A175304 -id:A175304 - cp's OEIS frontend

A282231 First term of A175304 with a given prime signature.

3, 6, 12, 60, 70, 72, 96, 125, 128, 250, 264, 450, 480, 756, 1152, 1380, 1458, 1980, 2030, 2048, 3640, 4860, 6552, 7776, 10648, 11448, 11907, 12348, 14960, 17664, 18432, 27540, 31620, 34200, 40500, 42978, 58140, 65000, 75776, 102240, 131328, 146529, 153120

Offset: 1

Vladimir Shevelev, Feb 09 2017

nonn

Conjecturally the sequence is infinite.
The sequence of the corresponding prime signatures begins  p, p*q, p^2*q, p^2*q*r, p*q*r, p^3*q^2, p^5*q, p^3, p^7, ...
There are no prime signatures of perfect squares. Indeed, A175304 contains no squares (see our comment there). - Vladimir Shevelev, Feb 10 2017
A037916(a(n)) gives a numerical version of the second comment: {1,11,21,211,111,32,51,3,7,31,311,221,511,321,72,2111,61,2211,1111,...}, however due to the limitations of the notation in A037916, we cannot represent a(20)=2048 since A037916(2^10)=digit 10, which is not a valid decimal digit. A037916 is useful if we refrain from rendering the multiplicities as decimal digits, instead maintaining them as a list. - Michael De Vlieger, Feb 10 2017

			From _Michael De Vlieger_, Feb 10 2017: (Start)
a(1) = 3 since 3 is prime and has a prime signature of "1"; it is the very first prime in the sequence, followed by {5,11,17,29,41,...}. The prime signature "1" is the first distinct signature encountered in the sequence
a(2) = 6 since it is a squarefree semiprime with prime signature "11"; it is the very first such number in the sequence, followed by {10,22,34, 35,51,...}. This prime signature is the second distinct signature encountered in the sequence.
a(3) = 12 since it has a prime signature of "21" (i.e., the exponents of  p^2*q^1, A037916(12)=21) and this signature is the third distinct signature encountered. It is the very first number with this signature, followed by {44,92,147,236,332,...}. (End)

Charles R Greathouse IV, Table of n, a(n) for n = 1..234

Cf. A025487, A037916, A175304, A282175.

Map[#[[1, 1]] &, GatherBy[#, Last]] &@ Map[{#, Reverse@ Sort@ FactorInteger[#][[All, -1]]} &, Select[Range[10^6], Function[n, DivisorSigma[0, n + #] == # &@ DivisorSigma[0, n]]]] (* Michael De Vlieger, Feb 10 2017 *)

sig(n)=vecsort(factor(n)[,2]~,,4)
has(n)=my(d=numdiv(n)); d==numdiv(n+d)
try(n)=my(t); has(n) && !mapisdefined(m,t=sig(n)) && (mapput(m,t,0) || 1)
v=List();for(n=3,1e9,if(try(n), listput(v,n); print(#v" "n))) \\ Charles R Greathouse IV, Feb 20 2017

More terms from Peter J. C. Moses, Feb 09 2017

A259934 Infinite sequence starting with a(0)=0 such that A049820(a(k)) = a(k-1) for all k>=1, where A049820(n) = n - (number of divisors of n).

Original entry on oeis.org

0, 2, 6, 12, 18, 22, 30, 34, 42, 46, 54, 58, 62, 70, 78, 90, 94, 102, 106, 114, 118, 121, 125, 129, 144, 152, 162, 166, 174, 182, 190, 194, 210, 214, 222, 230, 236, 242, 250, 254, 270, 274, 282, 294, 298, 302, 310, 314, 330, 342, 346, 354, 358, 366, 374, 390, 394, 402, 410, 418, 426, 434, 442, 446, 462, 466, 474, 486, 494, 510, 522, 530, 546, 558, 562, 566, 574, 582, 590

