A002183 -id:A002183 - cp's OEIS frontend

A263250 Even bisection of A263087; number of solutions to x - d(x) = 4(n^2), where d(x) is the number of divisors of x (A000005).

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

Offset: 0

Antti Karttunen, Nov 07 2015

nonn

Antti Karttunen, Table of n, a(n) for n = 0..10000

Cf. A000005, A060990, A263087, A263251.

Cf. also A263252 (partial sums).

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);
A263250(n) = A263087(2*n);
p = 0; for(n=0, 10000, k = A263250(n); p += k; write("b263250.txt", n, " ", k); write("b263252.txt", n, " ", p)); \\ Compute A263250 and A263252 at the same time.

Scheme
```
(define (A263250 n) (A263087 (+ n n)))
```

a(n) = A263087(2*n).

A063072 Sum of divisors of Ramanujan's highly composite numbers, or sigma(A002182(n)).

Original entry on oeis.org

1, 3, 7, 12, 28, 60, 91, 124, 168, 360, 546, 744, 1170, 2418, 2880, 4368, 5952, 9360, 19344, 28800, 39312, 59520, 79248, 99944, 112320, 180048, 203112, 232128, 345600, 471744, 714240, 950976, 1199328, 1451520, 2160576, 2437344, 2926080

Offset: 1

Jason Earls, Aug 02 2001

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..50 from Harry J. Smith)

Cf. A000005, A004394, A002182, A002183, A002473, A000203.

s={}; dm=0; Do[d = DivisorSigma[0,n]; If[d > dm, dm=d; AppendTo[s, DivisorSigma[1,n]]], {n, 1, 10^5}]; s (* Amiram Eldar, Jun 28 2019 *)

a=0; j=[]; for(n=1,200000,b=numdiv(n); if(b>a,a=b; j=concat(j, sigma(n)))); j

{ n=a=0; for (m=1, 10^9, b=numdiv(m); if(b>a, a=b; write("b063072.txt", n++, " ", sigma(m)); if (n==50, break)) ) } \\ Harry J. Smith, Aug 16 2009

a(n) = A000203(A002182(n)). - Michel Marcus, Jun 28 2018

More terms from Reiner Martin, Dec 22 2001

A189394 Highly composite numbers whose number of divisors is also highly composite.

Original entry on oeis.org

1, 2, 6, 12, 60, 360, 1260, 2520, 5040, 55440, 277200, 720720, 3603600, 61261200, 2205403200, 293318625600, 6746328388800, 195643523275200

Offset: 1

Krzysztof Ostrowski, Apr 21 2011

Both n and d(n) are highly composite numbers.
It is extremely likely that this sequence is complete. The highly composite numbers have a very special form. The number of divisors of a large HCN has a high power of 2 in its factorization -- which is not the form of an HCN. - T. D. Noe, Apr 21 2011
All but a(7) and a(12) are a multiple of the previous term: ratios a(n+1) / a(n) are (2, 3, 2, 5, 6, 7/2, 2, 2, 11, 5, 13/5, 5, 17, 36, 133, 23, 29, ...?). - M. F. Hasler, Jun 20 2022

			d(60) = 12; both 60 and 12 are highly composite numbers

Achim Flammenkamp, Highly composite numbers
Lars Magnus Øverlier, Highly Composite Numbers, arXiv:2305.14350 [math.NT], 2023.

Cf. A002182, A002183, A141320.

