A046704 -id:A046704 - cp's OEIS frontend

A088133 Sum of first and last digits of n. Different from A115299.

0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 8, 9, 10, 11, 12, 13, 14

Offset: 0

Zak Seidov, Sep 20 2003

Cf. A000030 (first digit of n), A010879 (last digit of n).

Cf. A046704, A007953, A088134, A088135, A088136.

Total[{First[IntegerDigits[#]],Last[IntegerDigits[#]]}]&/@Range[90] (* Harvey P. Dale, Aug 21 2018 *)

apply( {A088133(n)=n\10^logint(n+!n, 10)+n%10}, [0..99]) \\ M. F. Hasler, Apr 22 2024

list(map(A088133 := lambda n: int(str(n)[0])+n%10, range(99))) # M. F. Hasler, Apr 22 2024

a(n) = A000030(n) + A010879(n). - M. F. Hasler, Apr 22 2024

Extended to a(0) = 0 by M. F. Hasler, Apr 22 2024

A088134 Numbers n such that sum of first and last digits is prime.

Original entry on oeis.org

1, 11, 12, 14, 16, 20, 21, 23, 25, 29, 30, 32, 34, 38, 41, 43, 47, 49, 50, 52, 56, 58, 61, 65, 67, 70, 74, 76, 83, 85, 89, 92, 94, 98, 101, 102, 104, 106, 111, 112, 114, 116, 121, 122, 124, 126, 131, 132, 134, 136, 141, 142, 144, 146, 151, 152, 154, 156, 161, 162, 164

Offset: 1

Zak Seidov, Sep 20 2003

Cf. A046704, A007953, A088133, A088135, A088136.

A104213 Primes with nonprime sums of digits.

Original entry on oeis.org

13, 17, 19, 31, 37, 53, 59, 71, 73, 79, 97, 103, 107, 109, 127, 149, 163, 167, 181, 211, 233, 239, 251, 257, 271, 277, 293, 307, 347, 349, 367, 383, 389, 419, 431, 433, 439, 457, 479, 491, 499, 503, 509, 521, 523, 541, 547, 563, 569, 587, 613, 617, 619, 631

Offset: 1

Cino Hilliard, Mar 13 2005

Primes with nonprime digital sums. [Juri-Stepan Gerasimov, Apr 23 2010]

Subsequence of primes of A104211. - Michel Marcus, May 03 2015

			Sum of digits of prime 13 = 4, which is not prime, so 13 is in the sequence.

Jason Yuen, Table of n, a(n) for n = 1..10000

Cf. A046704 (primes with prime sums of digits), A104211.

[p: p in PrimesUpTo(600) | not IsPrime(&+Intseq(p))]; // Vincenzo Librandi, May 03 2015

Select[ Prime[ Range[115]], !PrimeQ[Plus @@ IntegerDigits[ # ]] &] (* Robert G. Wilson v, Mar 16 2005 *)

select(p->!isprime(sumdigits(p)),primes(100)) \\ Joerg Arndt, May 03 2015

Definition clarified by Jonathan Sondow, Jun 11 2012

A088135 Sum of first and last digits of n-th prime.

Original entry on oeis.org

4, 6, 10, 14, 2, 4, 8, 10, 5, 11, 4, 10, 5, 7, 11, 8, 14, 7, 13, 8, 10, 16, 11, 17, 16, 2, 4, 8, 10, 4, 8, 2, 8, 10, 10, 2, 8, 4, 8, 4, 10, 2, 2, 4, 8, 10, 3, 5, 9, 11, 5, 11, 3, 3, 9, 5, 11, 3, 9, 3, 5, 5, 10, 4, 6, 10, 4, 10, 10, 12, 6, 12, 10, 6, 12, 6, 12, 10, 5, 13, 13, 5, 5, 7, 13, 7, 13

Offset: 1

Zak Seidov, Sep 20 2003

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A046704, A007953, A088133, A088134, A088136.

sfl[p_]:=Module[{idn=IntegerDigits[p]},idn[[1]]+idn[[-1]]]; sfl/@Prime[Range[90]] (* Harvey P. Dale, Jan 31 2023 *)

A091365 Primes p such that the sum of the digits of p is not prime, but the sum of the cubes of the digits of p is prime.