Offset: 0

Max Alekseyev, Jul 09 2015

Equivalently, satisfies the property: A000005(a(n)) = a(n)-a(n-1). The first differences a(n)-a(n-1) are given in A259935.
V. S. Guba (2015) proved that such an infinite sequence exists. Numerical evidence suggests that it may also be unique -- is it? All terms below 10^10 are defined uniquely.
If the current definition does not uniquely define the sequence, the "lexicographically earliest" condition may be added to make the sequence well-defined.
From Vladimir Shevelev, Jul 21 2015: (Start)
If a(k), a(k+1), a(k+2) is an arithmetic progression, then a(k+1) is in A175304.
Indeed, by the definition of this sequence, a(n)-a(n-1) = d(a(n)), for all n>=1, where d(n) = A000005(n). Hence, have a(k+1) - a(k) = a(k+2) - a(k+1) = d(a(k+1)) = d(a(k+2)). So a(k+1) + d(a(k+2)) = a(k+2) and a(k+1) + d(a(k+1)) = a(k+2).
Therefore, d(a(k+1) + d(a(k+1))) = d(a(k+2))= d(a(k+1)), i.e., a(k+1) is in A175304. Thus, if there are infinitely many pairs of the same consecutive terms of A259935, then A175304 is infinite (see there my conjecture). (End)
From Antti Karttunen, Nov 27 2015: (Start)
If multiple apparently infinite branches would occur at some point of computing, then even if the "lexicographically earliest" condition were then added to the definition, it would not help us much (when computing the sequence), as we would still not know which of the said branches were truly infinite. [See also Max Alekseyev's latter Jul 9 2015 posting on SeqFan-list, where he notes the same thing.] Note that many of the derived sequences tacitly assume that the uniqueness-conjecture is true. See also comments at A262693 and A262896.
One sufficient (but not a necessary) condition for the uniqueness of this sequence is that the sequence A262509 has infinite number of terms. Please see further comments there.
The graph of sequence exhibits two markedly different slopes, depending on whether it is on the "fast lane" of A049820 (even numbers) or the "slow lane" [odd numbers, for example when traversing the 1356 odd terms from 123871 to 113569 at range a(9859) .. a(8504)]. See A263086/A263085 for the "average cumulative speed difference" between the lanes. In general, slow and fast lane stay separate, except when they terminate into one of the squares (A262514) that work as "exchange ramps", forcing the parity (and thus the speed) to change. In average, the odd squares are slightly better than the even squares in attracting lanes going towards smaller numbers (compare A263253 to A263252). The cumulative effect of this bias is that the odd terms are much rarer in this sequence than the even terms (compare A263278 to A262516).
(End)

Robert Israel, Table of n, a(n) for n = 0..64800
M. Alekseyev et al., Apparently unique infinite sequences related to the sum of divisors, Discussion in SeqFan mailing list, 2015.
Michael De Vlieger, Poster illustrating A259934 and A263267
V. S. Guba et al., Sequence and divisors, 2015. (in Russian)

Cf. A000005, A049820, A060990, A259935 (first differences).
Topmost row of A263255. Cf. also irregular tables A263267 & A263265 and array A262898.
Cf. A262693 (characteristic function).
Cf. A155043, A262694, A262904 (left inverses).
Cf. A262514 (squares present), A263276 (their positions), A263277.
Cf. A262517 (odd terms).
Cf. A262509, A262510, A262897 (other subsequences).
Cf. also A175304, A260257, A262680.
Cf. A261089, A261103, A262503, A262506, A262516, A263279, A263280, A263085, A263086, A263253, A263257, A263278.
Cf. also A262679, A262896 (see the C++ program there).
No common terms with A045765 or A262903.
Positions of zeros in A262522, A262695, A262696, A262697, A263254.
Various metrics concerning finite side-trees: A262888, A262889, A262890.
Cf. also A262891, A262892 and A262895 (cf. its graph).
Cf. A260084, A260124 (variants).
Cf. also A179016 (a similar "beanstalk trunk sequence" but with more tractable and regular behavior).

