A002473 -id:A002473 - cp's OEIS frontend

A106600 Number of positive integers <= 10^n that are divisible by no prime exceeding 7.

1, 10, 46, 141, 338, 694, 1273, 2155, 3427, 5194, 7575, 10688, 14672, 19674, 25861, 33406, 42487, 53302, 66061, 80988, 98311, 118271, 141124, 167139, 196597, 229785, 267007, 308575, 354820, 406074, 462692, 525030, 593467, 668384, 750177, 839260

Offset: 0

Robert G. Wilson v, May 27 2005

nonn

Position of 10^n among the 7-smooth numbers (A002473). Note that all powers of 10 are in A002473. - Zak Seidov, Nov 18 2013

			A002473(a(2)) = A002473(46)=100.

Zak Seidov, Table of n, a(n) for n = 0..200

Row 4 of A253635.

Cf. A002473, A011557.

n = 35; t = Select[ Flatten[ Table[ 7^d*Select[ Flatten[ Table[ 5^c*Select[ Flatten[ Table[2^a*3^b, {a, 0, n*Log[2, 10]}, {b, 0, n*Log[3, 10]}]], # <= 10^n &], {c, 0, n*Log[5, 10]}]], # <= 10^n &], {d, 0, n*Log[5, 10]}]], # <= 10^n &]; Table[ Length[ Select[t, # <= 10^n &]], {n, 0, 35}]

A252492 The largest prime factor of n*(n+1) equals 17. (Related to the abc conjecture.)

Original entry on oeis.org

16, 17, 33, 34, 50, 51, 84, 119, 135, 153, 169, 220, 255, 272, 288, 374, 441, 560, 594, 714, 832, 935, 1088, 1155, 1224, 1274, 1700, 2057, 2430, 2499, 2600, 4913, 5831, 12375, 14399, 28560, 31212, 37179, 194480, 336140

Offset: 1

M. F. Hasler, Jan 16 2015

Equivalently, the prime factors of n and n+1 are not larger than 17, but not all smaller than 17 (in which case n is in A252493).
This sequence is complete by a theorem of Stormer, cf. A002071 and sequences A085152, A085153, A252494, A252493.
This is row 7 of A145605. It has A145604(7)=40 terms and ends with A002072(7)=336140.

OEIS Index entries for sequences related to the abc conjecture

Cf. A002473, A086247, A252493, A252494, A085153, A085152.

Select[Range[345678], FactorInteger[ # (# + 1)][[ -1,1]] == 17 &]

for(n=1,9e6,vecmax(factor(n++)[,1])<18 && vecmax(factor(n*n--)[,1])==17 && print1(n",")) \\ Skips 2 if n+1 is not 17-smooth: Twice as fast as the naïve version.

A350186 Numbers of multiplicative persistence 7 which are themselves the product of digits of a number.

Original entry on oeis.org

338688, 826686, 2239488, 3188646, 6613488, 14224896, 3416267673274176, 6499837226778624

Offset: 1

Daniel Mondot, Jan 15 2022

The multiplicative persistence of a number mp(n) is the number of times the product of digits function p(n) must be applied to reach a single digit, i.e., A031346(n).
The product of digits function partitions all numbers into equivalence classes. There is a one-to-one correspondence between values in this sequence and equivalence classes of numbers with multiplicative persistence 8.
There are infinitely many numbers with mp of 1 to 11, but the classes of numbers (p(n)) are postulated to be finite for sequences A350181....
Equivalently:
This sequence consists of the numbers A007954(k) such that A031346(k) = 8,
These are the numbers k in A002473 such that A031346(k) = 7,
Or:
- they factor into powers of 2, 3, 5 and 7 exclusively.
- p(n) goes to a single digit in 7 steps.
Postulated to be finite and complete.
a(9), if it exists, is > 10^20000, and likely > 10^119000.

			338688 is in this sequence because:
- 338688 goes to a single digit in 7 steps: p(338688) = 27648, p(27648) = 2688, p(2688)=768, p(768)=336, p(336)=54, p(54)=20, p(20)=0.
- p(4478976) = p(13477889) = 338688, etc.

Daniel Mondot, Multiplicative Persistence Tree
Eric Weisstein's World of Mathematics, Multiplicative Persistence

Cf. A002473, A003001 (smallest number with multiplicative persistence n), A031346 (multiplicative persistence), A031347 (multiplicative digital root), A046516 (all numbers with mp of 7).

Cf. A350180, A350181, A350182, A350183, A350184, A350185, A350187 (numbers with mp 1 to 6 and 8 to 10 that are themselves 7-smooth numbers).

mx=10^16;lst=Sort@Flatten@Table[2^i*3^j*5^k*7^l,{i,0,Log[2,mx]},{j,0,Log[3,mx/2^i]},{k,0,Log[5,mx/(2^i*3^j)]},{l,0,Log[7,mx/(2^i*3^j*5^k)]}];
Select[lst,Length@Most@NestWhileList[Times@@IntegerDigits@#&,#,#>9&]==7&]  (* code for 7-smooth numbers from A002473. - Giorgos Kalogeropoulos, Jan 16 2022 *)

A080672 Numbers having divisors 2 or 3 or 5 or 7.

