A002183 -id:A002183 - cp's OEIS frontend

A070319 a(n) = Max_{k=1..n} tau(k) where tau(x)=A000005(x) is the number of divisors of x.

1, 2, 2, 3, 3, 4, 4, 4, 4, 4, 4, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12

Offset: 1

Benoit Cloitre, May 11 2002

Is this the same as A068509? - David Scambler, Sep 10 2012
They are different even asymptotically: A068509(n)=O(sqrt(n)), while a(n) does not have polynomial growth. One example where the sequences differ: a(625) = 24 < A068509(625). (The inequality is implied by the set {1,2,..,25} where each pair of the elements has lcm <= 625.) - Max Alekseyev, Sep 11 2012
The two sequences first differ when n = 336, due to the set of 21 elements {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 16, 18, 21, 24, 30, 36, 42, 48} where each pair of elements has lcm <= 336, while no positive integer <= 336 has more than 20 divisors. Therefore A068509(336) = 21 and A070319(336) = 20. - William Rex Marshall, Sep 11 2012
Indices of records give A002182. - Omar E. Pol, Feb 18 2023

Sándor, J., Crstici, B., Mitrinović, Dragoslav S. Handbook of Number Theory I. Dordrecht: Kluwer Academic, 2006, p. 44.
S. Wigert, Sur l'ordre de grandeur du nombre des diviseurs d'un entier, Arkiv. for Math. 3 (1907), 1-9.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
S. Ramanujan, Highly composite numbers, Proceedings of the London Mathematical Society, 2, XIV, 1915, 347 - 409.

Cf. A000005, A002182, A002183, A261100, A261104.

a070319 n = a070319_list !! (n-1)
a070319_list = scanl1 max $ map a000005 [1..]
-- Reinhard Zumkeller, Apr 01 2011

a = {0}; Do[AppendTo[a, Max[DivisorSigma[0, n], a[[n]]]], {n, 120}]; Rest@ a (* Michael De Vlieger, Sep 29 2015 *)

PARI
```
a(n)=vecmax(vector(n,k,numdiv(k)))
```

v=vector(100);v[1]=1;for(n=2,#v,v[n]=max(v[n-1],numdiv(n))); v \\ Charles R Greathouse IV, Sep 12 2012

A070319(n,m=1,s=2)={for(k=s,n,mM. F. Hasler, Sep 12 2012

{a=0;for(n=1,100,print1(a=A070319(n,a,n),","))} /* Using this pattern, computation of a(1..10^6) is faster than "normal" computation of a(1..3000). */

a(n) = exp(log(2) log(n) / log(log(n)) + O(log(n) log(log(log(n))) / (log(log(n)))^2)). (See Sándor reference for more formulas.) - Eric M. Schmidt, Jun 30 2013

a(n) = A002183(A261100(n)). - Antti Karttunen, Jun 06 2017

A009287 a(1) = 3; thereafter a(n+1) = least k with a(n) divisors.

Original entry on oeis.org

3, 4, 6, 12, 60, 5040, 293318625600, 670059168204585168371476438927421112933837297640990904154667968000000000000

Offset: 1

David W. Wilson and James Kilfiger (jamesk(AT)maths.warwick.ac.uk)

