A097223 -id:A097223 - cp's OEIS frontend

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

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

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}]

A107120 Numbers m such that pi(m) = prime(d_1d_2...*d_k) where d_1 d_2 ... d_k is the decimal expansion of m.

Original entry on oeis.org

162, 242, 291, 371, 461, 515, 2419, 2815, 11874, 64751, 81927, 264961, 276184, 757155, 2537825, 7717729, 9548491, 14738827, 19728438, 19728446, 19728464, 23695527, 77362954, 269776516, 269776523, 269776532, 358399327, 2385883646, 59955748691, 67893872935, 848472784869

Offset: 1

Farideh Firoozbakht, May 13 2005

A107121 is a subsequence of this sequence (see the comments line of A107121).
a(32) > 7*10^14, if it exists. - Giovanni Resta, Jun 01 2020
The sequence is finite as pi(m) >= pi(10^(k-1)) grows faster than prime(9^k) >= prime(d_1*d_2*...*d_k). - Max Alekseyev, Dec 30 2024

			23695527 is in the sequence because pi(23695527)=prime(2*3*6*9*5*5*2*7).

Cf. A097223, A107121.

Do[h = IntegerDigits[m]; l = Length[h]; If[Min[h] > 0 && PrimePi[m] == Prime[Product[h[[k]], {k, l}]], Print[m]], {m, 52000000}]

a(23)-a(28) from Donovan Johnson, Jul 12 2010

a(29)-a(31) from Giovanni Resta, Jun 01 2020

A107121 Numbers m such that m=prime(prime(d_1d_2...*d_k)) where d_1 d_2 ... d_k is the decimal expansion of m.

Original entry on oeis.org

461, 264961, 9548491, 14738827

Offset: 1

Farideh Firoozbakht, May 13 2005

If m is in the sequence then pi(m) = prime(d_1*d_2*...*d_k) where d_1 d_2 ... d_k is the decimal expansion of m, so this sequence is a subsequence of A107120.

The sequence is finite as m >= 10^(k-1) grows faster than prime(prime(9^k)) >= prime(prime(d_1*d_2*...*d_k)). If it exists, a(5) > 10^14. - Max Alekseyev, Dec 30 2024

			14738827 is in the sequence because 14738827=prime(prime(1*4*7*3*8*8*2*7)).

Cf. A097223, A107120, A107122.

Do[h= IntegerDigits[Prime[m]];l = Length[h];If[Min[h] > 0 && m == Prime[Product[h[[k]], {k, l}]], Print[Prime [m]]], {m, 20000000}]

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.

A306766 Primes whose index is divisible by the product of its digits.

Original entry on oeis.org

11, 13, 17, 61, 73, 113, 223, 541, 571, 1151, 1213, 1321, 1511, 1811, 2111, 2267, 3221, 3271, 4211, 4621, 5443, 11251, 11813, 12211, 12553, 13163, 17123, 17351, 19211, 21143, 21713, 24137, 28181, 29921, 31511, 32213, 34141, 34361, 41141, 61129, 63211, 71263, 95231

Offset: 1

William C. Laursen, Mar 08 2019

It is unknown whether this sequence is finite or not. For instance, if the index is exactly the product of the digits, A097223, it is known that only three such primes exist.

			A000040(21)=73 and 7*3 divides 21.
A000040(30)=113 and 1*1*3 divides 30.

Giovanni Resta, Table of n, a(n) for n = 1..10000 (first 600 terms from Robert Israel)

A097223 is a subset of this sequence where k=1, k being the above integer found after dividing.

A004022, the prime repunits, is a subsequence, because the product of the digits for all of them is 1, which trivially divides every index that the prime could hold.

p:= 2: count:= 0: Res:= NULL:
for i from 2 while count < 100 do
  p:= nextprime(p);
  pd:= convert(convert(p,base,10),`*`);
  if pd > 0 and i mod pd = 0 then
    count:= count+1; Res:= Res, p
  fi
od:
Res; # Robert Israel, Mar 10 2019

seqQ[n_] := PrimeQ[n] && (prod=Times@@IntegerDigits[n])>0 && Divisible[PrimePi[n], prod]; Select[Range[100000], seqQ] (* Amiram Eldar, Mar 11 2019 *)

isok(n) = isprime(n) && (pd=vecprod(digits(n))) && !(primepi(n) % pd); \\ Michel Marcus, Mar 09 2019

If a prime is to be in this sequence, its index q must obey A007954(A000040(q))/q = k, where k is an integer.

