A046704 -id:A046704 - cp's OEIS frontend

A089056 Primes whose sum of digits [s(d)] is a prime and where sum of two such consecutive s(d) values is a square.

7, 11, 23, 29, 41, 131, 137, 197, 199, 467, 487, 557, 577, 593, 757, 773, 827, 829, 863, 881, 883, 937, 953, 1013, 1019, 1031, 1103, 1109, 1277, 1279, 1567, 1583, 1637, 1657, 1871, 1873, 2003, 2027, 2087, 2089, 2267, 2269, 2377, 2393, 2447, 2467, 2591, 2593

Offset: 1

Enoch Haga, Dec 20 2003

Subsequence of A046704.

			a(2)=23; a(3)=29: 2+3=5; 2+9=11; 5+11=16, a square.

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

Cf. A046704.

Module[{sdp=Select[Prime[Range[500]],PrimeQ[Total[IntegerDigits[#]]]&]},Flatten[Select[ Partition[ sdp,2,1],IntegerQ[ Sqrt[ Total[ Flatten[ IntegerDigits/@ #]]]]&]]]//Union (* Harvey P. Dale, Oct 09 2023 *)

From the sequence of primes whose s(d) values are prime, select those where sum of two consecutive s(d) values is a square.

Edited, corrected and extended by Ray Chandler, Feb 14 2004

A091368 Primes p such that the sum of the digits of p is not prime, but the sum of each digit raised to the 4th power is prime.

Original entry on oeis.org

1699, 2689, 6199, 6829, 6991, 7477, 8089, 8269, 8629, 9619, 12589, 15289, 19069, 19609, 20599, 20959, 21589, 21859, 23857, 25189, 25819, 25873, 25981, 27259, 27529, 27583, 28069, 28537, 28573, 28591, 28753, 29059, 29527, 29581, 29851

Offset: 1

Chuck Seggelin, Jan 03 2004

Apparently, for primes such that each digit raised to the 4th power 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 760 primes whose digits raised to the 4th power sum to a prime. Of these, only 106 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 A091367 (primes whose digits raised to the 4th power sum to a prime) that do not also appear in A046704 (primes whose digits sum to a prime).

			a(1)=1699 because 1+6+9+9 = 25 which is not prime, but 1^4 + 6^4 + 9^4 + 9^4 = 14419 which is prime.

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

Cf. A046704 (primes whose digits sum to a prime) A091367 (primes whose digits raised to the 4th power sum to a prime) A052034 and A091362 (same observation for digits squared) A091366 and A091365 (same observation for digits cubed).

pnpQ[n_]:=Module[{idn=IntegerDigits[n]},!PrimeQ[Total[idn]]&&PrimeQ[ Total[ idn^4]]]; Select[Prime[Range[4000]],pnpQ] (* Harvey P. Dale, Apr 26 2018 *)

A141640 Additive nonprimes: odd sum of digits is a nonprime.

Original entry on oeis.org

1, 9, 10, 18, 27, 36, 45, 54, 63, 69, 72, 78, 81, 87, 90, 96, 100, 108, 117, 126, 135, 144, 153, 159, 162, 168, 171, 177, 180, 186, 195, 207, 216, 225, 234, 243, 249, 252, 258, 261, 267, 270, 276, 285, 294, 306, 315, 324, 333, 339, 342, 348, 351, 357, 360, 366

Offset: 1

Juri-Stepan Gerasimov, Sep 03 2008

A141468(n)=n-th nonprime.

			Nonprime (7)=10 is additive nonprime because 1+0=1 (odd) is nonprime (2).
Nonprime (12)=18 is additive nonprime because 1+8=9 (odd) is nonprime (6), etc.

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

Cf. A046704, A141468.

osdnpQ[n_]:=Module[{s=Total[IntegerDigits[n]]},OddQ[s]&&!PrimeQ[s]]; Select[Range[400],osdnpQ] (* Harvey P. Dale, Aug 04 2013 *)

45 and 207 inserted and extended by R. J. Mathar, Sep 05 2008

A158328 Lessers p1 of twin primes with prime sums of digits of p1 and p2.

