A052213 -id:A052213 - cp's OEIS frontend

A034173 a(n) is minimal such that prime factorizations of a(n), ..., a(n)+n-1 have same exponents.

1, 2, 33, 19940, 204323, 380480345, 440738966073

Offset: 1

Dean Hickerson, Oct 01 1998

a(8) > 10^13. - Donovan Johnson, Oct 20 2009
Don Reble has shown that a(8) < 1.9*10^42, cf. link.
From David Wasserman, Jan 05 2019: (Start)
a(8) <= 108111092880293127811946663766147737122,
a(9) <= 6850672946809600696044301071559918192380244,
a(10) <= 96037988156124494415303285590850571857698741869620,
a(11) <= 9044737840075556371215937303485030235666252755947862558252154847122. (End)

			a(4) = 19940 because 19940, ..., 19943 all have the form p^2 q r.

Don Reble, A034173(8) exists, SeqFan list, Oct 23 2012.

Cf. A034174.
Cf. A006558, A083790.
Cf. A052213, A052214, A175590, A218448. This sequence is the first column of A083785 and first row of A113456. The latter generalizes to arithmetic progressions with step d>=1. - M. F. Hasler, Oct 28 2012

A034173(n)={my(f);for(k=1,oo,f=0;for(i=1,n, f==(f=vecsort(factor(k+n-i)[,2])) || i==1 || [k+=n-i; next(2)]);return(k))} \\ M. F. Hasler, Oct 23 2012

a(n) = A034174(n) - n + 1. - Max Alekseyev, Nov 10 2009

a(n) = A083785(n,1) = A113456(1,n); a(2) = A052213(1), a(3) = A052214(1), a(4) = A175590(1), a(5) = A218448(1), a(6) = A218448(62) = A218448(63)-1. - M. F. Hasler, Oct 28 2012

a(7) from Donovan Johnson, Oct 20 2009

Don Reble link repaired by N. J. A. Sloane, Oct 24 2024

A175590 Numbers k with prime signature(k) = prime signature(k+1) = prime signature(k+2) = prime signature(k+3).

Original entry on oeis.org

19940, 49147, 54585, 118923, 136825, 183554, 204323, 204324, 262932, 304675, 361275, 361322, 476377, 486962, 506905, 619722, 668211, 734948, 854018, 937025, 938203, 999649, 1062025, 1118275, 1335572, 1336075, 1356324, 1466225, 1541491

Offset: 1

Charles R Greathouse IV, Jul 19 2010

nonn

			a(1) = 2^2 * 5 * 997; a(1)+1 = 3 * 17^2 * 23; a(1)+2 = 2 * 13^2 * 59; a(1)+3 = 7^2 * 11 * 37. All have prime signature {2, 1, 1}.

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

Cf. A052213, A052214, A218448. Subsequence of A070284.

SequencePosition[Table[Sort[FactorInteger[n][[All,2]]],{n,1542000}],{x_,x_,x_,x_}][[All,1]] (* Requires Mathematica version 10 or later *) (* The program will take a long time to run. *) (* Harvey P. Dale, Jun 09 2021 *)

sig(n)={vecsort(factor(n)[,2])}; s=sig(1);for(n=1,1e6,t=sig(n+1);if(s==t&t==sig(n+2)&t==sig(n+3),print1(n-1,","));s=t)

is_A175590(n)={my(f(n)=vecsort(factor(n)[,2]),t=f(n));!for(i=1,3,f(n+i)!=t & return)}  \\ M. F. Hasler, Nov 01 2012

A218448 First of a run of 5 consecutive numbers with same prime signature.

Original entry on oeis.org

204323, 3252571, 5205074, 7201674, 20182921, 28387953, 36193650, 43216722, 51049537, 56155074, 57070850, 61961315, 62167075, 65425473, 76647074, 82507473, 92658049, 95943321, 100498849, 107236449, 109751473, 110899321, 112198075, 112477849, 116736323

Offset: 1

M. F. Hasler, Oct 28 2012

nonn

A number n is in this sequence iff n and n+1 is in A175590; also: iff n and n+2 are in A052214 (in which case n+1 is in A052214, too); and also: iff {n,n+1,n+2,n+3} are in A052213.

A034173(6) = A218448(62) = A218448(63)-1 is the least term n such that n+1 is also in the sequence.