A107122 Numbers m such that m=prime(prime(prime(d_1d_2...*d_k))) where d_1 d_2 ... d_k is the decimal expansion of m.

Original entry on oeis.org

31, 161159, 2935241, 12393851, 25792148743, 8378273888129

Offset: 1

Farideh Firoozbakht, May 13 2005, May 27 2008

a(7) > 7*10^14, if it exists. - Giovanni Resta, Jun 01 2020

The sequence is finite as m > 10^(k-1) grows faster than prime(prime(prime(9^k))) >= prime(prime(prime(d_1*d_2*...*d_k))). - Max Alekseyev, Dec 30 2024

			12393851 is in the sequence because 12393851 = prime(prime(prime(1*2*3*9*3*8*5*1))).

Cf. A097223, A107120, A107121.

Do[h= IntegerDigits[Prime[m]];l = Length[h];If[Min[h] > 0 && m == Prime[Prime[Prime[Product[h[[k]], {k, l}]]], Print[Prime[m]]]], {m, 10000000}]

a(6) from Giovanni Resta, Jun 01 2020

A140175 Numbers n such that n=prime((d_1d_2...d_k)(d_1+d_2+...+d_k)) where d_1d_2...d_k is the decimal expansion of n.

Original entry on oeis.org

4621, 34361, 2881861, 63882851

Offset: 1

Farideh Firoozbakht, Jun 09 2008

			63882851=prime((6*3*8*8*2*8*5*1)*(6+3+8+8+2+8+5+1)).

Cf. A097223.

fQ[n_] := Block[{id = IntegerDigits@ n}, n == Prime[Plus @@ id Times @@ id]]; k = 1; lst = {}; While[ k < 10^8, If[ fQ@n, AppendTo[lst, k]; Print@ n]; k++ ]; lst (* Robert G. Wilson v *)

A307851 Prime numbers prime(k) with a zeroless decimal representation such that (product of decimal digits of prime(k)) / k is an integer.

Original entry on oeis.org

2, 17, 73, 89, 2475989

Offset: 1

Ctibor O. Zizka, May 01 2019

			For k = 21, prime(21) = 73, product of decimal digits of prime(k) / k = 7 * 3 / 21 = 1 so prime(21) = 73 is in the sequence.

C. Pomerance and Ch. Spicer, Proof of the Sheldon Conjecture.

Cf. A000040, A007954, A052382, A097220, A097223, A306766.

lista(nn) = {my(ip=0, d); forprime(p=2, nn, ip++; d = digits(p); if (vecmin(d) && !(frac(vecprod(d)/ip)), print1(p, ", ")););} \\ Michel Marcus, May 02 2019

from math import prod
from sympy import nextprime
def aupton(terms):
  p, k, t = 2, 1, 0
  while t < terms:
    strp = str(p)
    if '0' not in strp and prod(int(d) for d in strp)%k == 0:
      t += 1; print(p, end=", ")
    p, k = nextprime(p), k+1
aupton(5) # Michael S. Branicky, Feb 17 2021

a(5) from Alois P. Heinz, May 01 2019

cp's OEIS Frontend

A097220 Numbers n such that pi(n) = product of 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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A107120 Numbers m such that pi(m) = prime(d_1*d_2*...*d_k) where d_1 d_2 ... d_k is the decimal expansion of m.

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A107121 Numbers m such that m=prime(prime(d_1*d_2*...*d_k)) where d_1 d_2 ... d_k is the decimal expansion of m.

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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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A306766 Primes whose index is divisible by the product of its digits.

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A107122 Numbers m such that m=prime(prime(prime(d_1*d_2*...*d_k))) where d_1 d_2 ... d_k is the decimal expansion of m.

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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.

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.

A107120 Numbers m such that pi(m) = prime(d_1d_2...*d_k) where d_1 d_2 ... d_k is the decimal expansion of m.

A107121 Numbers m such that m=prime(prime(d_1d_2...*d_k)) where d_1 d_2 ... d_k is the decimal expansion of m.

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.

A107122 Numbers m such that m=prime(prime(prime(d_1d_2...*d_k))) where d_1 d_2 ... d_k is the decimal expansion of m.

A140175 Numbers n such that n=prime((d_1d_2...d_k)(d_1+d_2+...+d_k)) where d_1d_2...d_k is the decimal expansion of n.