Original entry on oeis.org

3, 5, 41, 137, 191, 197, 227, 281, 311, 461, 599, 641, 821, 827, 881, 1031, 1091, 1277, 1301, 1451, 1721, 1871, 2027, 2081, 2087, 2111, 2267, 2591, 2711, 2801, 3167, 3251, 3257, 3299, 3527, 3581, 3671, 3851, 4001, 4157, 4241, 4337, 4421, 4481, 4517, 4799

Offset: 1

Juri-Stepan Gerasimov, Mar 16 2009

Or, numbers n such that n and n+2 are terms in A046704. [Zak Seidov, Feb 02 2010]

			a(3)=41 is in the sequence because 41, 41+2=43, 4+1=5 and 4+3=7 are primes.
a(4)=137 is in the sequence because 137, 137+2=139, 1+3+7=11 and 1+3+9=13 are primes.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A001359, A046704.

sd:= n -> convert(convert(n,base,10),`+`):
p:= 1: q:= 2: count:= 0: Res:= NULL:
while count < 100 do
  if q = p+2 and isprime(sd(p)) and isprime(sd(q)) then
      count:= count+1; Res:= Res, p
  fi;
  p:= q; q:= nextprime(q);
od:
Res; # Robert Israel, Apr 08 2018

sd[n_]:=Plus@@IntegerDigits[n]; Select[Prime[Range[650]],And@@PrimeQ[{#+2,sd[#],sd[#+2]}] &] (* Jayanta Basu, May 25 2013 *)

Corrected by Juri-Stepan Gerasimov, Mar 24 2009
3299 and 4481 inserted by R. J. Mathar, Mar 27 2009
Example edited by Robert Israel, Apr 08 2018

A162658 Primes such that the sum of its smallest and largest decimal digits is an odd prime.

Original entry on oeis.org

23, 29, 41, 47, 61, 67, 83, 103, 107, 163, 211, 223, 229, 233, 241, 269, 293, 307, 383, 421, 431, 433, 443, 449, 457, 461, 467, 479, 491, 499, 503, 509, 523, 547, 587, 607, 613, 631, 641, 661, 677, 701, 829, 853, 857, 863, 883, 929, 947

Offset: 1

Parthasarathy Nambi, Jul 09 2009

			947 is a prime in which the sum of the smallest digit (4) and the largest digit (9) is an odd prime (13).

A046704, A070027, A086244, A088136, A091939, A109981

A166975 Four-digit primes such that, if the digits are ABCD, then AB+CD and A+B+C+D are also primes.

Original entry on oeis.org

1013, 1019, 1031, 1033, 1051, 1091, 1093, 1097, 1217, 1231, 1259, 1277, 1291, 1297, 1433, 1439, 1453, 1459, 1493, 1499, 1613, 1637, 1657, 1693, 1697, 1811, 1871, 2003, 2027, 2063, 2069, 2081, 2083, 2087, 2089, 2207, 2221, 2267, 2281, 2287, 2423, 2447

Offset: 1

Ray G. Opao, Oct 26 2009

The last term of this sequence is a(179) = 9859.
Subsequence of A046704. - R. J. Mathar, Oct 28 2009
Prime digit sums 5, 7, 11, 13, 17, 19, 23, 29, 31 occur 5, 8, 23, 19, 37, 37, 38, 9, 3 times, respectively. Sequence contains ten twin prime pairs. - Rick L. Shepherd, Feb 19 2013

			1217 is in the list since 1217, 12+17=29, and 1+2+1+7=11 are all primes.

Nathaniel Johnston, Table of n, a(n) for n = 1..179 (full sequence)

p:=1009: while p<10000 do d:=convert(p,base,10): if(isprime(add(d[j],j=1..4)) and isprime(d[1]+d[3]+10*(d[2]+d[4])))then printf("%d, ", p): fi: p:=nextprime(p): od: # Nathaniel Johnston, Jun 03 2011

apQ[n_]:=Module[{idn=IntegerDigits[n],a,b,c,d},a=idn[[1]];b=idn[[2]];c= idn[[3]];d=idn[[4]];AllTrue[{10a+b+10c+d,Total[idn]},PrimeQ]]; Select[ Prime[Range[PrimePi[1000]+1,PrimePi[9999]]],apQ] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Mar 14 2015 *)

from sympy import isprime
def ok(n):
     if n < 1000 or n > 9999 or not isprime(n): return False
     return isprime(n//100 + n%100) and isprime(sum(map(int, str(n))))
afull = [k for k in range(1001, 10000, 2) if ok(k)]
print(afull[:42]) # Michael S. Branicky, Aug 20 2022

Numbers in the range 1000 to 1200 inserted by R. J. Mathar, Oct 28 2009

Many terms corrected by Nathaniel Johnston, Jun 03 2011

A172216 Smallest k such that sum of digits of prime(n)^k is prime.

