A097227 -id:A097227 - cp's OEIS frontend

A098683 Numbers n such that pi(n) = prime(d_1)prime(d_2)...*prime(d_k) where d_1 d_2 ... d_k is the decimal expansion of n.

123, 5224, 11166, 51174, 172451, 546322, 14355351, 23539612, 23539621, 24322837, 122924349, 4575242147, 42256772524, 283186883151, 623286236455, 665318971119, 665318971191, 5257788212426, 27452719198281, 273643846355134, 787812731751347, 787812731751374

Offset: 1

Farideh Firoozbakht, Sep 24 2004

a(n) must necessarily be a zeroless number, i.e., the sequence is a subsequence of A052382. - Chai Wah Wu, Mar 04 2019

			122924349 is in the sequence because pi(122924349) = P(1)*P(2)*P(2)*P(9)*P(2)*P(4)*P(3)*P(4)*P(9) where P(i) is i-th prime.

Cf. A052382, A097227, A098684, A098685, A160040.

Do[d=IntegerDigits[n];k=Length[d];If[ !MemberQ[d, 0]&&PrimePi[n]==Product[Prime[d[[j]]], {j, k}], Print[n]], {n, 230000000}]

Entries corrected by Robert G. Wilson v, May 04 2009
a(13)-a(17) from Donovan Johnson, Jul 12 2010
a(18) from Giovanni Resta, Apr 01 2017
a(19) from Chai Wah Wu, Mar 05 2019
a(20)-a(22) from Chai Wah Wu, Mar 06 2019

A097228 Numbers n such that the product of digits of n equals the concatenation of pi(d)'s where d runs through the digits of n.

Original entry on oeis.org

27, 38, 127, 138, 289, 298, 1127, 1138, 1289, 1298, 11127, 11138, 11289, 11298, 111127, 111138, 111289, 111298, 1111127, 1111138, 1111289, 1111298, 11111127, 11111138, 11111289, 11111298, 111111127, 111111138, 111111289, 111111298

Offset: 1

Farideh Firoozbakht, Aug 15 2004

This sequence is infinite because if n is in the sequence then the number with an arbitrary number of 1's in front of n is also in the sequence. Are 27, 38, 289 and 298 the only nontrivial terms (i.e., terms whose first digit is not 1) in this sequence? The next term is greater than 3*10^8.

There are no more nontrivial terms; i.e., all terms in the sequence are 27, 38, 289, 298 prepended with zero or more 1's. To see this, note that a nontrivial term must have at most 21 digits since 9^22 < 10^21, i.e., has 21 digits. Searching through all numbers in A009994 of at most 21 digits that do not start with 1 shows that there are no more nontrivial terms. - Chai Wah Wu, Aug 10 2017

			298 is in the sequence because 2*9*8 = 144 = concatenate(pi(2), pi(9), pi(8)) = concatenate(1, 4, 4).

Chai Wah Wu, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (0, 0, 0, 11, 0, 0, 0, -10).

Cf. A097227.

h[a_]:=(v1={};Do[l=Length[a];v1=Join[v1, IntegerDigits[a[[n]]]], {n, l}]; FromDigits[v1]);v={};Do[h1=IntegerDigits[n];l=Length[h1]; p=Product[h1[[k]], {k, l}];s=Sum[h1[[k]], {k, l}];If[p>0&& p==h[PrimePi[h1]], v=Append[v, n];Print[v]], {n, 300000000}]
LinearRecurrence[{0,0,0,11,0,0,0,-10},{27,38,127,138,289,298,1127,1138,1289,1298},30] (* Harvey P. Dale, Jan 01 2019 *)

from _future_ import division
A097228_list = [27,38] + [1000*(10**k-1)//9+d for k in range(20) for d in [127,138,289,298]] # Chai Wah Wu, Aug 10 2017

From Chai Wah Wu, Aug 10 2017: (Start)
a(n) = 11*a(n-4) - 10*a(n-8) for n > 10.
G.f.: x*(-1600*x^9 - 1620*x^8 - 380*x^7 - 270*x^6 - 120*x^5 - 8*x^4 + 138*x^3 + 127*x^2 + 38*x + 27)/(10*x^8 - 11*x^4 + 1). (End)

A099068 Numbers n such that n=P(d_1)P(d_2)...*P(d_k)+(P(d_1)+P(d_2)+...+P(d_k)) where d_1 d_2 ... d_k is the decimal expansion of n and P(i) is the i-th prime.