N:= 10^4: # to get "guaranteed unique" terms <= N
S:= Vector(N,datatype=integer[1]):
for n from N+1 to 2*N do
  k:= n - numtheory:-tau(n);
  if k <= N then S[k]:= S[k]+1; B[k]:= n; fi;
od:
for n from N to 3 by -1 do
  if S[n] >= 1 then
    k:= n - numtheory:-tau(n);
    S[k]:= S[k]+1; B[k]:= n;
  fi
od:
A[0]:= 0: A[1]:= 2:
for n from 2 do
  b:= B[A[n-1]];
  if b > N or S[b] > 1 then break fi;
  A[n]:= b;
od:
seq(A[i],i=0..n-1); # Robert Israel, Jul 09 2015

NN = 10^4; (* to get "guaranteed unique" terms <= NN *)
Clear[A, B, S]; S[]=0; For[n = NN+1, n <= 2*NN, n++, k = n-DivisorSigma[0, n]; If[k <= NN, S[k] = S[k]+1; B[k]=n]]; For[n=NN, n >= 3, n--, If[S[n] >= 1 , k = n-DivisorSigma[0, n]; S[k] = S[k]+1; B[k]=n]]; A[0]=0; A[1]=2; For[n=2, True, n++, b = B[A[n-1]]; If[b>NN || S[b]>1, Break[]]; A[n]=b]; Table[A[i], {i, 0, n-1}] (* _Jean-François Alcover, Jul 22 2015, after Robert Israel *)

From Antti Karttunen, Nov 27 2015: (Start)
Other identities and observations. For all n >= 0:
a(n) = A262679(A262896(n)).
A155043(a(n)) = A262694(a(n)) = A262904(a(n)) = n.
A261089(n) <= a(n) <= A262503(n). [A261103 and A262506 give the distances of a(n) to these bounds.]
(End)

A259935 Infinite sequence of positive integers such that a(n) = A000005(a(1) + a(2) + ... + a(n)) for all n >= 1.

Original entry on oeis.org

2, 4, 6, 6, 4, 8, 4, 8, 4, 8, 4, 4, 8, 8, 12, 4, 8, 4, 8, 4, 3, 4, 4, 15, 8, 10, 4, 8, 8, 8, 4, 16, 4, 8, 8, 6, 6, 8, 4, 16, 4, 8, 12, 4, 4, 8, 4, 16, 12, 4, 8, 4, 8, 8, 16, 4, 8, 8, 8, 8, 8, 8, 4, 16, 4, 8, 12, 8, 16, 12, 8, 16, 12, 4, 4, 8, 8, 8, 8, 8, 24, 8, 12, 8, 4, 8, 8, 8, 16, 8, 6, 6, 8, 4, 8, 4, 8, 8, 12, 8, 18, 8, 32, 24, 18, 4, 8, 16, 4, 16, 4, 8, 12, 8, 8, 8, 8, 8, 8, 12

Offset: 1

Max Alekseyev, Jul 09 2015

V. S. Guba (2015) proved that such an infinite sequence exists. Numerical evidence suggests that it may also be unique (cf. A259934).

If there are infinitely many n with a(n) = a(n+1), then A175304 is infinite (see comment in A259934). - Vladimir Shevelev, Jul 21 2015

N. J. A. Sloane, Table of n, a(n) for n = 1..64800 (Based on Robert Israel's b-file for A259934.)
V. S. Guba et al., Sequence and divisors, 2015. (in Russian)

First differences of A259934.

Cf. A000005, A260257, A260085, A260123, A286540.

a(n) = A000005(A259934(n)) = A259934(n) - A259934(n-1).

gcd(a(n), A259934(n)) = A286540(n) = A009191(A259934(n)). - Antti Karttunen, Nov 26 2017

A286529 a(n) = d(n+d(n)), where d(n) is the number of divisors of n (A000005).

