A097221 -id:A097221 - cp's OEIS frontend

A097223 Prime numbers p such that p = prime(n) and n = product of the digits of p.

17, 73, 2475989

Offset: 1

Farideh Firoozbakht, Aug 06 2004

This sequence is a subsequence of A097220, so the sequence is also a subsequence of A097221.

There is no further term. - Farideh Firoozbakht, Jul 15 2009

			2475989 is in the sequence because 2475989 is (2*4*7*5*9*8*9)-th prime.

Jessie Byrnes, Chris Spicer and Alyssa Turnquist, The Sheldon Conjecture. Math Horizons, Vol. 23, No. 2 (November 2015), pp. 12-15 (4 pages); alternate link.
Chris K. Caldwell and G. L. Honaker, Jr., 2475989
Carl Pomerance, What we still don't know about addition and multiplication, Trjitzinsky Lecture 1, U. Illinois Urbana-Champaign, November 27, 2018. See slides 22 & 24.
Carl Pomerance and Chris Spicer, Proof of the Sheldon Conjecture, The American Mathematical Monthly, September 2019, 126(8), 688-698; alternate link.

Cf. A097220, A097221.

v={}; Do[If[h=IntegerDigits[Prime[n]]; l=Length[h]; p=Product[h[[k]], {k, l}]; p==n, v=Append[v, Prime[n]]; Print[v]], {n, 205000000}]

isok(p) = isprime(p) && (primepi(p) == vecprod(digits(p))); \\ Michel Marcus, Jan 27 2019

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

A097222 Numbers n such that for some k there exist k numbers a1,a2, ...,ak that concatenations of them is equal to n and sum of them is equal to Pi(n).

Original entry on oeis.org

15, 27, 39, 130, 131, 252, 370, 489, 1195, 2345, 3484, 4619, 5752, 6879

Offset: 1

Farideh Firoozbakht, Aug 05 2004

			9642459 is in the sequence because Pi(9642459)=9+642459
in fact a1=9 & a2=642459.

Cf. A097220, A097221.

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A097223 Prime numbers p such that p = prime(n) and n = product of the digits of p.

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

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A097222 Numbers n such that for some k there exist k numbers a1,a2, ...,ak that concatenations of them is equal to n and sum of them is equal to Pi(n).

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