A006049 -id:A006049 - cp's OEIS frontend

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

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);}

A045934 Numbers n such that n through n+5 have the same number of distinct prime factors.

Original entry on oeis.org

91, 141, 142, 143, 212, 213, 214, 323, 324, 2302, 2303, 6850, 9061, 10280, 10281, 15740, 16130, 16164, 16682, 16683, 19052, 19053, 20212, 20213, 21195, 21196, 21790, 22055, 23064, 25779, 25780, 25991, 28608, 28674, 29971, 31442, 33084

Offset: 1

David W. Wilson

nonn

			The numbers from 91 to 96 all have 2 distinct prime factors: 91=7*13, 92=2^2*23, 93=3*31, 94=2*47, 95=5*19, and 96=2^5*3.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..2620 from Zak Seidov)
Roger B. Eggleton and James A. MacDougall, Consecutive integers with equally many principal divisors, Math. Mag. 81 (2008), 235-248.

Cf. A001221, A006049, A006073, A045932-A045933, A045935-A045938, A088983.

Select[Range[35000],Length[Union[Length/@FactorInteger[Range[#,#+5]]]]==1&]  (* Harvey P. Dale, Feb 27 2011 *)

A045935 Numbers n such that n through n+6 are divisible by the same number of distinct primes.

Original entry on oeis.org

141, 142, 212, 213, 323, 2302, 10280, 16682, 19052, 20212, 21195, 25779, 33332, 35118, 35164, 35202, 39693, 39694, 40269, 41390, 41780, 42342, 42410, 44360, 44361, 44362, 48919, 48920, 48921, 48922, 53734, 54349, 54350, 56014, 56015

Offset: 1

David W. Wilson

nonn

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

Cf. A001221, A006049, A006073, A045932-A045934, A045936-A045938.

Offset corrected by Amiram Eldar, Oct 26 2019

A321489 Numbers m such that both m and m+1 have at least 7 distinct prime factors.

Original entry on oeis.org

965009045, 1068044054, 1168008204, 1177173074, 1209907985, 1218115535, 1240268490, 1338753129, 1344185205, 1408520805, 1477640450, 1487720234, 1509981395, 1663654629, 1693460405, 1731986894, 1758259425, 1819458354, 1821278459, 1826445984, 1857332840

Offset: 1

Amiram Eldar and M. F. Hasler, Nov 12 2018

nonn

The first 300 terms of this sequence are such that m and m+1 both have exactly 7 prime divisors. See A321497 for the terms m such that m or m+1 has more than 7 prime factors: the smallest such term is 5163068910.
Numbers m and m+1 can never have a common prime factor (consider them mod p), therefore the terms are > sqrt(p(7+7)#) = A003059(A002110(7+7)). (Here we see that sqrt(p(7+8)#) is a more realistic estimate of a(1), but for smaller values of k we may have sqrt(p(2k+1)#) > m(k) > sqrt(p(2k)#), where m(k) is the smallest of two consecutive integers each having at least k prime divisors. For example, A321503(1) < sqrt(p(3+4)#) ~ A321493(1).)
From M. F. Hasler, Nov 28 2018: (Start)
The first 100 terms and beyond are all congruent to one of {14, 20, 35, 49, 50, 69, 84, 90, 104, 105, 110, 119, 125, 129, 134, 140, 144, 170, 174, 189, 195} mod 210. Here, 35, 195, 189, 14 140, 20 and 174 (in order of decreasing frequency) occur between 6 and 13 times, and {49, 50, 110, 129, 134, 144, 170} occur only once.
However, as observed by Charles R Greathouse IV, one can construct a term of this sequence congruent to any given m > 0, modulo any given n > 0.
The first terms of this sequence which are multiples of 210 are in A321497. An example of a term that is a multiple of 210 but not in A321497 is 29759526510, due to Charles R Greathouse IV. Such examples can be constructed by solving A*210 + 1 = B for A having 3 distinct prime factors not among {2, 3, 5, 7}, B having 7 distinct prime factors and gcd(B, 210*A) = 1. (End)

			a(1) = 5 * 7 * 11 * 13 * 23 * 83 * 101, a(1)+1 = 2 * 3 * 17 * 29 * 41 * 73 * 109.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000 (first 102 terms from M. F. Hasler)

Cf. A255346, A321503 .. A321506 (analog for k = 2, ..., 6 prime divisors).
Cf. A321502, A321493 .. A321497 (m and m+1 have at least but not both exactly k = 2, ..., 7 prime divisors).
Cf. A074851, A140077, A140078, A140079 (m and m+1 both have exactly k = 2, 3, 4, 5 prime divisors).
Cf. A006049, A006549, A093548.
Cf. A002110.