Original entry on oeis.org

2, 3, 2, 2, 2, 4, 3, 6, 6, 4, 2, 6, 4, 6, 2, 4, 2, 8, 4, 4, 3, 4, 3, 6, 6, 8, 2, 4, 2, 4, 4, 4, 2, 4, 4, 6, 4, 8, 2, 10, 2, 6, 6, 6, 4, 6, 3, 4, 6, 8, 4, 4, 4, 4, 2, 7, 2, 4, 2, 12, 6, 8, 4, 2, 4, 4, 4, 4, 2, 8, 2, 12, 6, 8, 5, 4, 5, 4, 5, 12, 4, 4, 4, 12, 2, 12, 4, 12, 4, 8, 4, 6, 2, 6, 6, 12, 6, 8, 8, 2, 2, 8, 8, 10, 2, 8, 2, 16, 4, 4, 4, 4, 4, 4, 4, 4, 4

Offset: 1

Antti Karttunen, May 21 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000
Aleksandar Ivić, An asymptotic formula involving the enumerating function of finite Abelian groups, Publikacije Elektrotehničkog fakulteta, Serija Matematika 3 (1992), pp. 61-66.
Imre Kátai, On a problem of A. Ivic, Mathematica Pannonica, Vol. 18, No. 1 (2007), pp. 11-18.

Cf. A000005, A062249, A175304, A286479, A286530.

Table[DivisorSigma[0, n + DivisorSigma[0, n]], {n, 117}] (* Michael De Vlieger, May 21 2017 *)

PARI
```
A286529(n) = numdiv(n+numdiv(n));
```

from sympy import divisor_count as d
def a(n): return d(n + d(n)) # Indranil Ghosh, May 21 2017

(define (A286529 n) (A000005 (+ n (A000005 n))))

a(n) = A000005(A062249(n)) = A000005(n+A000005(n)).

Sum_{k=1..n} a(k) ~ D*n*log(n) + O(n*log(n)/log(log(n))), where D > 0 is a constant (conjectured with an error O(n) by Ivić, 1992; proven by Kátai, 2007). - Amiram Eldar, Jul 08 2020

A286530 a(n) = d(n+d(n)) - d(n), where d(n) is the number of divisors of n (A000005).

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, May 21 2017

sign

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A000005, A286529, A286480.

Cf. A175304 (the positions of zeros).

Table[DivisorSigma[0, n + DivisorSigma[0, n]] - DivisorSigma[0, n], {n, 109}] (* Michael De Vlieger, May 21 2017 *)

A286530(n) = (numdiv(n+numdiv(n)) - numdiv(n));

from sympy import divisor_count as d
def a(n): return d(n + d(n)) - d(n) # Indranil Ghosh, May 21 2017

(define (A286530 n) (- (A286529 n) (A000005 n)))

a(n) = A286529(n) - A000005(n) = A000005(n+A000005(n)) - A000005(n).

A260577 Numbers n for which d(n+d(n)) < d(n), where d(n) is the number of divisors of n.

Original entry on oeis.org

4, 15, 16, 20, 21, 24, 27, 28, 30, 32, 33, 36, 39, 42, 45, 48, 52, 54, 55, 56, 57, 63, 64, 66, 68, 69, 75, 76, 78, 81, 85, 90, 93, 100, 105, 110, 112, 114, 116, 117, 120, 123, 126, 133, 135, 138, 140, 144, 145, 150, 153, 159, 160, 162, 165, 168, 170, 171, 172

Offset: 1

Vladimir Shevelev, Jul 29 2015

nonn

All terms are composite.
Indeed, if p is prime then d(p)=2 will never be larger than d(p+d(p)) = d(p+2). - M. F. Hasler, Jul 30 2015
Conjecture: for every x>=6, among the first x terms, the terms divisible by 3 are never in the minority.
Let A(y) be the number of terms <= y, y>=1. If the conjecture is true, then, taking into account the initials, we conclude that always A(y) < (2/3)*y. - Vladimir Shevelev, Jul 31 2015

			75 is in the sequence, since d(75) = 6 > d(75+6) = 5.

