34, 36, 38, 40, 42, 44, 46, 72, 75, 78, 81, 84, 87, 90, 93, 128, 132, 136, 140, 144, 148, 152, 156, 200, 205, 210, 215, 220, 225, 230, 235, 288, 294, 300, 306, 312, 318, 324, 330, 392, 399, 406, 413, 420, 427, 434

Offset: 1

Hieronymus Fischer, Aug 20 2007

nonn

If n is written in base-8 as n=d(m)d(m-1)d(m-2)...d(2)d(1)d(0) (where d(k) is the digit at position k) then a(n) is also the product d(m)d(m-1)d(m-2)...d(2)d(1)d(0)*d(m)d(m-1)d(m-2)...d(2)d(1)*d(m)d(m-1)d(m-2)...d(2)*...*d(m)d(m-1)d(m-2)*d(m)d(m-1)*d(m).

			a(70)=floor(70/8^0)*floor(70/8^1)*floor(70/8^2)=70*8*1=560;
For n=75, 75=113(base-8) and so a(75)=113*11*1(base-8)=75*9*1=675.

Cf. A048651, A132024, A132036, A000217.
For formulas regarding a general parameter p (i.e. terms floor(n/p^k)) see A132264.
For the product of terms floor(n/p^k) for p=2 to p=12 see A098844(p=2), A132027(p=3)-A132033(p=9), A067080(p=10), A132263(p=11), A132264(p=12).
For the products of terms 1+floor(n/p^k) see A132269-A132272, A132327, A132328.

Recurrence: a(n)=n*a(floor(n/8)); a(n*8^m)=n^m*8^(m(m+1)/2)*a(n).

a(k*8^m)=k^(m+1)*8^(m(m+1)/2), for 0

Asymptotic behavior: a(n)=O(n^((1+log_8(n))/2)); this follows from the inequalities below.

a(n)<=b(n), where b(n)=n^(1+floor(log_8(n)))/8^((1+floor(log_8(n)))*floor(log_8(n))/2); equality holds for n=k*8^m, 0=0. b(n) can also be written n^(1+floor(log_8(n)))/8^A000217(floor(log_8(n))).

Also: a(n)<=3^((1-log_8(3))/2)*n^((1+log_8(n))/2) = 1.295758534...*8^A000217(log_8(n)), equality holds for n=3*8^m, m>=0.

a(n)>c*b(n), where c = 0.46456888368647639098... (see constant A132024).

Also: a(n)>c*2^(1/3)*n^((1+log_8(n))/2)=0.4645688836...*1.25992105...*8^A000217(log_8(n)).

lim inf a(n)/b(n)=0.46456888368647639098..., for n-->oo.

lim sup a(n)/b(n)=1, for n-->oo.

lim inf a(n)/n^((1+log_8(n))/2)=0.46456888368647639098...*2^(1/3), for n-->oo.

lim sup a(n)/n^((1+log_8(n))/2)=sqrt(3)/3^log_8(sqrt(3))=1.295758534..., for n-->oo.

lim inf a(n)/a(n+1)=0.46456888368647639098... for n-->oo (see constant A132024).

A226233 Ten copies of each positive integer.

Original entry on oeis.org

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

Offset: 1

Sam Vaseghi, Jun 01 2013

Class of well and totally ordered sequences of (p-1)-tuples of natural numbers for p = 11.
Given a prime p the class of sequences a(n,p) can be constructed. The above example is for p=11. The class of well and totally ordered sequences of (prime-1)-tuples of natural numbers contains all sequences a(n) according to FORMULA for primes p. The class is crucial and will be applied to define other sequences, that will be submitted to OEIS as well a posterior.
a(n) = A132272(n-1) for n<=200, but the two sequences start to differ then. - R. J. Mathar, Jun 13 2025

S. Vaseghi (alias al-Hwarizmi), Combination of positive integers in terms of primes (sophisticated version 2)
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,0,0,0,1,-1).

Cf. A000027, A004526, A002265.

Cf. A059995 (10 copies of nonnegative integers).

A226233 := proc(n)
    option remember ;
    if n <= 10 then
        1;
    elif n <=20 then
        2;
    else
        procname(n-1)+procname(n-10)-procname(n-11) ;
    end if;
end proc:
seq(A226233(n),n=1..120) ; # R. J. Mathar, Jun 13 2025

p=11; k = (p - 1); alpha = (k + n - 1 - (Mod[(n - 1), k]))/k; Table[alpha, {n, 100}]
Table[PadRight[{},10,n],{n,10}]//Flatten (* Harvey P. Dale, May 24 2021 *)

a(n)=(n+9)\10 \\ Charles R Greathouse IV, Jun 05 2013

a(n,p) = ((p-1) + n - (1 + ((n-1) mod (p-1))))/(p-1); p is a prime and n positive integer; for this sequence p = 11.