Select[Range[36000000], PrimeNu[#] > 6 && PrimeNu[# + 1] > 6 &]

is(n)=omega(n)>6&&omega(n+1)>6
A321489=List();for(n=965*10^6,1.8e9,is(n)&&listput(A321489,n))

a(n) ~ n. - Charles R Greathouse IV, Nov 29 2018

A348345 Number k such that k and k+1 have the same positive number of noninfinitary divisors (A348341).

Original entry on oeis.org

44, 75, 98, 116, 147, 171, 242, 243, 244, 332, 387, 507, 548, 603, 604, 724, 735, 819, 844, 908, 931, 963, 1035, 1075, 1083, 1196, 1251, 1274, 1275, 1324, 1412, 1449, 1467, 1556, 1587, 1665, 1675, 1772, 1924, 1925, 1952, 1988, 2324, 2331, 2511, 2523, 2524, 2540

Offset: 1

Amiram Eldar, Oct 13 2021

nonn

First differs from A049103 at n=17.
Numbers k such that A348341(k) = A348341(k+1) > 0.
The terms are restricted to have a positive number of noninfinitary divisors, since there are many consecutive numbers without noninfinitary divisors (these are the terms of A036537).

			44 is a term since A348341(44) = A348341(45) = 2 > 0.

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

Cf. A036537, A049103, A348341, A348346.
Subsequence of A162643.
Similar sequences: A005237, A006049, A343819, A344312, A344313, A344314.

nid[1] = 0; nid[n_] := DivisorSigma[0, n] - Times @@ Flatten[2^DigitCount[#, 2, 1] & /@ FactorInteger[n][[;; , 2]]]; Select[Range[2500],(nid1 = nid[#]) > 0 && nid1 == nid[# + 1] &]

A348341(n) = (numdiv(n)-factorback(apply(a -> 2^hammingweight(a), factorint(n)[, 2])));
isA348345(n) = { my(u=A348341(n)); (u>0&&(A348341(1+n)==u)); }; \\ Antti Karttunen, Oct 13 2021

A355710 Numbers k such that k and k+1 have the same number of 5-smooth divisors.

Original entry on oeis.org

2, 21, 33, 34, 38, 57, 85, 86, 93, 94, 104, 116, 122, 141, 145, 146, 154, 158, 171, 177, 182, 189, 201, 205, 213, 214, 218, 237, 265, 266, 273, 296, 302, 321, 326, 332, 334, 338, 344, 357, 362, 381, 385, 387, 393, 394, 398, 417, 445, 446, 453, 454, 475, 476, 482

Offset: 1

Amiram Eldar, Jul 15 2022

nonn

Numbers k such that A355583(k) = A355583(k+1).

			2 is a term since A355583(2) = A355583(3) = 2.

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

Cf. A355583, A355709 (3-smooth analog).
Subsequences: A355711, A355712.
Similar sequences: A005237, A006049, A332386, A343819, A344312, A344313, A344314, A348345.

s[n_] := Times @@ (1 + IntegerExponent[n, {2, 3, 5}]); Select[Range[500], s[#] == s[#+1] &]

s(n) = (valuation(n, 2) + 1) * (valuation(n, 3) + 1) * (valuation(n, 5) + 1);
s1 = s(1); for(k = 2, 500, s2 = s(k); if(s1 == s2, print1(k-1,", ")); s1 = s2);

A045936 Numbers n such that n through n+7 are divisible by the same number of distinct primes.

Original entry on oeis.org

141, 212, 39693, 44360, 44361, 48919, 48920, 48921, 54349, 56014, 56015, 56791, 60044, 65721, 72650, 72651, 73292, 73293, 76581, 76582, 82324, 82325, 86331, 86332, 87758, 87759, 90092, 91814, 91815, 99843, 106249, 112142, 112143, 121594

Offset: 1

David W. Wilson

nonn

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

Cf. A001221, A006049, A006073, A045932-A045935, A045937-A045938.

npQ[n_]:=Length[Union[Length[FactorInteger[#]]&/@Range[n,n+7]]]==1
Select[Range[125000],npQ]  (* Harvey P. Dale, Feb 23 2011 *)

Offset corrected by Amiram Eldar, Oct 26 2019

A045937 Numbers n such that n through n+8 are divisible by the same number of distinct primes.