Original entry on oeis.org

23, 119, 428, 918, 1637682, 652827658771

Offset: 1

Farideh Firoozbakht, Oct 29 2004

There is no other term up to 15000000.
a(7) > 10^12. [Donovan Johnson, Mar 26 2010]
There are no other terms < 10^44. - Chai Wah Wu, Aug 12 2017

			1637682 is in the sequence because 1637682=
P(1)*P(6)*P(3)*P(7)*P(6)*P(8)*P(2)+(P(1)+P(6)+P(3)+P(7)+P(6)+P(8)+P(2)).

C. Rivera, Puzzle 279The Prime Puzzles & Problems connection.

Cf. A099067, A099069, A097223, A097227.

Do[h=IntegerDigits[n];l=Length[h];If[ !MemberQ[h, 0]&&n==Product[Prime[h[[k]]], {k, l}]+Sum[Prime[h[[k]]], {k, l}], Print[n]], {n, 15000000}]

Definition corrected by D. S. McNeil, Mar 14 2009

a(6) from Donovan Johnson, Mar 26 2010

A099069 Numbers n such that n = prime(d_1d_2...*d_k) - phi(d_1 + d_2 + ... + d_k) where d_1 d_2 ... d_k is the decimal expansion of n.

Original entry on oeis.org

1, 2, 3, 19, 35497

Offset: 1

Farideh Firoozbakht, Oct 29 2004

Sequence is finite since prime(d_1*d_2*...*d_k) <= prime(9^k) <= 9^k(k log 9 + log k + log log 9) < 10^(k-1) for large enough k, i.e., it will have fewer than k digits. In particular, a(n) < 10^69. - Chai Wah Wu, Aug 10 2017

			35497 is in the sequence because 35497 = prime(3*5*4*9*7) - phi(3 + 5 + 4 + 9 + 7).

C. Rivera, Puzzle 279The Prime Puzzles & Problems connection.

Cf. A097223, A097227, A099067, A099068.

Do[h=IntegerDigits[n];l=Length[h];If[ !MemberQ[h, 0]&&n==Prime[Product[h[[k]], {k, l}]]-EulerPhi[Sum[h[[k]], {k, l}]], Print[n]], {n, 6000000}]

A110070 Numbers n such that n=pi(d_1!d_2!...*d_k!) where d_1 d_2 ... d_k is the decimal expansion of n.

Original entry on oeis.org

0, 3, 34, 52, 2800414

Offset: 1

Farideh Firoozbakht, Jul 22 2005

No other terms below 10^15. - Max Alekseyev, Jul 21 2024

			2800414 is in the sequence because 2800414=pi(2!*8!*0!*0!*4!*1!*4!).

Cf. A110071, A110072, A097220, A097223, A097227, A098683, A098684, A098685.

A290675 For a given n, the nonzero digits of n are the prime indices for the factorization of n.

Original entry on oeis.org

14, 154, 1196, 66079, 279174, 302768895, 2249805789

Offset: 1

Charles Ronco, Aug 08 2017

a(8) > 10^35, if it exists. - Giovanni Resta, Aug 09 2017
a(n) != 1 mod 10. a(8) > 10^44, if it exists. - Chai Wah Wu, Aug 10 2017
Fixed points of A113581. - Alois P. Heinz, May 11 2023

			14 = 2*7, which are the 1st and 4th primes. 154 = 2*11*7 which are the 1st, 5th, and 4th primes, respectively. So use the digits of n (excluding zero) to find the corresponding primes, and the product of those primes equals n.

Supersequence of A097227.

Cf. A113581.

x = 10^7; (* this number is the upper end of the search *) Do[If[n == Times @@ Prime /@ DeleteCases[RealDigits[n][[1]], 0], Print[n]], {n, x}] (* or *)
up = 3*^9; ric[n_, e_, k_] := Block[{m=n, j=0}, If[k == 10, If[Most@ DigitCount[n] == e, Print@n; Sow@n], While[m < up, ric[m, Append[e, j], k+1]; j++; m *= Prime[k] ]]]; Sort@ Reap[ric[1, {}, 1]][[2, 1]] (* faster, Giovanni Resta, Aug 09 2017 *)

is(n) = my(d=digits(n), prd=1); for(k=1, #d, if(d[k]!=0, prd=prd*prime(d[k]))); prd==n \\ Felix Fröhlich, Aug 09 2017