Peter J. C. Moses, Table of n, a(n) for n = 1..2000

Cf. A000005, A175304.

is(n)=numdiv(n+n=numdiv(n))M. F. Hasler, Jul 30 2015

A154689 Numbers n such that sigma_0(n-sigma_0(n))= sigma_0(n).

Original entry on oeis.org

5, 7, 10, 13, 14, 18, 19, 26, 31, 38, 39, 43, 50, 55, 61, 62, 69, 72, 73, 78, 84, 86, 91, 95, 96, 98, 103, 108, 109, 110, 115, 119, 122, 123, 129, 133, 136, 138, 139, 145, 146, 151, 153, 159, 181, 182, 187, 190, 193, 199, 205, 206, 209, 213, 217, 218, 219, 221, 229

Offset: 1

Ctibor O. Zizka, Jan 14 2009

If n is a term, then n - d(n) is in A175304. Indeed, if N = n - d(n), then d(N) = d(n). Now d(N+d(N)) = d(N+d(n)) = d(n) = d(N). - Vladimir Shevelev, Jul 29 2015

Numbers n such that A000005(n-A000005(n)) = A000005(n). - Typo fixed by Ivan N. Ianakiev, Oct 08 2016

Harvey P. Dale, Table of n, a(n) for n = 1..1000
J.-M. de Koninck, F. Luca, Positive integers n such that sigma(phi(n))=sigma(n), J. Int. Seq. vol 11 (2008), #08.1.5. [From _R. J. Mathar_, Jan 15 2009]

Cf. A000005, A006512 (a subsequence).

A000005 := proc(n) numtheory[tau](n) ; end: for n from 1 to 1000 do a05 := A000005(n) ; if A000005(n-a05) = a05 then printf("%d,",n) ; fi; od: # R. J. Mathar, Jan 15 2009

snQ[n_]:=Module[{nd=DivisorSigma[0,n]},DivisorSigma[0,n-nd]==nd]; Select[ Range[300],snQ] (* Harvey P. Dale, Nov 10 2011 *)

is(n)=my(d=numdiv(n));numdiv(n-d)==d \\ Charles R Greathouse IV, Feb 04 2013

Extended by R. J. Mathar, Jan 15 2009

A260257 Numbers n such that A259935(n) = A259935(n-1).

Original entry on oeis.org

4, 12, 14, 23, 29, 30, 35, 37, 45, 54, 58, 59, 60, 61, 62, 75, 77, 78, 79, 80, 87, 88, 92, 98, 115, 116, 117, 118, 119, 122, 123, 125, 127, 144, 147, 154, 155, 158, 159, 160, 163, 171, 175, 179, 183, 184, 190, 192, 206, 207, 217, 221, 222, 225, 228, 232, 234, 236, 238, 242, 249, 255, 256, 257, 258, 260, 268, 269, 270, 271, 272, 278, 281, 284, 288, 294, 301

Offset: 1

Vladimir Shevelev, Jul 21 2015

nonn

For every n, A259934(a(n)-2), A259934(a(n)-1), A259934(a(n)) is arithmetic progression, such that A259934(a(n)-1) is in A175304 (see comment in A259934).

Cf. A175304, A259934, A259935.

A282175 a(n) is the smallest product M=p_1p_2...p_n with distinct primes p_i such that M+2^n=B, where B=q_1q_2...q_n with distinct primes q_i, or a(n)=0 if there is no such M.

Original entry on oeis.org

3, 6, 70, 2030, 42978, 1788710, 63905142, 5705962314, 888081948858, 120056591419170

Offset: 1

Vladimir Shevelev and Peter J. C. Moses, Feb 07 2017

Conjecturally all a(n)>0.

