A051270 -id:A051270 - cp's OEIS frontend

A364265 The first term in a chain of at least 3 consecutive numbers each with exactly 6 distinct prime factors (i.e., belonging to A074969).

323567034, 431684330, 468780388, 481098980, 577922904, 639336984, 715008644, 720990620, 726167154, 735965384, 769385252, 808810638, 822981560, 831034918, 839075510, 847765554, 879549670, 895723268, 902976710, 903293468, 904796814, 918520420, 940737005, 944087484, 982059364

Offset: 1

R. J. Mathar, Jul 16 2023

nonn

To distinguish this from A259349: "Numbers n with exactly k distinct prime factors" means numbers with A001221(n) = omega(n) = k, which specifies that in the prime factorization n = Product_{i>=1} p_i^(e_i), e_i >= 1, the exponents are ignored, and only the size of the set of the (distinct) p_i is considered. In A259349, the numbers n are products of k distinct primes, which means in the prime factorization of n, all exponents e_i are equal to 1. (If all exponents e_i = 1, the n are squarefree, i.e., in A005117.) Rephrased: the n which are products of k distinct primes have A001221(n) = omega(n) = A001222(n) = bigomega(n) = k, whereas the n which have exactly k distinct prime factors are the superset of (weaker) requirement A001221(n) = omega(n) = k. - R. J. Mathar, Jul 18 2023

David A. Corneth, Table of n, a(n) for n = 1..10000

Cf. A259349 (requires squarefree). Subsequence of A273879.
Cf. A364266 (5 distinct factors).
See also A001221, A001222, A005117.
Numbers divisible by d distinct primes: A246655 (d=1), A007774 (d=2), A033992 (d=3), A033993 (d=4), A051270 (d=5), A074969 (d=6), A176655 (d=7), A348072 (d=8), A348073 (d=9).

omega := proc(n)
    nops(numtheory[factorset](n)) ;
end proc:
for k from 1 do
    if omega(k) = 6 then
        if omega(k+1) = 6 then
            if omega(k+2) = 6 then
                print(k) ;
            end if;
        end if;
    end if;
end do:

upto(n) = {my(res = List(), streak = 0); forfactored(i = 2, n, if(#i[2]~ == 6, streak++; if(streak >= 3, listput(res, i[1] - 2)), streak = 0)); res} \\ David A. Corneth, Jul 18 2023

a(1) = A138206(3).

{k: A001221(k) = A001221(k+1) = A001221(k+2) = 6}.

More terms from David A. Corneth, Jul 18 2023

A046391 Odd numbers with exactly 5 distinct prime factors.

Original entry on oeis.org

15015, 19635, 21945, 23205, 25935, 26565, 31395, 33495, 33915, 35805, 36465, 39585, 40755, 41055, 42315, 42735, 45885, 47355, 49335, 49665, 50505, 51051, 51765, 53295, 54285, 55335, 55965, 57057, 57855, 58695, 61215, 61845, 62205

Offset: 1

Patrick De Geest, Jun 15 1998

nonn

			50505 = 3 * 5 * 7 * 13 * 37.

Karl-Heinz Hofmann, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)

Cf. A046318, A046407.

Intersection of A051270 and A005408.

isA046391 := proc(n)
    type(n,'odd') and (A001221(n) = 5 ) ;
end proc:
for n from 1 do
    if isA046391(n) then
        print(n);
    end if;
end do: # R. J. Mathar, Nov 10 2014

f[n_]:=Last/@FactorInteger[n]=={1,1,1,1,1}&&FactorInteger[n][[1,1]]>2; lst={};Do[If[f[n],AppendTo[lst,n]],{n,9!}];lst (* Vladimir Joseph Stephan Orlovsky, Nov 23 2009 *)

from sympy import primefactors, factorint
print([n for n in range(1, 100000, 2) if len(primefactors(n)) == 5 and max(list(factorint(n).values())) < 2]) # Karl-Heinz Hofmann, Mar 01 2023

A323056 Numbers with exactly five distinct exponents in their prime factorization, or five distinct parts in their prime signature.

Original entry on oeis.org

174636000, 206388000, 244490400, 261954000, 269892000, 274428000, 288943200, 291060000, 301644000, 309582000, 343980000, 349272000, 365148000, 366735600, 377848800, 383292000, 404838000, 411642000, 412776000, 422301600, 433414800, 449820000, 452466000, 457380000

Offset: 1

Gus Wiseman, Jan 03 2019

nonn

The first term is A006939(5) = 174636000.
Positions of 5's in A071625.
Numbers k such that A001221(A181819(k)) = 5.

			174636000 = 2^5 * 3^4 * 5^3 * 7^2 * 11^1 has five distinct exponents so belongs to the sequence.

David A. Corneth, Table of n, a(n) for n = 1..10000

One distinct exponent: A062770 or A072774.
Two distinct exponents: A323055.
Three distinct exponents: A323024.
Four distinct exponents: A323025.
Five distinct exponents: A323056.
Cf. A001221, A001222, A006939, A051270, A059404, A071625, A118914, A181819, A182855, A323014, A323022, A323024.