(* First run program at A002182 *) Select[A002182, MemberQ[A002182, DivisorSigma[0, #]] &] (* Alonso del Arte, Apr 21 2011 *)

is_A189394(n)={is_A002182(numdiv(n)) && is_A002182(n)}
M189394=[1,2]/*for memoization*/; A189394(n)={if(#M189394A189394(k++*s),); M189394=concat(M189394,k*s)); M189394[n]} \\ M. F. Hasler, Jun 20 2022

Typo in a(15) corrected by Ben Beer, Jul 20 2016

Keywords fini and full, following Øverlier's thesis, added by Michel Marcus, May 25 2023

A226900 Record values of Hooley's Delta function A226898.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 12, 14, 15, 17, 18, 19, 20, 21, 22, 23, 25, 27, 28, 29, 31, 32, 33, 34, 39, 44, 48, 51, 55, 60, 62, 66, 69, 77, 80, 83, 88, 91, 92, 98, 100, 106, 107, 111, 118, 121, 122, 124, 137, 138, 142, 150, 156

Offset: 1

Charles R Greathouse IV, Jun 21 2013

nonn

C. Hooley, On a new technique and its applications to the theory of numbers, Proc. London Math. Soc. 3 38:1 (1979), pp. 115-151.

Cf. A226898, A226899, A002183.

Delta(n)=my(d=divisors(n), m=1); for(i=1, #d-1, my(t=exp(1)*d[i]); m=max(sum(j=i, #d, d[j]r,r=t;print1(t", ")))

a(38)-a(54) from Charles R Greathouse IV, Jun 24 2013

a(55)-a(56) from Charles R Greathouse IV, Jul 01 2013

A263092 Numbers whose squares are in A236562; numbers n such that there is at least one such k for which k - d(k) = n^2, where d(k) is the number of divisors of k (A000005).

Original entry on oeis.org

0, 1, 2, 3, 4, 9, 11, 12, 13, 15, 17, 19, 21, 23, 24, 25, 29, 30, 31, 32, 33, 36, 38, 39, 40, 41, 42, 43, 44, 45, 48, 49, 51, 52, 53, 55, 57, 61, 63, 64, 65, 66, 68, 69, 70, 71, 72, 74, 75, 76, 77, 79, 80, 81, 83, 84, 86, 87, 88, 91, 92, 93, 96, 97, 99, 101, 102, 105, 107, 108, 109

Offset: 0

Antti Karttunen, Oct 11 2015

nonn

Starting offset is zero, because a(0)=0 is a special case in this sequence.
Numbers n for which A060990(n^2) = A263087(n) > 0.
Numbers n for which A049820(x) = n^2 has a solution.

Antti Karttunen, Table of n, a(n) for n = 0..20763

Complement: A263093.
Cf. A000005, A000196, A049820, A060990, A236562, A263087, A263098.
Cf. A263094 (the squares of these numbers).
Cf. A262515 (a subsequence).

\\ Compute A263092 and A263094 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 good for at least up to A002182(67) = 1102701600 as A002183(67) = 1440.
n = 0; k = 0; while((n^2)<1102701600, if((A060990(n*n) > 0), write("b263092.txt", k, " ", n); write("b263094.txt", k, " ", (n*n)); k++; ); n++; if(!(n%8192),print1(n,",k=", k, ", ")); );

;; With Antti Karttunen's IntSeq-library.
(define A263092 (MATCHING-POS 0 0 (lambda (n) (not (zero? (A060990 (* n n)))))))
(define A263092 (NONZERO-POS 0 0 A263087))

A263251 Odd bisection of A263087; number of solutions to x - d(x) = (2n+1)^2, where d(x) is the number of divisors of x (A000005).

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, Nov 07 2015

nonn

Antti Karttunen, Table of n, a(n) for n = 0..10000

Cf. A000005, A060990, A263087, A263250.

Cf. also A263253 (partial sums).

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);
A263251(n) = A263087((2*n)+1);
p = 0; for(n=0, 10000, k = A263251(n); p += k; write("b263251.txt", n, " ", k); write("b263253.txt", n, " ", p)); \\ Compute A263251 and A263253 at the same time.

(define (A263251 n) (A263087 (+ n n 1)))

a(n) = A263087(2*n + 1).

A188591 Records of A188550.