nonn

The sequence must start with 3, since a(1)=1 or a(1)=2 would lead to a constant sequence. - M. F. Hasler, Sep 02 2008
The calculation of a(7) and a(8) is based upon the method in A037019 (which, apparently, is the method previously used by the authors of A009287). So a(7) and a(8) are correct unless n=a(6)=5040 or n=a(7)=293318625600 are "exceptional" as described in A037019. - Rick L. Shepherd, Aug 17 2006
a(7) is correct because 5040 is not exceptional (see A072066). - T. D. Noe, Sep 02 2008
Terms from a(2) to a(7) are highly composite (that is, found in A002182), but a(8) is not. - Ivan Neretin, Mar 28 2015 [Equivalently, the first 6 terms are in A002183, but a(7) is not. Note that the smallest number with at least a(7) divisors is A002182(695) ~ 1.77 * 10^59 with 293534171136 divisors, which is much smaller than a(8) ~ 6.70 * 10^75. - Jianing Song, Jul 15 2021]
Grime reported that Ramanujan unfortunately missed a(7) with 5040 divisors. - Frank Ellermann, Mar 12 2020
It is possible to prepend 2 to this sequence as follows. a(0) = 2; for n > 0, a(n) = the smallest natural number greater than a(n-1) with a(n-1) divisors. - Hal M. Switkay, Jul 03 2022

			5040 is the smallest number with 60 divisors.

Amarnath Murthy, Pouring a few more drops in the ocean of Smarandache Sequences and Conjectures (to be published in the Smarandache Notions Journal) [Note: this author submitted two erroneous versions of this sequence to the OEIS, A036460 and A061080, entries which contained invalid conjectures.]

Jason Earls, A note on the Smarandache divisors of divisors sequence and two similar sequences, in Smarandache Notions Journal (2004), Vol. 14.1, page 274.
James Grime and Brady Haran, Infinite Anti-Primes, Numberphile video (2016).

Coincides with A251483 for 1 <= n <= 7 (only).

Cf. A000005, A005179, A037019, A251483.

f[n_] := Block[{k = 3, s = (Times @@ (Prime[Range[Length@ #]]^Reverse[# - 1])) & @ Flatten[FactorInteger[#] /. {a_Integer, b_} :> Table[a, {b}]] & /@ Range@ 10000}, Reap@ Do[Sow[k = s[[k]]], {n}] // Flatten // Rest]; f@ 6 (* Michael De Vlieger, Mar 28 2015, after Wouter Meeussen at A037019 *)

a(n) = A005179(a(n-1)).

Entry revised by N. J. A. Sloane, Aug 25 2006

A112781 Number of highly composite numbers (definition 1, A002182) < 10^n.

Original entry on oeis.org

4, 9, 15, 20, 29, 38, 47, 56, 66, 76, 86, 95, 106, 117, 125, 135, 146, 156, 167, 177, 186, 196, 209, 219, 231, 241, 254, 267, 280, 292, 305, 316, 330, 343, 356, 368, 381, 396, 409, 423, 436, 450, 463, 476, 491, 503, 517, 530, 547, 561, 577, 593, 608, 625, 640

Offset: 1

Ray Chandler, Nov 11 2005

nonn

			a(1) = 4 since there are four highly composite numbers < 10^1 {1,2,4,6}.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Highly Composite Number
A. Flammenkamp, First 1200 highly composite numbers

Cf. A002182, A002183, A108602, A112778, A112779, A112780.

Partial sums of A112780. - Lekraj Beedassy, Sep 02 2006

A181808 Numbers that set a record for number of even divisors: a(n) = 2*A002182(n).

Original entry on oeis.org

2, 4, 8, 12, 24, 48, 72, 96, 120, 240, 360, 480, 720, 1440, 1680, 2520, 3360, 5040, 10080, 15120, 20160, 30240, 40320, 50400, 55440, 90720, 100800, 110880, 166320, 221760, 332640, 443520, 554400, 665280, 997920, 1108800, 1330560, 1441440, 2162160, 2882880, 4324320

Offset: 1

Matthew Vandermast, Nov 27 2010

nonn

In other words, a positive integer n appears in the sequence iff more even numbers divide n than divide any positive integer smaller than n.

For all positive integer values (j,k) such that jk = n, the number of divisors of n that are multiples of j equals A000005(k). Therefore, n sets a record for the number of its divisors that are multiples of j iff k=n/j is highly composite (A002182). Cf. A181803, A181809, A181810.

			a(4)=12 has exactly four even divisors (2, 4, 6 and 12).  (Note that these are precisely the numbers that are twice a divisor of A002182(4)=6; see row 6 of A027750.)  No positive integer smaller than 12 has as many as four even divisors; hence, 12 is a member of the sequence.