M. F. Hasler and Donovan Johnson, Table of n, a(n) for n = 1..1000 (first 140 terms from _M. F. Hasler_)

Cf. A034173, A034174, A006558, A083790, A052213, A052214, A175590.

is_A218448(n)={my(f);!for(i=0,4,f!=(f=vecsort(factor(n+i)[,2])) & i & return)}

f(k)=vecsort(factor(k)[,2]~,,4)
t=f(n=2);while(n<1e8, for(i=n+1, n+4, tt=f(i); if(tt!=t, n=i; t=tt; next(2))); print1(n", "); n++) \\ Charles R Greathouse IV, Oct 28 2012

a(6)-a(8) from Charles R Greathouse IV, Oct 28 2012
a(9)-a(25) from Donovan Johnson, Oct 28 2012
Values up to a(140) computed using b175590.txt from Charles R Greathouse IV  - M. F. Hasler, Oct 28 2012

A358817 Numbers k such that A046660(k) = A046660(k+1).

Original entry on oeis.org

1, 2, 5, 6, 10, 13, 14, 21, 22, 29, 30, 33, 34, 37, 38, 41, 42, 44, 46, 49, 57, 58, 61, 65, 66, 69, 70, 73, 75, 77, 78, 80, 82, 85, 86, 93, 94, 98, 101, 102, 105, 106, 109, 110, 113, 114, 116, 118, 122, 129, 130, 133, 135, 137, 138, 141, 142, 145, 147, 154, 157

Offset: 1

Amiram Eldar, Dec 02 2022

nonn

First differs from its subsequence A007674 at n=18.

The numbers of terms not exceeding 10^k, for k = 1, 2, ..., are 5, 38, 369, 3655, 36477, 364482, 3644923, 36449447, 364494215, 3644931537, ... . Apparently, the asymptotic density of this sequence exists and equals 0.36449... .

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A046660.
Subsequences: A007674, A052213, A085651, A358818.
Similar sequences: A002961, A005237, A006049, A045920.

seq[kmax_] := Module[{s = {}, e1 = 0, e2}, Do[e2 = PrimeOmega[k] - PrimeNu[k]; If[e1 == e2, AppendTo[s, k - 1]]; e1 = e2, {k, 2, kmax}]; s]; seq[160]

e(n) = {my(f = factor(n)); bigomega(f) - omega(f)};
lista(nmax) = {my(e1 = e(1), e2); for(n=2, nmax, e2=e(n); if(e1 == e2, print1(n-1,", ")); e1 = e2);}

A260143 Runs of consecutive integers with same prime signature.

Original entry on oeis.org

2, 3, 14, 15, 21, 22, 33, 34, 35, 38, 39, 44, 45, 57, 58, 75, 76, 85, 86, 87, 93, 94, 95, 98, 99, 116, 117, 118, 119, 122, 123, 133, 134, 135, 136, 141, 142, 143, 145, 146, 147, 148, 158, 159, 171, 172, 177, 178, 201, 202, 203, 205, 206, 213, 214, 215, 217, 218, 219, 230, 231, 244, 245

Offset: 1

Jean-François Alcover, Jul 17 2015

This sequence is infinite, see A189982 and Theorem 4 in Goldston-Graham-Pintz-Yıldırım. - Charles R Greathouse IV, Jul 17 2015

			Runs begin:
(terms)         (prime signature)
{2, 3},         [1]
{14, 15},       [1,1]
{21, 22},       [1,1]
{33, 34, 35},   [1,1]
{38, 39},       [1,1]
{44, 45},       [1,2]
{57, 58},       [1,1]
{75, 76},       [1,2]
{85, 86, 87},   [1,1]
{93, 94, 95},   [1,1]
{98, 99},       [1,2]
...

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
D. A. Goldston, S. W. Graham, J. Pintz, and C. Y. Yıldırım, Small gaps between almost primes, the parity problem, and some conjectures of Erdos on consecutive integers, arXiv:0803.2636 [math.NT], 2008.
MathOverflow, Question on consecutive integers with similar prime factorizations
Eric Weisstein's World of Mathematics, Prime Signature
OEIS Wiki, Prime signatures

Main sequence is A052213.

Split[Range[2,250], Sort[FactorInteger[#1][[All, 2]]] === Sort[FactorInteger[#2][[All, 2]]]&] // Select[#, Length[#] > 1&]& // Flatten

is(n)=my(f=vecsort(factor(n)[,2])); f==vecsort(factor(n-1)[,2]) || f==vecsort(factor(n+1)[,2]) \\ Charles R Greathouse IV, Jul 17 2015

from sympy import factorint
def aupto(limit):
    aset, prevsig = {2}, [1]
    for k in range(3, limit+2):
        sig = sorted(factorint(k).values())
        if sig == prevsig: aset.update([k - 1, k])
        prevsig = sig
    return sorted(aset)
print(aupto(250)) # Michael S. Branicky, Sep 20 2021

A276553 Numbers n such that n^2 and (n + 1)^2 have the same number of divisors.