Select[Range[300000000],Length[Union[Last/@FactorInteger[#]]]==5&]

is(n) = #Set(factor(n)[, 2]) == 5 \\ David A. Corneth, Jan 12 2019

a(13)-a(24) from Daniel Suteu, Jan 12 2019

A046395 Palindromes that are the product of 5 distinct primes.

Original entry on oeis.org

6006, 8778, 20202, 28182, 41514, 43134, 50505, 68586, 87978, 111111, 141141, 168861, 202202, 204402, 209902, 246642, 249942, 262262, 266662, 303303, 323323, 393393, 399993, 438834, 454454, 505505, 507705, 515515, 516615, 519915, 534435, 535535, 543345

Offset: 1

Patrick De Geest, Jun 15 1998

No exponent of the distinct prime factors can be greater than one, i.e., no prime powers are permitted. - Harvey P. Dale, Apr 09 2021 at the suggestion of Sean A. Irvine

See A373465 for the similar sequence where only distinct prime divisors are counted, but may occur to higher powers. - M. F. Hasler, Jun 06 2024

			505505 = 5 * 7 * 11 * 13 * 101.

Harvey P. Dale, Table of n, a(n) for n = 1..500

Cf. A002113 (palindromes), A051270 (omega(.) = 5).

Cf. A046331 (palindromes with 5 prime factors counted with multiplicity), A373465 (counting only distinct prime divisors).

Select[Range[550000],PalindromeQ[#]&&PrimeNu[#]==PrimeOmega[#]==5&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Apr 09 2021 *)

Intersection of A002113 and A046387.

Corrected at the suggestion of Sean A. Irvine by Harvey P. Dale, Apr 09 2021

Name edited to avoid confusion by M. F. Hasler, Jun 06 2024

A064040 Integers whose number of distinct prime divisors is prime.

Original entry on oeis.org

6, 10, 12, 14, 15, 18, 20, 21, 22, 24, 26, 28, 30, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 46, 48, 50, 51, 52, 54, 55, 56, 57, 58, 60, 62, 63, 65, 66, 68, 69, 70, 72, 74, 75, 76, 77, 78, 80, 82, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95, 96, 98, 99, 100, 102, 104, 105

Offset: 1

Lior Manor, Aug 23 2001

For all terms below 210 this sequence and A024619 are identical.

			210 = 2*3*5*7 has 4 prime factors, hence it is not here, but it is part of A024619.

Harry J. Smith, Table of n, a(n) for n = 1..1000

Cf. A001221, A024619, A000977, A033992, A007774, A033993, A051270, A063989.

q:= n-> isprime(nops(ifactors(n)[2])):
select(q, [$1..210])[];  # Alois P. Heinz, Apr 18 2024

Select[Range[200], PrimeQ[PrimeNu[#]] &] (* Paolo Xausa, Mar 28 2024 *)

n=0; for (m=1, 10^9, if (isprime(omega(m)), write("b064040.txt", n++, " ", m); if (n==1000, break))) \\ Harry J. Smith, Sep 06 2009

is(n)=isprime(omega(n)) \\ Charles R Greathouse IV, Sep 18 2015

Edited by Charles R Greathouse IV, Mar 18 2010

Name edited by Michel Marcus, Oct 16 2023

A136154 Composites one larger than a prime, with exactly five distinct prime factors.

Original entry on oeis.org

2310, 2730, 3990, 4290, 6090, 6270, 7590, 7854, 8610, 8970, 9030, 9240, 9282, 9690, 10010, 10710, 11550, 11970, 12012, 12540, 12810, 13110, 13260, 13398, 13650, 13860, 14322, 14490, 14630, 15330, 15810, 15960, 16302, 16422, 16530, 16830

Offset: 1

Enoch Haga, Dec 16 2007

			a(0)=2310 because 2310 follows the prime 2309 and has five factors 2, 3, 5, 7 and 11.

Cf. A136151, A136152, A136153, A136155.

isok(n) = (omega(n)==5) && isprime(n-1); \\ Michel Marcus, Jun 08 2014

Equals A008864 INTERSECT A051270. - R. J. Mathar, Feb 20 2008

Edited by R. J. Mathar, Feb 20 2008

Typo in a(36) corrected by Seth A. Troisi, May 13 2022

A373465 Palindromes with exactly 5 distinct prime divisors.

Original entry on oeis.org

6006, 8778, 20202, 28182, 40404, 41514, 43134, 50505, 60606, 63336, 66066, 68586, 80808, 83538, 86268, 87978, 111111, 141141, 168861, 171171, 202202, 204402, 209902, 210012, 212212, 219912, 225522, 231132, 232232, 239932, 246642, 249942, 252252, 258852, 262262, 266662, 272272

Offset: 1

M. F. Hasler, Jun 06 2024

			a(1) = 6006 = 2 * 3 * 7 * 11 * 13 is a palindrome (A002113) with 5 prime divisors.
a(5) = 40404 = 2^2 * 3 * 7 * 13 * 37 also is a palindrome with 5 prime divisors, although the divisor 2 occurs twice as a factor in the factorization.

Cf. A002113 (palindromes), A051270 (omega(.) = 5).