Eric Weisstein's World of Mathematics, Highly composite number

Numbers n such that 2 appears in row n of A181803. See also A181809, A181810.
A002183(n) gives number of even divisors of a(n).
A053624 gives numbers that set records for number of odd divisors. No number sets records both for its number of odd divisors and its number of even divisors.

a(n)=2*A002182(n).

A263077 a(n) = greatest k where A155043(k) < A155043(n).

Original entry on oeis.org

0, 0, 2, 2, 6, 2, 12, 6, 6, 6, 12, 6, 18, 12, 18, 18, 22, 12, 30, 18, 30, 18, 34, 22, 22, 22, 42, 22, 48, 22, 60, 30, 60, 30, 72, 48, 84, 34, 84, 34, 96, 34, 108, 42, 96, 42, 108, 42, 48, 48, 120, 48, 132, 48, 132, 48, 140, 60, 140, 48, 140, 72, 140, 140, 140, 72, 140, 84, 140, 84, 140, 60, 140, 96, 140, 96, 150, 96, 156, 96, 108, 108, 120, 72, 120, 120, 132, 108, 140, 108, 140, 132, 140, 120, 140, 84

Offset: 1

Antti Karttunen, Oct 09 2015

nonn

Antti Karttunen, Table of n, a(n) for n = 1..124340

Cf. A155043, A261089, A262503, A263078, A263079, A263080, A263082.

a[0] = 0; a[n_] := a[n] = 1 + a[n - DivisorSigma[0, n]]; Table[k = 3 n;
While[a@ k >= a@ n, k--]; k, {n, 96}] (* Michael De Vlieger, Oct 13 2015 *)

allocatemem((2^31)+(2^30));
uplim1 = 36756720 + 640; \\ = A002182(53) + A002183(53).
uplim2 = 36756720; \\ = A002182(53).
uplim3 = 32432400; \\ = A002182(52). Really just some Ad Hoc value smaller than above.
v155043 = vector(uplim1);
vother = vector(uplim3); \\ Contains A262503 and A263082 in succession.
v155043[1] = 1; v155043[2] = 1;
for(i=3, uplim1, v155043[i] = 1 + v155043[i-numdiv(i)]; if(!(i%1048576),print1(i,", ")));
A155043 = n -> if(!n,n,v155043[n]);
maxlen = 0; for(i=1, uplim2, len = v155043[i]; vother[len] = i; maxlen = max(maxlen,len); if(!(i%1048576),print1(i,", "))); \\ First it will be A262503.
print("uplim2=", uplim2, " uplim3=", uplim3, " maxlen=", maxlen);
\\ Then we convert it to A263082:
m = 0; for(i=1, maxlen, m = max(m, vother[i]); vother[i] = m; if(!(i%1048576),print1(i,", ")));
A263082 = n -> if(!n,n,vother[n]);
A263077 = n -> A263082(A155043(n)-1);
\\ Finally we can compute A263077:
for(i=1, uplim3, write("b263077.txt", i, " ", A263077(i)); );

a(n) = A263082(A155043(n)-1).

A263087 a(n) = A060990(n^2); number of solutions to x - d(x) = n^2, where d(x) is the number of divisors of x (A000005).

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, Oct 12 2015

nonn

Antti Karttunen, Table of n, a(n) for n = 0..10082

Cf. A000005, A000290, A002182, A002183, A060990.
Cf. A263093 (positions of zeros), A263092 (nonzeros).
Cf. A263250, A263251 (bisections) and A263252, A263253 (their partial sums).
Cf. also A261088, A263088.