Original entry on oeis.org

997, 2797, 3499, 4993, 7297, 7477, 7927, 8089, 8999, 9277, 9349, 9439, 9907, 11689, 12697, 12967, 14479, 14767, 14929, 14947, 16189, 16477, 16729, 16747, 16927, 16981, 17449, 17467, 18169, 18691, 19249, 19267, 19429, 19447, 19681, 19861

Offset: 1

Chuck Seggelin, Jan 03 2004

Apparently if the cubes of the digits of a prime sum to a prime, it is more likely that the digits themselves also sum to a prime. In the first 10,000 primes there are 1969 primes p such that the cubes of the digits of p sum to a prime. Of these, only 358 are such that the sums of the digits are not prime. Interestingly, all of these primes have a digit sum of 25 or 35. Essentially this sequence is the terms of A091366 (primes whose digits cubed sum to a prime) that do not also appear in A046704 (primes whose digits sum to a prime).

			a(1)=997 because 9+9+7 = 25 which is not prime, but 9^3+9^3+7^3 = 1801 which is prime.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A046704 (primes whose digits sum to a prime) A091366 (primes whose digits squared sum to a prime).

ssdQ[n_]:= Module[{idn = IntegerDigits[n]}, !PrimeQ[Total[idn]]&&PrimeQ[Total[idn^3]]]; Select[Prime[Range[4000]], ssdQ] (* Vincenzo Librandi, Apr 17 2013 *)

A046713 Multiplicative and additive primes: primes where the product and sum of digits are also prime.

Original entry on oeis.org

2, 3, 5, 7, 113, 131, 151, 311, 2111, 11113, 11117, 11131, 11171, 11311, 111121, 111211, 112111, 1111151, 1111711, 1117111, 1171111, 111111113, 111111131, 111113111, 115111111, 131111111, 1111111121, 1111211111, 1121111111, 11111111113, 11111111131, 11113111111

Offset: 1

Felice Russo

Any term of this sequence has one prime digit and all other digits are 1. - Sean A. Irvine, Apr 17 2021

James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 113.

Alois P. Heinz, Table of n, a(n) for n = 1..1000

Intersection of A028834, A028842, and A000040.

Intersection of A046703 and A046704.

d[n_]:=IntegerDigits[n]; t={}; Do[p=Prime[n]; If[PrimeQ[Plus@@(x=d[p])]&&PrimeQ[Times@@x],AppendTo[t,p]],{n,2*10^5}]; t (* Jayanta Basu, May 18 2013 *)
Select[Prime[Range[5033*10^5]],AllTrue[{Total[IntegerDigits[ #]],Times@@ IntegerDigits[ #]},PrimeQ]&] (* or -- much faster *) Select[Union[ Flatten[ Table[FromDigits/@Permutations[PadRight[{p},n,1]],{p,{2,3,5,7}},{n,11}]]],AllTrue[{#,Total[ IntegerDigits[#]],Times@@ IntegerDigits[ #]},PrimeQ]&] (* Harvey P. Dale, Feb 28 2022 *)

More terms from Harvey P. Dale, Aug 23 2000
Corrected by Jud McCranie, Jan 03 2001
Edited by Charles R Greathouse IV, Aug 02 2010

A225534 Numbers whose sum of cubed digits is prime.