Original entry on oeis.org

2, 14, 15, 21, 33, 34, 38, 44, 57, 75, 81, 85, 86, 93, 94, 98, 116, 118, 122, 133, 135, 141, 142, 145, 147, 158, 171, 177, 201, 202, 205, 213, 214, 217, 218, 230, 244, 253, 272, 285, 296, 298, 301, 302, 326, 332, 334, 375, 381, 387, 393, 394, 405, 429, 434, 445

Offset: 1

K. D. Bajpai, Apr 10 2017

nonn

Except for a(1), all the terms are composite.

			We see that 14^2 = 196, the divisors of which are 1, 2, 4, 7, 14, 28, 49, 98, 196, and there are nine of them. And we see that 15^2 = 225, the divisors of which are 1, 3, 5, 9, 15, 25, 45, 75, 225, and there are nine of them. Both 14^2 and 15^2 have the same number of divisors, hence 14 is in the sequence.
And we see that 16^2 = 256, the divisors of which are the powers of 2 from 2^0 to 2^8, that's nine divisors. Both 15^2 and 16^2 have the same number of divisors, hence 15 is also in the sequence.
But 16 is not in the sequence, since 17 is prime and 17^2 consequently only has three divisors.

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

Cf. A000290, A048691, A062832, A276542, A284378.
Cf. A052213 (a subsequence).
Positions of zeros in A284570.

N:= 1000: # to get all terms <= N
T:= map(t -> numtheory:-tau(t^2), [$1..N+1]):
select(t -> T[t]=T[t+1], [$1..N]); # Robert Israel, Apr 10 2017

Select[Range[1000], DivisorSigma[0, #^2] == DivisorSigma[0, (# + 1)^2] &]

k=[]; for(n=1, 1000, a=numdiv(n^2); b=numdiv((n+1)^2); if(a==b, k=concat(k, n))); k

from sympy.ntheory import divisor_count
print([n for n in range(1, 501) if divisor_count(n**2) == divisor_count((n + 1)**2)]) # Indranil Ghosh, Apr 10 2017
(Scheme, with Antti Karttunen's IntSeq-library) (define A276553 (ZERO-POS 1 1 A284570)) ;; Antti Karttunen, Apr 15 2017

A333055 Numbers k such that k and k+1 have different (ordered) prime signatures and d(k) = d(k+1), where d(k) is the number of divisors of k (A000005).

Original entry on oeis.org

26, 104, 189, 231, 242, 243, 344, 374, 663, 664, 735, 776, 782, 874, 903, 1015, 1029, 1095, 1106, 1112, 1161, 1208, 1269, 1335, 1374, 1544, 1625, 1809, 1832, 1917, 1952, 1970, 2055, 2133, 2241, 2247, 2264, 2343, 2344, 2504, 2655, 2696, 2726, 2781, 2874, 2936

Offset: 1

Amiram Eldar, Mar 06 2020

nonn

Apparently most of the numbers k such that k and k+1 have the same number of divisors (A005237) also have the same prime signature, i.e., they are also terms of A052213 which is a subsequence of A005237.

For example, up to 10^8 there are 9593611 terms in A005237, of them only 1573778 (about 16.4%) are not in A052213. This sequence in the complement of A052213 with respect to A005237.

			26 is a term since 26 = 2 * 13 and 27 = 3^3 have different prime signatures, and d(26) = d(27) = 4.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000005, A005237, A052213, A124010, A333056, A333057.

Select[Range[3000], DivisorSigma[0, #] == DivisorSigma[0, #+1] && Sort[FactorInteger[#][[;;,2]]] != Sort[FactorInteger[#+1][[;;,2]]] &]

A085073 Smallest k such that n+k and n*k have the same prime signature, or 0 if no such number exists.

Original entry on oeis.org

2, 1, 7, 41, 15, 134, 3, 127, 11, 2, 3, 548, 2, 1, 3, 389, 5, 582, 2, 316, 1, 38, 3, 2216, 3, 2, 13, 212, 5, 2742, 2, 1669, 1, 1, 31, 2764, 2, 1, 13, 1094, 4, 2298, 3, 1, 123, 14, 11, 8912, 3, 202, 17, 2, 2, 1146, 23, 904, 1, 26, 3, 11028, 13, 22, 57, 3581, 37, 1194, 2, 172, 15

Offset: 1

Amarnath Murthy, Jul 01 2003

nonn

			a(6) = 379 as 6*379 = 2*3*379 and 6+379 = 385 = 5*7*11 both have prime signature p*q*r.