Original entry on oeis.org

44360, 48919, 48920, 56014, 72650, 73292, 76581, 82324, 86331, 87758, 91814, 112142, 143491, 147951, 158719, 184171, 184172, 197588, 202498, 205244, 215300, 218972, 218973, 218974, 229728, 230628, 241129, 250933, 253204, 253665, 287492

Offset: 1

David W. Wilson

nonn

Primes counted without multiplicity. - Harvey P. Dale, May 05 2015

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

Cf. A001221, A006049, A006073, A045932-A045936, A045938.

Flatten[Position[Partition[PrimeNu[Range[300000]],9,1],?(Length[ Union[ #]] == 1&),{1},Heads->False]] (* _Harvey P. Dale, May 05 2015 *)

Offset corrected by Amiram Eldar, Oct 26 2019

A076252 Integers k such that omega(k) = omega(k-1) + omega(k-2) + omega(k-3), where omega(n) is the number of distinct prime factors of n.

Original entry on oeis.org

2310, 3990, 4290, 6090, 6270, 10010, 11550, 12810, 13650, 17094, 17940, 18270, 19380, 21930, 22110, 22770, 23100, 24990, 25410, 27300, 28644, 30090, 32214, 32604, 34034, 34314, 35340, 35880, 37310, 38190, 38570, 38640, 39270, 39780

Offset: 1

Joseph L. Pe, Nov 04 2002

nonn

			omega(2310) = 5 = 1 + 2 + 2 = omega(2309) + omega(2308) + omega(2307), so 2310 belongs to the sequence.

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

Cf. A001221, A006049, A076251, A076253.

omega[n_] := Length[FactorInteger[n]]; a = {}; Do[If[omega[n] == omega[n - 1] + omega[n - 2] + omega[n - 3], a = Append[a, n]], {n, 1, 10^5}]; a
Flatten[Position[Partition[PrimeNu[Range[40000]],4,1],?(#[[4]] == Total[ Take[ #,3]]&), {1}, Heads->False]]+3 (* _Harvey P. Dale, Oct 31 2016 *)

A322840 Positive integers n with fewer prime factors (counted with multiplicity) than n + 1.

Original entry on oeis.org

1, 3, 5, 7, 11, 13, 15, 17, 19, 23, 26, 29, 31, 35, 37, 39, 41, 43, 47, 49, 51, 53, 55, 59, 61, 62, 63, 65, 67, 69, 71, 73, 74, 77, 79, 83, 87, 89, 91, 95, 97, 99, 101, 103, 107, 109, 111, 113, 115, 119, 123, 125, 127, 129, 131, 134, 137, 139, 143, 146, 149

Offset: 1

Gus Wiseman, Dec 28 2018

nonn

			49 = 7*7 has two prime factors, while 50 = 2*5*5 has three, so 49 belongs to the sequence.

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

Cf. A001222, A006049, A045920, A294277, A294278, A302242, A322837, A322838, A322839.

Select[Range[100],PrimeOmega[#]?(#[[1]]< #[[2]]&),1,Heads->False]//Flatten (* _Harvey P. Dale, Sep 23 2021 *)

isok(n) = bigomega(n) < bigomega(n+1); \\ Michel Marcus, Dec 29 2018

cp's OEIS Frontend

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

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A045934 Numbers n such that n through n+5 have the same number of distinct prime factors.

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A045935 Numbers n such that n through n+6 are divisible by the same number of distinct primes.

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A321489 Numbers m such that both m and m+1 have at least 7 distinct prime factors.

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A348345 Number k such that k and k+1 have the same positive number of noninfinitary divisors (A348341).

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A355710 Numbers k such that k and k+1 have the same number of 5-smooth divisors.

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A045936 Numbers n such that n through n+7 are divisible by the same number of distinct primes.

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A045937 Numbers n such that n through n+8 are divisible by the same number of distinct primes.

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