Cf. A046331 (same but counting prime factors with multiplicity), A046395 (same but squarefree), A373466 (same with omega = 6), A373467 (with omega = 7).

Select[Range[300000],PalindromeQ[#]&&Length[FactorInteger[#]]==5&] (* James C. McMahon, Jun 08 2024 *)
Select[Range[300000],PalindromeQ[#]&&PrimeNu[#]==5&] (* Harvey P. Dale, Sep 01 2024 *)

A373465_upto(N, start=1, num_fact=5)={ my(L=List()); while(N >= start = nxt_A002113(start), omega(start)==num_fact && listput(L, start)); L}

Intersection of A002113 and A051270.

A348266 k-digit numbers whose digit(s) are the number of distinct prime factors in each of the preceding k integers.

Original entry on oeis.org

22, 313, 2232, 2323, 2333, 32215, 432152, 2434332, 4222423, 43332543, 332325334, 2535332433, 4532543535234, 5435433351423

Offset: 1

Metin Sariyar, Oct 09 2021

a(12) <= 2535332433. - David A. Corneth, Oct 10 2021

a(12) >= 10^9. - Michel Marcus, Oct 11 2021

			22 is a term because omega(20) = 2 and omega(21) = 2, whose concatenation is 22.
313 is a term because preceding it omega(310) = 3, omega(311) = 1 and omega(312) = 3, and their concatenation is 313.
32215 is a term because, the number of distinct prime divisors of 32210, 32211, 32212, 32213 and 32214 are 3, 2, 2, 1, 5 and their ordered concatenation gives the next number 32215.

Cf. A001221, A000961, A007774, A033992, A033993, A051270, A074969, A176655.

Cf. A323083.

Select[Range[33000], FromDigits[PrimeNu /@ (# - Range[IntegerLength[#], 1, -1])] == # &] (* Amiram Eldar, Oct 09 2021 *)

isok(m) = {my(s="", k=m, i=1); while(1, s = concat(s, Str(omega(k))); if (eval(s) == m+i, return (i)); if (eval(s) > m+i, return(0)); k++; i++;);}
lista(nn) = my(nb); for(n=1, nn, if (nb=isok(n), print1(n+nb, ", "))); \\ Michel Marcus, Oct 09 2021

a(8)-a(9) from Amiram Eldar, Oct 09 2021
a(10)-a(11) from Michel Marcus, Oct 10 2021
a(12) confirmed by Martin Ehrenstein, Oct 28 2021
a(13)-a(14) from Martin Ehrenstein, Oct 30 2021

A324206 Numbers with exactly six distinct exponents in their prime factorization, or six distinct parts in their prime signature.

Original entry on oeis.org

5244319080000, 6197831640000, 6857955720000, 7342046712000, 7664774040000, 7866478620000, 8241072840000, 8676964296000, 8740531800000, 9278410680000, 9296747460000, 9578467080000, 9601138008000, 10286933580000, 10329719400000, 10488638160000, 10598658840000, 10705345560000

Offset: 1

David A. Corneth, Feb 17 2019

			6197831640000 = 2^6 * 3^5 * 5^4 * 7^3 * 11 * 13^2 is in the sequence as there are 6 distinct exponents; 1 through 6.

David A. Corneth, Table of n, a(n) for n = 1..10000

Cf. A001221, A001222, A006939, A051270, A059404, A071625, A118914, A181819, A182855, A323014, A323022, A323024, A323025, A323056.

PARI
```
is(n) = #Set(factor(n)[, 2]) == 6
```

A324207 Numbers with exactly seven distinct exponents in their prime factorization, or seven distinct parts in their prime signature.

Original entry on oeis.org

2677277333530800000, 2992251137475600000, 3164055030536400000, 3501054974617200000, 3536296798834800000, 3622198745365200000, 3748188266943120000, 4015916000296200000, 4189151592465840000, 4207150095548400000, 4280780335431600000, 4373290124002800000, 4429677042750960000

Offset: 1

David A. Corneth, Feb 17 2019

			2677277333530800000 = 2^7 * 3^6 * 5^5 * 7^4 * 11^3 * 13^2 * 17 is in the sequence. There are exactly 7 distinct exponents; 1 through 7 in it.

David A. Corneth, Table of n, a(n) for n = 1..10000

Cf. A001221, A001222, A006939, A051270, A059404, A071625, A118914, A181819, A182855, A323014, A323022, A323024, A323025, A323056.

PARI
```
is(n) = #Set(factor(n)[, 2]) == 7
```

cp's OEIS Frontend

A364265 The first term in a chain of at least 3 consecutive numbers each with exactly 6 distinct prime factors (i.e., belonging to A074969).

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A046391 Odd numbers with exactly 5 distinct prime factors.

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A323056 Numbers with exactly five distinct exponents in their prime factorization, or five distinct parts in their prime signature.

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A046395 Palindromes that are the product of 5 distinct primes.

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A064040 Integers whose number of distinct prime divisors is prime.

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A136154 Composites one larger than a prime, with exactly five distinct prime factors.

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A373465 Palindromes with exactly 5 distinct prime divisors.

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Mathematica