from functools import reduce
from operator import mul
from itertools import combinations_with_replacement
A290675_list, lmax, ptuple = [], 12, (2,3,5,7,11,13,17,19,23)
for l in range(1,lmax+1):
    for d in combinations_with_replacement(range(1,10),l):
        n = reduce(mul,(ptuple[i-1] for i in d))
        if n < 10**lmax and tuple(sorted((int(x) for x in str(n) if x != '0'))) == d:
            A290675_list.append(n) # Chai Wah Wu, Aug 10 2017

a(6)-a(7) from Giovanni Resta, Aug 09 2017

A318298 Numbers whose set of decimal digits coincides with the set of the indices of their prime factors.

Original entry on oeis.org

12, 14, 154, 1196, 14112, 21888, 53625, 226512, 279174, 358435, 821142, 1222452, 1665664, 2228814, 2454375, 2614248, 2872116, 4425729, 5751746, 8653645, 9551256, 15261246, 19427226, 19644898, 19775998, 21271488, 27676935, 29591892, 29956212, 41878242, 45574144

Offset: 1

Michel Lagneau, Aug 24 2018

It is impossible to find a number with 9 distinct decimal digits because the prime factors 2 and 5 generate d_k = 0.

The finite subsequence containing the smallest numbers having at least j distinct digits for j = 2, 3, ..., 8, is 12, 154, 53625, 279174, 19427226, 82447365 and 41762985264.

			1196 is in the sequence because the prime factors are {2, 13, 23} = {prime(1), prime(6), prime(9)}, and 1196 contains the decimal digits 1, 6, 9.

Giovanni Resta, Table of n, a(n) for n = 1..10000

Cf. A001221, A080683, A097227, A290675.

with(numtheory):nn:=10^8:
for n from 1 to nn do:
lst:={}:d:=factorset(n):n0:=nops(d):
q:=convert(n,base,10):n1:=nops(q):
p:=product(‘q[i]’, ‘i’=1..n1):
if p<>0
  then
  for i from 1 to n1 do :
   lst:=lst union {ithprime(q[i])}:
  od:
   if lst = d
    then
     print(n):
     else
     fi:fi:
od:

ok[n_] := Block[{f = First /@ FactorInteger[n], d}, Last@f < 24 && Min[d = Union@ IntegerDigits@ n] > 0 && Prime[d] == f]; Select[Range[10^6], ok] (* Giovanni Resta, Aug 24 2018 *)

A329711 Numbers n such that n = prime(d_1) * prime(d_2) * ... * prime(d_k), where n is a concatenation of d_1, d_2, ..., d_k.

Original entry on oeis.org

14, 154, 1196, 2127, 61411, 172482, 223227, 279174, 291318, 1233822, 1346235, 2681318, 3127010, 6541482, 9105217, 14216826, 15136418, 15454362, 17211896, 22442133, 24174129, 32693925, 35219085, 35523825, 51157348, 51431138, 57121662, 58935162, 91242978, 101721214

Offset: 1

Bartlomiej Pawlik, Mar 07 2020

			14 = prime(1)*prime(4) = 2*7, so 14 is a term.
154 = prime(1)*prime(5)*prime(4) = 2*11*7, so 154 is a term.
2127 = prime(2)*prime(127) = 3*709, so 2127 is a term.
9105217 = prime(9)*prime(10)*prime(5)*prime(21)*prime(7), so 9105217 is a term.

Giovanni Resta, Table of n, a(n) for n = 1..191
Bartlomiej Pawlik, Table of concatenations and prime factorizations of a(n) for n = 1..191

Cf. A097227 (a subsequence), A318298.

ok[n_] := Block[{d = DigitCount@ n}, AllTrue[Range@ 9, IntegerExponent[n, Prime@ #] <= d[[#]] &]]; ric[v_, d_] := If[PrimeQ@ v, PrimePi@ v == FromDigits@ d, Block[ {r=False, p, m = Length@ d}, Do[ If[ d[[i + 1]] > 0, p = Prime@ FromDigits@ Take[d, i]; If[Mod[v, p] == 0 && (r = ric[v/p, Take[d, i - m]]), Break[]]], {i, m - 1}]; r]]; Select[ Range@ 300000, If[ok@# && ric[#, IntegerDigits@ #], Print@#; True, False] &] (* Giovanni Resta, Mar 12 2020 *)

More terms from Giovanni Resta, Mar 12 2020

A113735 Let prime(0) = 1 and f(n) = product prime(d), where d ranges over all the decimal digits of n. The sequence gives numbers n such that f(n) == 0 (mod n).