Original entry on oeis.org

2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28, 30, 32, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 46, 48, 49, 50, 51, 52, 54, 55, 56, 57, 58, 60, 62, 63, 64, 65, 66, 68, 69, 70, 72, 74, 75, 76, 77, 78, 80, 81, 82, 84, 85, 86, 87, 88, 90, 91, 92, 93

Offset: 1

Cino Hilliard, Mar 02 2003

A020639(a(n)) <= 7; A210679(a(n)) > 0. - Reinhard Zumkeller, Apr 02 2012

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, order 162.

Cf. A020639, A008364 (complement).

Subsequences: A002473, A343597.

a080672 n = a080672_list !! (n-1)
a080672_list = filter ((<= 7) . a020639) [2..]
-- Reinhard Zumkeller, Apr 02 2012

Select[Range[100],Length[Intersection[Divisors[#],{2,3,5,7}]]>0&] (* Harvey P. Dale, Apr 03 2024 *)

div2357(n)= for(x=1,n, if(gcd(x,210)<>1,print1(x" ")) )

is(n)=gcd(n,210)>1 \\ Charles R Greathouse IV, Sep 14 2015

From Charles R Greathouse IV, Sep 14 2015: (Start)
a(n) = 35n/27 + O(1).
For n > 162, a(n) = a(n-162) + 210. [Corrected by Peter Munn, Apr 22 2021]
(End)
For n < 162, a(n) = 210 - a(162-n). - Peter Munn, Apr 22 2021

Offset fixed by Reinhard Zumkeller, Apr 02 2012

A097220 Numbers n such that pi(n) = product of digits of n.

Original entry on oeis.org

16, 17, 63, 73, 364, 437, 545, 573, 963, 6475, 23797, 67458, 2475989, 2475998

Offset: 1

Farideh Firoozbakht, Aug 02 2004

The only numbers with the property that pi(n) = sum of the digits of n, are the three numbers 15, 27 & 39.
When n exceeds approximately 10^44, then pi(n) is consistently greater than the product of digits of n. So no term of this sequence exceeds 10^44. In particular, this sequence is finite. - Jeppe Stig Nielsen, Nov 04 2018
Products of digits of terms are in A002473. Term by term up to some bound (such that the bounds on primes hold), one could check terms t in A002473 on some known bounds. See example below. - David A. Corneth, Nov 06 2018
There are no other terms below 10^17. - Max Alekseyev, Nov 07 2024

			2475998 is in the sequence because pi(2475998)=2*4*7*5*9*9*8.
1152 is in A002473. As 8643 <= prime(1152) <= 9794. Examples of the 13 numbers with product of digits is 1152 in that interval are: 8944, 9288, 9448, 9484 none of which are terms. - _David A. Corneth_, Nov 06 2018

Cf. A002473, A007954, A000720.

Cf. A097221, A097222, A097223.

[n: n in [1..10^5] | &*Intseq((n)) eq #PrimesUpTo(n)]; // Vincenzo Librandi, Nov 06 2018

v={}; Do[If[h=IntegerDigits[n]; l=Length[h]; p=Product[h[[k]], {k, l}]; PrimePi[n]==p, v=Append[v, n]; Print[v], If[Mod[n, 1000000]==0, Print[ -n]]], {n, 200000000}]
Select[Range[2500000],PrimePi[#]==Times@@IntegerDigits[#]&] (* Harvey P. Dale, Dec 04 2012 *)

isok(n) = primepi(n) == factorback(digits(n)); \\ Michel Marcus, Apr 23 2018

Keyword fini from Jeppe Stig Nielsen, Nov 04 2018

A210679 Number of distinct prime factors <= 7 of n.

Original entry on oeis.org

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

Offset: 1

Reinhard Zumkeller, Apr 01 2012

Periodic with period length 210. - Amiram Eldar, Sep 16 2023

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (-3,-5,-6,-6,-5,-3,0,3,5,6,6,5,3,1).

Cf. A001221, A002473, A008364, A027748, A080672, A165743.

Number of distinct prime factors <= p: A171182 (p=3), A178146 (p=5), this sequence (p=7).

a210679 = length . takeWhile (<= 7) . a027748_row

Join[{0},Table[Count[FactorInteger[n][[All,1]],?(#<8&)],{n,2,100}]] (* _Harvey P. Dale, Aug 18 2021 *)
a[n_] := PrimeNu[GCD[n, 210]]; Array[a, 100] (* Amiram Eldar, Sep 16 2023 *)

a(n) = omega(gcd(n, 210)); \\ Amiram Eldar, Sep 16 2023

a(n) <= 4.
a(A008364(n)) = 0; a(A080672(n)) > 0.
a(n) = A001221(n) iff n is 7-smooth: a(A002473(n)) = A001221(A002473(n)). [corrected by Amiram Eldar, Sep 16 2023]
From Amiram Eldar, Sep 16 2023: (Start)
Additive with a(p^e) = 1 if p <= 7, and 0 otherwise.
a(n) = A001221(A165743(n)).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 247/210. (End)

A337741 Numbers all of whose divisors are Niven numbers (A005349).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 18, 20, 21, 24, 27, 36, 40, 54, 63, 72, 81, 108, 162, 216, 243, 324, 486, 648, 972, 1944

Offset: 1

Amiram Eldar, Sep 17 2020

Since the only prime Niven numbers are the single-digit primes 2, 3, 5 and 7, all the terms are 7-smooth numbers (A002473).