G.f.: x / ( (1+x)*(x^4-x^3+x^2-x+1)*(x^4+x^3+x^2+x+1)*(x-1)^2 ). - R. J. Mathar, Jun 13 2025

A132031 Product{0<=k<=floor(log_7(n)), floor(n/7^k)}, n>=1.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 28, 30, 32, 34, 36, 38, 40, 63, 66, 69, 72, 75, 78, 81, 112, 116, 120, 124, 128, 132, 136, 175, 180, 185, 190, 195, 200, 205, 252, 258, 264, 270, 276, 282, 288, 343, 350, 357, 364, 371, 378, 385, 448, 456, 464, 472, 480, 488

Offset: 1

Hieronymus Fischer, Aug 20 2007

nonn

If n is written in base-7 as n=d(m)d(m-1)d(m-2)...d(2)d(1)d(0) (where d(k) is the digit at position k) then a(n) is also the product d(m)d(m-1)d(m-2)...d(2)d(1)d(0)*d(m)d(m-1)d(m-2)...d(2)d(1)*d(m)d(m-1)d(m-2)...d(2)*...*d(m)d(m-1)d(m-2)*d(m)d(m-1)*d(m).

			a(52)=floor(52/7^0)*floor(52/7^1)*floor(52/7^2)=52*7*1=364.
a(58)=464 since 58=112(base-7) and so a(58)=112*11*1(base-7)=58*8*1=464.

Cf. A048651, A132023, A132035, A000217.
For formulas regarding a general parameter p (i.e. terms floor(n/p^k)) see A132264.
For the product of terms floor(n/p^k) for p=2 to p=12 see A098844(p=2), A132027(p=3)-A132033(p=9), A067080(p=10), A132263(p=11), A132264(p=12).
For the products of terms 1+floor(n/p^k) see A132269-A132272, A132327, A132328.

Table[Times@@Floor[n/7^Range[0,Floor[Log[7,n]]]],{n,70}] (* Harvey P. Dale, Oct 11 2017 *)

Recurrence: a(n)=n*a(floor(n/7)); a(n*7^m)=n^m*7^(m(m+1)/2)*a(n).

a(k*7^m)=k^(m+1)*7^(m(m+1)/2), for 0

Asymptotic behavior: a(n)=O(n^((1+log_7(n))/2)this follows from the inequalities below.

a(n)<=b(n), where b(n)=n^(1+floor(log_7(n)))/7^((1+floor(log_7(n)))*floor(log_7(n))/2); equality holds for n=k*7^m, 0=0. b(n) can also be written n^(1+floor(log_7(n)))/7^A000217(floor(log_7(n))).

Also: a(n)<=3^((1-log_7(3))/2)*n^((1+log_7(n))/2)=1.270209197...*7^A000217(log_7(n)), equality holds for n=3*7^m, m>=0.

a(n)>c*b(n), where c=0.4587667266997689850200... (see constant A132023).

Also: a(n)>c*(sqrt(2)/2^log_7(sqrt(2)))*n^((1+log_7(n))/2)=0.4587667266...*1.249972544...*7^A000217(log_7(n)).

lim inf a(n)/b(n)=0.4587667266997689850200..., for n-->oo.

lim sup a(n)/b(n)=1, for n-->oo.

lim inf a(n)/n^((1+log_7(n))/2)=0.4587667266997689850200...*sqrt(2)/2^log_7(sqrt(2)), for n-->oo.

lim sup a(n)/n^((1+log_7(n))/2)=sqrt(3)/3^log_7(sqrt(3))=1.270209197..., for n-->oo.

lim inf a(n)/a(n+1)=0.4587667266997689850200... for n-->oo (see constant A132023).

Previous Showing 11-19 of 19 results.

cp's OEIS Frontend

A329624 Number of iterations of A329623 for starting value n before a repeated value appears, or -1 if this never happens.

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PARI

A132271 Product{k>=0, 1+floor(n/10^k)}.

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A132028 Product{0<=k<=floor(log_4(n)), floor(n/4^k)}, n>=1.

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A132325 Decimal expansion of Product_{k>=0} (1+1/10^k).

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A132029 Product{0<=k<=floor(log_5(n)), floor(n/5^k)}, n>=1.

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A132030 a(n) = Product_{k=0..floor(log_6(n))} floor(n/6^k), n>=1.

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Maple

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A132032 Product{0<=k<=floor(log_8(n)), floor(n/8^k)}, n>=1.

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A226233 Ten copies of each positive integer.

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