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 9, 10, 12, 16, 18, 20, 24, 30, 32, 36, 40, 48, 60, 64, 72, 80, 84, 90, 96, 100, 108, 120, 128, 144, 160, 168, 180, 192, 200, 216

Offset: 1

Vladimir Shevelev, Apr 04 2011

Is this the same sequence as A002183, number of divisors of n-th highly composite number?

Cf. A188550, A188579.

max = 10^6; (* b = A188550 *) b[n_] := Max @ Table[Length @ Select[ Table[ n-d, {d, Divisors[n-k] // Rest}], Mod[#, k] == 0&], {k, 2, Floor[ Sqrt[n] ]}]; A188591 = Reap[For[record = 0; k = 1; n = 1, n <= max, n++, bn = b[n]; If[bn > record, record = bn; Print["a(", k++, ") = b(", n, ") = ", bn]; Sow[bn]]]][[2, 1]] (* Jean-François Alcover, Feb 07 2016 *)

More terms from Jean-François Alcover, Feb 07 2016

A194095 Record values in A056595.

Original entry on oeis.org

0, 1, 3, 4, 6, 7, 10, 14, 15, 17, 20, 22, 24, 30, 32, 37, 44, 45, 46, 54, 60, 62, 66, 68, 74, 77, 78, 92, 93, 96, 114, 124, 138, 154, 160, 168, 186, 188, 189, 191, 204, 216, 234, 252, 282, 314, 328, 348, 378, 380, 391, 420, 440, 468, 474, 488, 504, 508, 564

Offset: 1

Reinhard Zumkeller, Aug 15 2011

nonn

a(n) = A056595(A194096(n)) and A056595(m) < a(n) for m < A194096(n).

Amiram Eldar, Table of n, a(n) for n = 1..655

Cf. A002183.

mx = -1; t = {}; Do[s = Length[Select[Divisors[n], ! IntegerQ[Sqrt[#]] &]]; If[s > mx, mx = s; AppendTo[t, {n, mx}]], {n, 1000000}]; Transpose[t][[2]] (* T. D. Noe, Aug 15 2011 *)
DeleteDuplicates[Table[Count[Divisors[n],?(#!=Floor[Sqrt[#]]^2&)],{n,10^6}],GreaterEqual] (* _Harvey P. Dale, Mar 16 2023 *)

A210523 Record values of Dedekind psi function.

Original entry on oeis.org

1, 3, 4, 6, 12, 18, 24, 36, 48, 72, 96, 108, 144, 168, 192, 216, 240, 288, 360, 384, 432, 576, 648, 672, 720, 864, 1008, 1152, 1296, 1344, 1440, 1728, 1800, 2016, 2304, 2592, 2880, 3024, 3456, 4032, 4320, 4608, 5184

Offset: 1

Enrique Pérez Herrero, Jan 27 2013

nonn

Record values of A001615.

Amiram Eldar, Table of n, a(n) for n = 1..808 (terms 1..373 from Enrique Pérez Herrero)

Cf. A002183, A006093, A330006 (the corresponding positions of records).

N:= 100: # to get a(1) to a(N)
A001615 := proc(n) n*mul((1+1/i[1]), i=ifactors(n)[2]) end:
count:= 0:
val:= -infinity:
for i from 1 while count < N do
  v:= A001615(i);
  if v > val then
     val:= v;
       count:= count+1;
       A[count]:=v;
  fi
od:
seq(A[i],i=1..N); # Robert Israel, Nov 19 2014

JordanTotient[n_,k_:1] := DivisorSum[n, #^k*MoebiusMu[n/#]&] /; (n>0) && IntegerQ[n]; DedekindPsi[n_] := JordanTotient[n,2]/EulerPhi[n]; a=1; lst={a}; Do[b=DedekindPsi[n]; If[b>a, a=b; AppendTo[lst,b]], {n,2000}]; lst
psi[1] = 1; psi[n_] := n * Times @@ (1 + 1/FactorInteger[n][[;; , 1]]); seq = {}; pmax = 0; Do[pmax = psi[n]; If[p > pmax, pmax = p; AppendTo[seq, p]], {n, 1, 10^5}]; seq (* Amiram Eldar, Nov 26 2019 *)

a(n) = A001615(A330006(n)). - Amiram Eldar, Nov 26 2019

A244053 Let m = A244052(n) = n-th highly regular number; a(n) = number of numbers r <= m, all of whose prime divisors p also divide m.