A060990(n) = { my(k = n + 2400, s=0); while(k > n, if(((k-numdiv(k)) == n),s++); k--;); s}; \\ Hard limit A002183(77)=2400 good for at least up to A002182(77) = 10475665200.
A263087(n) = A060990(n^2);
for(n=0, 10082, write("b263087.txt", n, " ", A263087(n)));

(define (A263087 n) (A060990 (A000290 n)))

a(n) = A060990(n^2) = A060990(A000290(n)).

A112780 Number of highly composite numbers (definition 1, A002182) with n decimal digits.

Original entry on oeis.org

4, 5, 6, 5, 9, 9, 9, 9, 10, 10, 10, 9, 11, 11, 8, 10, 11, 10, 11, 10, 9, 10, 13, 10, 12, 10, 13, 13, 13, 12, 13, 11, 14, 13, 13, 12, 13, 15, 13, 14, 13, 14, 13, 13, 15, 12, 14, 13, 17, 14, 16, 16, 15, 17, 15, 19, 15, 18, 15, 16, 17, 16, 17, 16, 15, 19, 15, 19, 14, 18, 14, 19, 17

Offset: 1

Ray Chandler, Nov 11 2005

			a(1) = 4 since there are four highly composite numbers with one decimal digit {1,2,4,6}.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Highly Composite Number
A. Flammenkamp, First 1200 highly composite numbers

Cf. A002182, A002183, A108602, A112778, A112779, A112781.

First differences of A112781. - Amiram Eldar, Jul 02 2019

A181809 Numbers n such that both n and n/2 are highly composite (A002182).

Original entry on oeis.org

2, 4, 12, 24, 48, 120, 240, 360, 720, 1680, 2520, 5040, 10080, 15120, 20160, 50400, 55440, 110880, 166320, 221760, 332640, 554400, 665280, 1441440, 2162160, 2882880, 4324320, 7207200, 8648640, 14414400, 17297280, 21621600, 43243200, 73513440

Offset: 1

Matthew Vandermast, Nov 27 2010

nonn

These are the numbers that set records both for total number of divisors and for number of even divisors; intersection of A002182 and A181808.

For all positive integer values (j,k) such that jk = n, the number of divisors of n that are multiples of j equals A000005(k). Therefore, n sets a record for the number of its divisors that are multiples of j iff k=n/j is highly composite (A002182).

			The number 12 is both highly composite (A002182(5) = 12) and twice another highly composite number (12 = 2*6 = 2*A002182(4)).  It therefore has more divisors (A002183(5)=6) than any smaller positive integer, and more even divisors (A002183(4)=4) than any smaller positive integer. Since 12 is the third positive integer with the properties that define this sequence, a(3)=12.

Amiram Eldar, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Highly composite number

Numbers n such that 1 and 2 both appear in row n of A181803. See also A181808, A181810.
A053624 gives numbers that set records for number of odd divisors. No number sets records both for its number of odd divisors and its number of even divisors. Only the number 1 sets a record for its number of odd divisors and its number of total divisors.
Subsequence of A025487.

A263091 Primes p for which A049820(x) = p has no solution.

Original entry on oeis.org

7, 13, 19, 37, 43, 67, 79, 103, 109, 113, 131, 163, 167, 193, 229, 241, 251, 257, 271, 293, 307, 313, 353, 359, 379, 383, 397, 401, 439, 463, 479, 487, 491, 499, 503, 509, 563, 571, 647, 653, 661, 673, 701, 739, 743, 757, 761, 773, 823, 859, 863, 883, 887, 911, 937, 941, 953, 967, 971, 977, 983, 1009, 1093, 1103, 1109, 1171, 1181, 1193, 1217, 1279, 1283, 1291, 1297, 1307, 1321, 1361

Offset: 1

Antti Karttunen, Oct 11 2015

nonn

Primes p that there is no such k for which k - d(k) = p, where d(k) is the number of divisors of k (A000005).