Original entry on oeis.org

11, 101, 110, 111, 113, 115, 122, 124, 128, 131, 139, 142, 146, 148, 151, 155, 164, 166, 182, 184, 193, 199, 212, 214, 218, 221, 223, 227, 232, 236, 238, 241, 245, 254, 256, 263, 265, 269, 272, 278, 281, 283, 287, 289, 296, 298, 311, 319, 322, 326, 328, 335

Offset: 1

Kevin L. Schwartz and Christian N. K. Anderson, May 10 2013

Note that 11 is the only two-digit number in the sequence.

a(n) ~ n. For 414 < n < 10000, 6.38*n - 528 provides an estimate of a(n) to within 6%.

			139 is in the sequence because 1^3 + 3^3 + 9^3 = 757, which is prime.

Christian N. K. Anderson, Table of n, a(n) for n = 1..10000

Cf. A197039, A055012, A046459, A225535.
Cf. A165330, A046197.
Cf. A046704, A023194.

Select[Range[350],PrimeQ[Total[IntegerDigits[#]^3]]&] (* Harvey P. Dale, Mar 16 2016 *)

digcubesum<-function(x) sum(as.numeric(strsplit(as.character(x),split="")[[1]])^3); library(gmp);
which(sapply(1:1000,function(x) isprime(digcubesum(x))>0))

A091367 Primes p such that the sum of the digits raised to the 4th power is prime.

Original entry on oeis.org

11, 23, 29, 41, 43, 47, 61, 67, 83, 89, 101, 113, 131, 179, 191, 197, 223, 269, 311, 313, 331, 353, 379, 397, 401, 443, 461, 601, 607, 641, 661, 719, 739, 809, 883, 911, 937, 971, 1013, 1019, 1031, 1033, 1091, 1097, 1103, 1109, 1181, 1301, 1303, 1367, 1433

Offset: 1

Chuck Seggelin, Jan 03 2004

			a(1) = 11 because 1^4 + 1^4 = 2 which is prime.
a(10) = 89 because 8^4 + 9^4 = 10657 which is prime.

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

Cf. A046704 (primes whose digits sum to a prime), A052034 (primes whose digits squared sum to a prime), A091366 (primes whose digits cubed sum to a prime).

upto=500;Select[Prime[Range[upto]],PrimeQ[Total[IntegerDigits[#]^4]]&] (* Paolo Xausa, Nov 23 2021 *)

from sympy import isprime, primerange
def ok(p): return isprime(sum(int(d)**4 for d in str(p)))
def aupto(limit): return [p for p in primerange(1, limit+1) if ok(p)]
print(aupto(1433)) # Michael S. Branicky, Nov 23 2021

A207293 Primes p whose digit sum s(p) is also prime but whose iterated digit sum s(s(p)) is not prime.

Original entry on oeis.org

67, 89, 139, 157, 179, 193, 197, 199, 229, 269, 283, 337, 359, 373, 379, 397, 409, 449, 463, 467, 487, 557, 571, 577, 593, 607, 643, 647, 661, 683, 719, 733, 739, 751, 757, 773, 809, 823, 827, 829, 863, 881, 883, 919, 937, 953, 971, 991, 1039, 1093, 1097, 1129, 1187

Offset: 1

Jonathan Sondow, Jun 09 2012

A046704 is primes p with s(p) also prime. A207294 is primes p with s(p) and s(s(p)) also prime. A070027 is primes p with all s(p), s(s(p)), s(s(s(p))), ... also prime. A104213 is primes p with s(p) not prime. A213354 is primes p with s(p) and s(s(p)) also prime but s(s(s(p))) not prime. A213355 is smallest prime p whose k-fold digit sum s(s(..s(p)..)) is also prime for all k < n, but not for k = n.