Original entry on oeis.org

1, 14, 17, 154, 1196, 26979, 66079, 279174, 1698619, 9397685, 302768895, 594963655, 2249805789, 6794867989, 9785759929, 75077778589, 67471872963495, 34976979277935695, 275776822479793635, 459267544887917766, 34678475798796583278525

Offset: 1

Amarnath Murthy and Robert G. Wilson v, Nov 08 2005

Cf. A113581. Superset of A097227.

f[n_] := Times @@ (IntegerDigits[n] /. {0 -> 1, 1 -> 2, 2 -> 3, 3 -> 5, 4 -> 7, 5 -> 11, 6 -> 13, 7 -> 17, 8 -> 19, 9 -> 23}); Do[ If[ Mod[ f[n], n] == 0, Print[n]], {n, 10^8}]

a(11)-a(16) from Giovanni Resta, Aug 09 2017

a(17)-a(21) from Chai Wah Wu, May 07 2019

A115078 Numbers k such that k = prime(1 + d_1)prime(1 + d_2)...*prime(1 + d_m), where d_1 d_2 ... d_m is the decimal expansion of k.

Original entry on oeis.org

171, 290, 2145, 3381, 74613, 10664845620, 14771330561681694, 2744819721528289762500

Offset: 1

Giovanni Resta, Jan 12 2006

a(9), if it exists, must have more than 32 digits.

a(9) > 10^37 if it exists. - Chai Wah Wu, Aug 12 2017

			290 is a term because 290 = p(1+2)*p(1+9)*p(1+0) = 5*29*2.

Cf. A097227.

Fixed points of A359802.

t={};Do[If[n==Times@@Prime[1+IntegerDigits@n], Print[n];AppendTo[t, n]], {n, 10^5}];t

is(n) = my(d=digits(n)); n==prod(i=1, #d, prime(1+d[i])) \\ Felix Fröhlich, Aug 12 2017

cp's OEIS Frontend

A098683 Numbers n such that pi(n) = prime(d_1)*prime(d_2)*...*prime(d_k) where d_1 d_2 ... d_k is the decimal expansion of n.

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A097228 Numbers n such that the product of digits of n equals the concatenation of pi(d)'s where d runs through the digits of n.

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A099068 Numbers n such that n=P(d_1)*P(d_2)*...*P(d_k)+(P(d_1)+P(d_2)+...+P(d_k)) where d_1 d_2 ... d_k is the decimal expansion of n and P(i) is the i-th prime.

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A099069 Numbers n such that n = prime(d_1*d_2*...*d_k) - phi(d_1 + d_2 + ... + d_k) where d_1 d_2 ... d_k is the decimal expansion of n.

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A110070 Numbers n such that n=pi(d_1!*d_2!*...*d_k!) where d_1 d_2 ... d_k is the decimal expansion of n.

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A290675 For a given n, the nonzero digits of n are the prime indices for the factorization of n.

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A318298 Numbers whose set of decimal digits coincides with the set of the indices of their prime factors.

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A329711 Numbers n such that n = prime(d_1) * prime(d_2) * ... * prime(d_k), where n is a concatenation of d_1, d_2, ..., d_k.

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A098683 Numbers n such that pi(n) = prime(d_1)prime(d_2)...*prime(d_k) where d_1 d_2 ... d_k is the decimal expansion of n.

A099068 Numbers n such that n=P(d_1)P(d_2)...*P(d_k)+(P(d_1)+P(d_2)+...+P(d_k)) where d_1 d_2 ... d_k is the decimal expansion of n and P(i) is the i-th prime.

A099069 Numbers n such that n = prime(d_1d_2...*d_k) - phi(d_1 + d_2 + ... + d_k) where d_1 d_2 ... d_k is the decimal expansion of n.

A110070 Numbers n such that n=pi(d_1!d_2!...*d_k!) where d_1 d_2 ... d_k is the decimal expansion of n.

A115078 Numbers k such that k = prime(1 + d_1)prime(1 + d_2)...*prime(1 + d_m), where d_1 d_2 ... d_m is the decimal expansion of k.