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 2, 5, 1, 1, 7, 2, 1, 1, 1, 2, 5, 1, 1, 6, 2, 2, 1, 1, 4, 1, 4, 2, 2, 1, 2, 1, 1, 1, 2, 1, 1, 4, 6, 1, 1, 2, 1, 1, 1, 1, 3, 1, 1, 1, 5, 6, 1, 4, 4, 1, 1, 2, 2, 1, 1, 4, 3, 1, 1, 1, 1, 1, 8, 2, 1, 1, 2, 1, 1, 5, 2, 1, 1, 1, 8, 1, 4, 2, 3, 1, 1, 2, 1, 1, 1, 4, 1, 8, 3, 2, 6, 2, 3, 6, 2, 1, 10, 8, 1

Offset: 1

Klaus Brockhaus, Jan 29 2010

For all n, prime(n)^0 = 1 has nonprime sum of digits 1.

a(n) = 1 iff prime(n) is in A046704, an additive prime. a(n) = 1 iff n is in A075177.

			prime(1) = 2; 2^1 = 2 has prime sum of digits 2. Hence a(1) = 1.
prime(6) = 13; 13^1 = 13 has nonprime sum of digits 4; 13^2 = 169 has nonprime sum of digits 16; 13^3 = 2197 has prime sum of digits 19. Hence a(6) = 3.

Cf. A046704, A075177, A172035.

S:=[]; for n in [1..105] do j:=1; while not IsPrime(&+Intseq(NthPrime(n)^j)) do j+:=1; end while; Append(~S, j); end for; S;

sdp[n_]:=Module[{k=1},While[!PrimeQ[Total[IntegerDigits[Prime[n]^k]]], k++]; k]; Array[sdp,110] (* Harvey P. Dale, Apr 13 2014 *)

A176196 Primes such that the sum of k-th powers of digits, for each of k = 1, 2, 3, and 4, is also a prime.

Original entry on oeis.org

11, 101, 113, 131, 223, 311, 353, 461, 641, 661, 883, 1013, 1031, 1103, 1301, 1439, 1451, 1471, 1493, 1697, 1741, 2111, 2203, 3011, 3347, 3491, 3659, 4139, 4337, 4373, 4391, 4733, 4931, 5303, 5639, 5693, 6197, 6359, 6719, 6791, 6917, 6971, 7411, 7433

Offset: 1

Michel Lagneau, Apr 11 2010

For k = 1, 2, and 3 see A176179

			For the prime number n=14549 we obtain :
1 + 4 + 5 + 4 + 9 = 23 ;
1^2 +4^2 + 5^2 +4^2 + 9^2 = 139 ;
1^3 +4^3 + 5^3 +4^3 + 9^3 = 983 ;
1^4 +4^4 + 5^4 +4^4 + 9^4 = 7699 ;

Charles W. Trigg, Journal of Recreational Mathematics, Vol. 20(2), 1988.

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Mike Mudge, Morph code, Hands On Numbers Count, Personal Computer World, May 1997, p. 290.

Cf. A176179, A109181, A046704, A052034, A091366.

with(numtheory):for n from 2 to 20000 do:l:=evalf(floor(ilog10(n))+1):n0:=n:s1:=0:s2:=0:s3:=0:s4:=0:for m from 1 to l do:q:=n0:u:=irem(q,10):v:=iquo(q,10):n0:=v :s1:=s1+u:s2:=s2+u^2:s3:=s3+u^3:s4:=s4+u^4:od:if type(n,prime)=true and type(s1,prime)=true and type(s2,prime)=true and type(s3,prime)=true and type(s4,prime)=true then print(n):else fi:od:

Select[Prime[Range[1000]],And@@PrimeQ[Total/@Table[IntegerDigits[#]^n,{n,4}]]&] (* Harvey P. Dale, Jun 16 2013 *)

from sympy import isprime, primerange
def ok(p):
    return all(isprime(sum(int(d)**k for d in str(p))) for k in range(1, 5))
def aupto(limit): return [p for p in primerange(1, limit+1) if ok(p)]
print(aupto(7443)) # Michael S. Branicky, Nov 23 2021

A180404 Primes p such that the sum of fifth power of their digits is a prime.