Original entry on oeis.org

1, 2, 3, 5, 6, 8, 10, 11, 18, 19, 26, 28, 32, 36, 41, 44, 68, 77, 80, 96, 115, 131, 145, 156, 166, 174, 183, 192, 283, 295, 313, 322, 382, 395, 452, 463, 505, 519, 551, 567, 593, 629, 660, 691, 717, 743, 766, 1161, 1224, 1253, 1257, 1285, 1306, 1526

Offset: 1

Michael De Vlieger, Jun 18 2014

nonn

Analogous to A002183.

Records transform of A010846. -Michael De Vlieger, Mar 08 2017

			a(5) = 6 since 6 is the fifth record value of A010846. The first record value is 1 set at position 1; the second is 2 set at position 2, the third is 3 set at position 4, the fourth is 5 set at position 6. Sequence A244052 records the positions of these record values.

Michael De Vlieger and David A. Corneth, Table of n, a(n) for n = 1..563 (terms 55-149 from David A. Corneth)
Michael De Vlieger, Multiplicities of primes in the prime decomposition of A244052 and values of A244053.

Cf. A244052, A002183, A010846.

a010846[n_] := Block[{pf, a}, a[x_] := First /@ FactorInteger@ x; pf = a@ n; If[n == 1, 1, 1 + Count[Range@ n, ?(SubsetQ[pf, a@ #] &)]]]; f[n] := Block[{t = {}, max = 0, x}, Do[If[(x = a010846@ i) > max, max = x; AppendTo[t, a010846[i]]], {i, n}]; t]; f@ 1000 (* Michael De Vlieger, Feb 10 2015 *)
Union@ Rest@ FoldList[Max, 0, Array[Count[Range@ #, k_ /; PowerMod[#, Floor@ Log2@ #, k] == 0] &, 10^3]] (* simplest, or *)
f[n_] := If[n == 1, 1, Length@ Function[w, ToExpression@ StringJoin["Module[{n = ", ToString@ n, ", k = 0}, Flatten@ Table[k++, ", Most@ Flatten@ Map[{#, ", "} &, #], "]]"] &@ MapIndexed[Function[p, StringJoin["{", ToString@ Last@ p, ", 0, Log[", ToString@ First@ p, ", n/(", ToString@ InputForm[Times @@ Map[Power @@ # &, Take[w, First@ #2 - 1]]], ")]}"]]@ w[[First@ #2]] &, w]]@ Map[{#, ToExpression["p" <> ToString@ PrimePi@ #]} &, FactorInteger[n][[All, 1]]]]; Union@ Rest@ FoldList[Max, 0, Array[f, 10^4]] (* Michael De Vlieger, Mar 08 2017, more efficient *)

cp's OEIS Frontend

A263250 Even bisection of A263087; number of solutions to x - d(x) = 4(n^2), where d(x) is the number of divisors of x (A000005).

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A063072 Sum of divisors of Ramanujan's highly composite numbers, or sigma(A002182(n)).

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A189394 Highly composite numbers whose number of divisors is also highly composite.

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A226900 Record values of Hooley's Delta function A226898.

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A263092 Numbers whose squares are in A236562; numbers n such that there is at least one such k for which k - d(k) = n^2, where d(k) is the number of divisors of k (A000005).

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A263251 Odd bisection of A263087; number of solutions to x - d(x) = (2n+1)^2, where d(x) is the number of divisors of x (A000005).

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A188591 Records of A188550.

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A194095 Record values in A056595.

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