Michel Marcus, Table of n, a(n) for n = 1..3000

Cf. A052213 (a(n)=1), A085072.

s:= proc(n) s(n):= sort(map(i-> i[2], ifactors(n)[2])) end:
a:= proc(n) option remember; local k; for k
       while s(n*k)<>s(n+k) do od; k
    end:
seq(a(n), n=1..70);  # Alois P. Heinz, Mar 06 2019

kmax = 10^6;
s[n_] := FactorInteger[n][[All, 2]] // Sort;
a[n_] := Module[{k}, If[n == 1, Return[2]]; For[k = 1, k <= kmax, k++, If[s[n k] == s[n+k], Return[k]]]; 0];
Array[a, 70] (* Jean-François Alcover, Nov 17 2020 *)

sgntr(n) = vecsort(factor(n)[, 2]~);
a(n) = {my(k=1); while (sgntr(n+k) != sgntr(n*k), k++); k; } \\ Michel Marcus, Nov 17 2020

Corrected by Jason Earls, Jul 10 2003

More terms from David Wasserman, Jan 12 2005

A218455 First of a run of 6 consecutive numbers with same prime signature.

Original entry on oeis.org

380480345, 2713001274, 6282718946, 7209536449, 9809067073, 10684724346, 12008728850, 14824913049, 17231547073, 17552118546, 17659180314, 18036555273, 20473171322, 21507097001, 23676804346, 24742649321, 25401767522, 25694056449, 27656894273, 28259097818

Offset: 1

M. F. Hasler, Oct 29 2012

nonn

A number n is in this sequence iff n and n+1 is in A218448; see the comment there for other characterizations in terms of membership in A175590 or A052214 or A052213.

Donovan Johnson, Table of n, a(n) for n = 1..1000

Cf. A034173, A034174, A006558, A083790, A052213, A052214, A175590, A218448, A218865.

is_A218455(n)={my(s(n)=vecsort(factor(n)[,2]),t=s(n));!for(m=n+1,n+5, t!=s(m) & return)}

a(2)-a(20) from Donovan Johnson, Oct 29 2012

A279767 Numbers m such that m and m+2 have the same prime signature.

Original entry on oeis.org

3, 5, 11, 17, 18, 29, 33, 41, 50, 54, 55, 59, 71, 85, 91, 93, 101, 107, 137, 141, 143, 149, 159, 179, 183, 185, 191, 197, 201, 203, 213, 215, 217, 219, 227, 235, 239, 242, 247, 248, 265, 269, 281, 299, 301, 303, 306, 311, 319, 321, 327, 339, 340, 347, 348, 391, 393, 411, 413

Offset: 1

Altug Alkan, Dec 18 2016

The sequence contains some terms such that m and m + 2k (k > 1) have the same prime signature. For some terms where m and m + 2k share the same prime signature this means that every alternate element between, and including m and m + 2k have the same prime signature. The first such example is where a(41951) = 402677, a(41953) = 402679, and a(41955) = 402681, share the same prime signature {1, 1}. Also the remaining alternate terms excluding endpoints share the same prime signature. Using the above example, a(41952) = 402678 and a(41954) = 402680 share the prime signature {1,1,3}. - Torlach Rush, Feb 25 2018

			18 is a term because 18 = 2 * 3^2 and 18 + 2 = 20 = 2^2 * 5.
19 is not a term because it is prime and 21 is the product of two primes, so the prime signatures are different.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..5585 from Michel Marcus)
Index to sequences related to prime signature.

Cf. A001359, A052213, A052214.

primeSignature[n_] := Sort[Transpose[FactorInteger[n]][[2]]]; Select[ Range[2, 1000], primeSignature[#] == primeSignature[# + 2] &] (* Adapted from A052213 *)

isok(n) = vecsort(factor(n)[,2]) == vecsort(factor(n+2)[,2]); \\ Michel Marcus, Feb 25 2018

cp's OEIS Frontend

A034173 a(n) is minimal such that prime factorizations of a(n), ..., a(n)+n-1 have same exponents.

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A175590 Numbers k with prime signature(k) = prime signature(k+1) = prime signature(k+2) = prime signature(k+3).

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A218448 First of a run of 5 consecutive numbers with same prime signature.

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A358817 Numbers k such that A046660(k) = A046660(k+1).

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A260143 Runs of consecutive integers with same prime signature.

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A276553 Numbers n such that n^2 and (n + 1)^2 have the same number of divisors.

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A333055 Numbers k such that k and k+1 have different (ordered) prime signatures and d(k) = d(k+1), where d(k) is the number of divisors of k (A000005).

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