If k is a term, all the divisors of k are also terms. Since all the terms are 7-smooth, every term is of the form p * k, where p is in {2, 3, 5, 7} and k is a smaller term. Thus it is easy to verify that there are only 31 terms in this sequence, and 1944 being the last term.

			6 is a term since all the divisors of 6, i.e., 1, 2, 3 and 6, are Niven numbers.

Subsequence of A002473 and A005349.
Cf. A332268, A335708, A335709.
Similar sequences: A062687, A190217, A329419.

nivenQ[n_] := Divisible[n, Plus @@ IntegerDigits[n]]; allQ[n_] := AllTrue[Divisors[n], nivenQ]; p = {1, 2, 3, 5, 7}; s = {1}; n = 0; While[Length[s] != n, n = Length[s]; s = Select[Union @ Flatten @ Outer[Times, s, p], allQ]]; s

A086288 Number of distinct prime factors of 7-smooth numbers.

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 1, 1, 1, 2, 2, 2, 2, 1, 2, 2, 2, 2, 1, 1, 2, 3, 1, 2, 2, 2, 3, 2, 2, 1, 2, 2, 2, 3, 2, 1, 3, 2, 2, 2, 1, 3, 3, 2, 2, 2, 3, 2, 2, 3, 1, 3, 1, 2, 3, 2, 2, 3, 2, 2, 3, 2, 3, 2, 2, 2, 2, 4, 2, 2, 2, 3, 1, 2, 2, 3, 1, 3, 3, 2, 3, 3, 3, 2, 2, 3, 1, 3, 3, 2, 3, 2, 2, 2, 2, 4, 2, 2, 2, 3

Offset: 1

Reinhard Zumkeller, Jul 15 2003

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from G. C. Greubel)

Cf. A001221, A002473, A086289, A086290.

PrimeNu[Select[Range[500], Max[Transpose[FactorInteger[#]][[1]]] <= 7 &]] (* G. C. Greubel, May 09 2017 *)

a(n) = A001221(A002473(n)).

a(n) <= 4.

A086289 Total number of prime factors of 7-smooth numbers.

Original entry on oeis.org

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

Offset: 1

Reinhard Zumkeller, Jul 15 2003

nonn

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

Cf. A001222, A002473, A086288, A086290, A086291.

PrimeOmega[Select[Range[500], Max[Transpose[FactorInteger[#]][[1]]] <= 7 &]] (* Amiram Eldar, Jan 06 2020 after G. C. Greubel at A086288 *)

a(n) = A001222(A002473(n)).

A086290(n) <= A086291(n) <= a(n).

A218013 Numbers that divide the product of the nonzero digits (in base 10) of their square.

Original entry on oeis.org

1, 2, 3, 5, 6, 8, 28, 36, 42, 54, 75, 192, 216, 288, 486, 525, 648, 768, 864, 882, 1728, 2160, 3024, 6048, 6075, 7056, 7680, 17280, 18144, 20736, 30240, 40824, 56448, 60480, 61236, 62208, 64512, 84672, 122472, 138915, 150528, 387072, 408240, 497664, 622080

Offset: 1

Nels Olson, Oct 18 2012

			For n=5, n^2 is 25; the product of the digits of 25 is 2 * 5 = 10, which is divisible by n=5.

David A. Corneth, Table of n, a(n) for n = 1..10919 (first 473 terms from Chai Wah Wu; terms <= 10^100)

Cf. A002473.

Related to A218072. Subsets of this sequence include A218029 and A218030.

isok(n) = digs = digits(n^2); (prod(i=1, #digs, if (digs[i], digs[i], 1)) % n) == 0; \\ Michel Marcus, Aug 12 2013

from operator import mul
from functools import reduce
from gmpy2 import t_mod, mpz
A218013 = [n for n in range(1,10**6) if not t_mod(reduce(mul,(mpz(d) for d in str(n**2) if d != '0')),n)] # Chai Wah Wu, Aug 23 2014

cp's OEIS Frontend

A106600 Number of positive integers <= 10^n that are divisible by no prime exceeding 7.

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A252492 The largest prime factor of n*(n+1) equals 17. (Related to the abc conjecture.)

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A350186 Numbers of multiplicative persistence 7 which are themselves the product of digits of a number.

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A080672 Numbers having divisors 2 or 3 or 5 or 7.

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A097220 Numbers n such that pi(n) = product of digits of n.

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A210679 Number of distinct prime factors <= 7 of n.

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A337741 Numbers all of whose divisors are Niven numbers (A005349).

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A086288 Number of distinct prime factors of 7-smooth numbers.

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