Since d(a(n)+2^n) = 2^n, where d(n) is the number of divisors of n, and d(a(n)+d(a(n)+2^n)) = d(a(n)), then it is a subsequence of sequence A175304.

			For n=3,...,8, we have the following numbers M, B=M+2^n and their prime divisors:
70 = 2 5 7; 78 = 2 3 13.
2030 = 2 5 7 29; 2046 = 2 3 11 31.
42978 = 2 3 13 19 29; 43010 = 2 5 11 17 23.
1788710 = 2 5 7 11 23 101; 1788774 = 2 3 13 17 19 71.
63905142 = 2 3 7 17 37 41 59; 63905270 = 2 5 11 13 23 29 67.
5705962314 = 2 3 13 17 19 23 43 229; 5705962570 = 2 5 7 11 29 59 61 71.

a(9)-a(10) from Giovanni Resta, Feb 28 2017

A282354 Positive j such that d(j) = d(j + 2*d(j)), where d(j) is the number of divisors of j.

Original entry on oeis.org

3, 6, 7, 13, 14, 19, 20, 24, 26, 27, 32, 37, 38, 40, 43, 54, 57, 60, 63, 67, 69, 72, 74, 77, 79, 84, 85, 86, 87, 88, 97, 103, 108, 109, 111, 114, 115, 125, 126, 127, 132, 133, 134, 136, 138, 154, 158, 163, 170, 174, 177, 193, 194, 200, 201, 204, 205, 206, 209

Offset: 1

Vladimir Shevelev, Feb 13 2017

nonn

The sequence contains the smaller member of every pair of cousin primes (A023200).

The sequence contains no perfect squares. Indeed, let a(m) = k^2 for some m. Then, by the definition, d(k^2 + 2*d(k^2)) = d(k^2). Note that d(k^2) is odd. On the other hand, it is known (cf. A046522) that d(k^2) < 2*k. Hence (k+2)^2 - k^2 = 4*k + 4 > 2*d(k^2). Thus k^2 < k^2 + 2*d(k^2) < (k+2)^2. Since, evidently, k^2 + 2*d(k^2) cannot be (k+1)^2, then k^2 + 2*d(k^2) cannot be a square. Therefore, d(k^2 + 2*d(k^2)) is even, which is a contradiction.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A000005, A025487, A037916, A175304, A282231.

Select[Range@ 210, Function[d, DivisorSigma[0, # + 2 d] == d]@ DivisorSigma[0, #] &] (* Michael De Vlieger, Feb 13 2017 *)

is(n)=my(d=numdiv(n)); d==numdiv(n+2*d) \\ Charles R Greathouse IV, Feb 14 2017

More terms from Peter J. C. Moses, Feb 13 2017

cp's OEIS Frontend

A282231 First term of A175304 with a given prime signature.

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A259934 Infinite sequence starting with a(0)=0 such that A049820(a(k)) = a(k-1) for all k>=1, where A049820(n) = n - (number of divisors of n).

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A259935 Infinite sequence of positive integers such that a(n) = A000005(a(1) + a(2) + ... + a(n)) for all n >= 1.

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A286529 a(n) = d(n+d(n)), where d(n) is the number of divisors of n (A000005).

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A286530 a(n) = d(n+d(n)) - d(n), where d(n) is the number of divisors of n (A000005).

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A260577 Numbers n for which d(n+d(n)) < d(n), where d(n) is the number of divisors of n.

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A154689 Numbers n such that sigma_0(n-sigma_0(n))= sigma_0(n).

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A260257 Numbers n such that A259935(n) = A259935(n-1).

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A282175 a(n) is the smallest product M=p_1p_2...p_n with distinct primes p_i such that M+2^n=B, where B=q_1q_2...q_n with distinct primes q_i, or a(n)=0 if there is no such M.