Antti Karttunen, Table of n, a(n) for n = 1..15383

Complement among primes: A263090.
Intersection of A000040 and A045765.
Subsequence of A067774 (A049591).
Cf. A000005, A049820, A060990.

lim = 10000; s = Select[Complement[Range@ lim, Sort@ DeleteDuplicates@ Table[n - DivisorSigma[0, n], {n, lim}]], PrimeQ]; Take[s, 76] (* Michael De Vlieger, Oct 13 2015 *)

allocatemem(123456789);
uplim1 = 2162160 + 320; \\ = A002182(41) + A002183(41).
v060990 = vector(uplim1);
for(n=3, uplim1, v060990[n-numdiv(n)]++);
A060990 = n -> if(!n,2,v060990[n]);
n=0; forprime(p=2, 524287, if((0 == A060990(p)), n++; write("b263091.txt", n, " ", p)));

;; With Antti Karttunen's IntSeq-library.
(define A263091 (MATCHING-POS 1 1 (lambda (n) (and (= 1 (A010051 n)) (zero? (A060990 n))))))

A263093 Numbers whose squares are in A045765.

Original entry on oeis.org

5, 6, 7, 8, 10, 14, 16, 18, 20, 22, 26, 27, 28, 34, 35, 37, 46, 47, 50, 54, 56, 58, 59, 60, 62, 67, 73, 78, 82, 85, 89, 90, 94, 95, 98, 100, 103, 104, 106, 110, 114, 116, 118, 122, 124, 125, 126, 127, 128, 130, 135, 140, 141, 142, 148, 150, 155, 158, 161, 164, 170, 172, 174, 177, 178, 182, 184, 188, 190, 199, 202, 205, 207

Offset: 1

Antti Karttunen, Oct 11 2015

nonn

Numbers n such that there is no such k for which k - d(k) = n^2, where d(k) is the number of divisors of k (A000005).

Numbers n for which A060990(n^2) = A263087(n) = 0.

Antti Karttunen, Table of n, a(n) for n = 1..12443
A. Karttunen, Ratio a(n)/A263092(n) drawn with the help of OEIS Plot2-utility

Cf. A000005, A000196, A045765, A049820, A060990, A263098.
Complement: A263092.
Positions of zeros in A263087 and positions of ones in A263088.
Cf. A263095 (the squares of these numbers).

\\ Compute A263093 and A263095 at the same time:
A060990(n) = { my(k = n + 1440, s=0); while(k > n, if(((k-numdiv(k)) == n),s++); k--;); s}; \\ Hard limit 1440 is good for at least up to A002182(67) = 1102701600 as A002183(67) = 1440.
n = 1; k = 0; while((n^2)<1102701600, if((0 == A060990(n*n)), k++; write("b263093.txt", k, " ", n); write("b263095.txt", k, " ", (n*n)); ); n++; if(!(n%8192),print1(n,",k=", k, ", ")); );

;; With Antti Karttunen's IntSeq-library.
(define A263093 (MATCHING-POS 1 1 (lambda (n) (zero? (A060990 (* n n))))))
(define A263093 (ZERO-POS 1 0 A263087))

a(n) = A000196(A263095(n)).

cp's OEIS Frontend

A070319 a(n) = Max_{k=1..n} tau(k) where tau(x)=A000005(x) is the number of divisors of x.

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A009287 a(1) = 3; thereafter a(n+1) = least k with a(n) divisors.

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A112781 Number of highly composite numbers (definition 1, A002182) < 10^n.

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A181808 Numbers that set a record for number of even divisors: a(n) = 2*A002182(n).

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A263077 a(n) = greatest k where A155043(k) < A155043(n).

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A263087 a(n) = A060990(n^2); number of solutions to x - d(x) = n^2, where d(x) is the number of divisors of x (A000005).

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A112780 Number of highly composite numbers (definition 1, A002182) with n decimal digits.

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A181809 Numbers n such that both n and n/2 are highly composite (A002182).

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