			67 is prime and s(67) = 6+7 = 13 is also prime, but s(s(67)) = s(13) = 1+3 = 4 is not prime. Since no smaller prime has this property, a(1) = 67.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A046704, A070027, A104213, A207294, A213354, A213355.

isA207293 := proc(n)
    local d;
    if isprime(n) then
        d := digsum(n) ;
        if isprime(d) then
            d := digsum(d) ;
            if isprime(d) then
                false ;
            else
                true ;
            end if;
        else
            false ;
        end if;
    else
        false;
    end if;
end proc:
A207293 := proc(n)
    option remember ;
    if n = 1 then
        67 ;
    else
        a := nextprime(procname(n-1)) ;
        while not isA207293(a) do
            a := nextprime(a) ;
        end do:
        a ;
    end if;
end proc: # R. J. Mathar, Feb 04 2021

Select[Prime[Range[300]],
PrimeQ[Apply[Plus, IntegerDigits[#]]] && !
    PrimeQ[Apply[Plus, IntegerDigits[Apply[Plus, IntegerDigits[#]]]]] &]
idsQ[n_]:=PrimeQ[Rest[NestList[Total[IntegerDigits[#]]&,n,2]]]=={True,False}; Select[Prime[Range[200]],idsQ] (* Harvey P. Dale, Dec 28 2013 *)

select(p->my(s=sumdigits(p));isprime(s)&&!isprime(sumdigits(s)), primes(1000)) \\ Charles R Greathouse IV, Jun 10 2012

A207294 Primes p whose digit sum s(p) and iterated digit sum s(s(p)) are also prime.

Original entry on oeis.org

2, 3, 5, 7, 11, 23, 29, 41, 43, 47, 61, 83, 101, 113, 131, 137, 151, 173, 191, 223, 227, 241, 263, 281, 311, 313, 317, 331, 353, 401, 421, 443, 461, 599, 601, 641, 797, 821, 887, 911, 977, 1013, 1019, 1031, 1033, 1051, 1091, 1103, 1109, 1123, 1163, 1181, 1213, 1217

Offset: 1

Jonathan Sondow, Jun 09 2012

Sum_{a(n) < x} 1/a(n) is asymptotic to (9/4)*log(log(log(log(x)))) as x -> infinity; see Harman (2012). Thus the sequence is infinite.
The first member not in A070027 is 59899999.
A046704 is primes p with s(p) also prime. A070027 is primes p with all s(p), s(s(p)), s(s(s(p))), ... also prime. A104213 is primes p with s(p) not prime. A207293 is primes p with s(p) also prime, but not s(s(p)). A213354 is primes p with s(p) and s(s(p)) also prime, but not s(s(s(p))). A213355 is smallest prime p whose k-fold digit sum s(s(..s(p)..)) is also prime for all k < n, but not for k = n.

			59899999 and s(59899999) = 5+9+8+9+9+9+9+9 = 67 and s(s(59899999)) = s(67) = 6+7 = 13 are all primes, so 59899999 is a member. But s(s(s(59899999))) = s(13) = 1+3 = 4 is not prime, so 59899999 is not a member of A070027.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
G. Harman, Counting Primes whose Sum of Digits is Prime, J. Integer Seq., 15 (2012), Article 12.2.2.

Cf. A046704, A070027, A104213, A207293, A213354, A213355.

Select[Prime[Range[200]], PrimeQ[Apply[Plus, IntegerDigits[#]]] && PrimeQ[Apply[Plus, IntegerDigits[Apply[Plus, IntegerDigits[#]]]]] &]

select(p->my(s=sumdigits(p));isprime(s)&&isprime(sumdigits(s)), primes(1000)) \\ Charles R Greathouse IV, Jun 10 2012

cp's OEIS Frontend

A088133 Sum of first and last digits of n. Different from A115299.

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A088134 Numbers n such that sum of first and last digits is prime.

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A104213 Primes with nonprime sums of digits.

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A088135 Sum of first and last digits of n-th prime.

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A091365 Primes p such that the sum of the digits of p is not prime, but the sum of the cubes of the digits of p is prime.

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A046713 Multiplicative and additive primes: primes where the product and sum of digits are also prime.

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A225534 Numbers whose sum of cubed digits is prime.

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A091367 Primes p such that the sum of the digits raised to the 4th power is prime.

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