A109016 -id:A109016 - cp's OEIS frontend

A108257 Numbers k such that concatenating k and the sum of factorials of the digits of k produces a prime.

1, 13, 15, 30, 31, 91, 101, 110, 128, 133, 136, 138, 144, 152, 156, 166, 175, 193, 199, 203, 215, 230, 250, 260, 280, 281, 303, 304, 306, 307, 309, 315, 320, 330, 331, 340, 361, 391, 412, 508, 520, 550, 606, 651, 661, 681, 708, 712, 717, 730, 750, 751, 780

Offset: 1

Jason Earls, Jun 18 2005

The largest prime I have found pertaining to this sequence is A109016(Fibonacci(9837)) with 2064 digits (not proved prime, only Fermat and Lucas PRP).

			193 is in the sequence because 1!+9!+3! = 362887 and 193362887 is prime.

Michael S. Branicky, Table of n, a(n) for n = 1..10000

Cf. A061602, A109016.

Select[Range[780],PrimeQ[FromDigits[Join[IntegerDigits[#],IntegerDigits[Total[IntegerDigits[#]!]]]]]&] (* James C. McMahon, Feb 22 2024 *)

from math import factorial
from sympy import isprime
def ok(n):
    return isprime(int((s:=str(n))+str(sum(factorial(int(d)) for d in s))))
print([k for k in range(999) if ok(k)]) # Michael S. Branicky, Feb 22 2024

A108220 Numbers n such that concatenating n and the sum of factorials of the digits of n produces a square.

Original entry on oeis.org

3, 102, 122, 202, 1102, 15031, 244134, 244233, 434412, 495918, 1267396, 1300035, 1324340, 1635150, 3442657, 3878374, 5064314, 6175125, 6181308, 10320734, 10560355, 10586434, 10730603, 10804230, 11544642, 16224135, 17523855

Offset: 1

Jason Earls, Jun 16 2005

			495918 is in the sequence because 4!+9!+5!+9!+1!+8! = 766225 and
495918766225 = 704215^2.

Cf. A061602, A109016.

sfdsQ[n_]:=Module[{f=Total[IntegerDigits[n]!]},IntegerQ[ Sqrt[ n*10^IntegerLength[ f]+ f]]]; Select[Range[176*10^5],sfdsQ] (* Harvey P. Dale, Jul 06 2020 *)

Extended by Don Reble, Nov 21 2006

A108241 Numbers n such that concatenating n and the sum of factorials of the digits of n produces a triangular number.

Original entry on oeis.org

0, 3, 18, 122, 2111, 12442, 40850, 113347, 202343, 233506, 332600, 435405, 525544, 546233, 554161, 716795, 1000334, 1016217, 1142240, 1244630, 2240515, 2414603, 3103412, 3203757, 3400440, 3413651, 3424213, 3461244, 4430222, 5226150

Offset: 1

Jason Earls, Jun 17 2005

			18 is in the sequence because 1!+8! = 40321 and 1918*1919/2 = 1840321.

Cf. A061602, A109016.

ctnQ[n_]:=Module[{idn=IntegerDigits[n],c},c=FromDigits[Join[idn, IntegerDigits[ Total[idn!]]]]; IntegerQ[(Sqrt[8c+1]-1)/2]]; Select[ Range[0,5300000],ctnQ] (* Harvey P. Dale, Sep 20 2012 *)

cp's OEIS Frontend

A108257 Numbers k such that concatenating k and the sum of factorials of the digits of k produces a prime.

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A108220 Numbers n such that concatenating n and the sum of factorials of the digits of n produces a square.

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A108241 Numbers n such that concatenating n and the sum of factorials of the digits of n produces a triangular number.

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