Original entry on oeis.org

11, 101, 191, 223, 227, 229, 281, 313, 331, 337, 359, 373, 379, 397, 463, 487, 557, 577, 593, 643, 683, 733, 739, 757, 773, 821, 863, 881, 911, 937, 953, 1019, 1033, 1091, 1109, 1123, 1129, 1181, 1213, 1231, 1259, 1277, 1291, 1303, 1321, 1381, 1433, 1439

Offset: 1

Carmine Suriano, Sep 02 2010

			a(5) = 227 since 2^5+2^5+7^5 = 32+32+16807 = 16871 is a prime.

Cf. A046704, A052034, A091366, A091367.

Select[Prime[Range[500]],PrimeQ[Total[IntegerDigits[#]^5]]&] (* Harvey P. Dale, May 25 2011 *)

If a prime p = abcdef... (each letter being a single digit) then sum = a^5+b^5+... belongs to this sequence if sum is a prime.

A260796 Numbers k such that digit sum of prime(k) plus digit sum of prime(k+1) is prime.

Original entry on oeis.org

1, 15, 37, 46, 47, 73, 91, 102, 107, 111, 118, 121, 123, 129, 161, 165, 187, 195, 197, 199, 203, 219, 239, 240, 242, 263, 275, 290, 292, 300, 326, 329, 357, 363, 388, 412, 416, 423, 426, 465, 470, 472, 504, 506, 539, 553, 565, 606, 611, 630, 641, 647, 660, 667

Offset: 1

K. D. Bajpai, Aug 08 2015

			a(2) = 15: digit sum of prime(15) + digit sum of prime (16) = (4+7) + (5+3) = 19 which is prime.
a(3) = 37: digit sum of prime(37) + digit sum of prime (38) = (1+5+7) + (1+6+3) = 23 which is prime.

K. D. Bajpai, Table of n, a(n) for n = 1..10000

Cf. A000040, A007605, A007953, A046704.

[n :  n in [1..500]  |  IsPrime(k) where k is (&+Intseq(NthPrime(n)) + &+Intseq(NthPrime(n+1))) ];

digsum:= n -> convert(convert(n, base, 10), `+`): select(n -> isprime(digsum(ithprime(n)) + digsum(ithprime(n+1))),[seq(k, k=1..10^3)]);

Select[Range[1000], PrimeQ[Plus @@ (IntegerDigits[Prime[#]]) + Plus @@ (IntegerDigits[Prime[# + 1]])] &]
Position[Total/@Partition[Total[IntegerDigits[#]]&/@Prime[Range[ 700]],2,1],?PrimeQ] // Flatten (* _Harvey P. Dale, Jun 09 2019 *)

for(n = 1, 500, if(isprime(sumdigits(prime(n)) + sumdigits(prime(n+1))), print1(n, ", ")));

from sympy import isprime, prime
A260796_list = [n for n in range(1,10**5) if isprime(sum(int(d) for d in str(prime(n))+str(prime(n+1))))] # Chai Wah Wu, Aug 09 2015

cp's OEIS Frontend

A089056 Primes whose sum of digits [s(d)] is a prime and where sum of two such consecutive s(d) values is a square.

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A091368 Primes p such that the sum of the digits of p is not prime, but the sum of each digit raised to the 4th power is prime.

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A141640 Additive nonprimes: odd sum of digits is a nonprime.

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A158328 Lessers p1 of twin primes with prime sums of digits of p1 and p2.

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A162658 Primes such that the sum of its smallest and largest decimal digits is an odd prime.

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A166975 Four-digit primes such that, if the digits are ABCD, then AB+CD and A+B+C+D are also primes.

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A172216 Smallest k such that sum of digits of prime(n)^k is prime.

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A176196 Primes such that the sum of k-th powers of digits, for each of k = 1, 2, 3, and 4, is also a prime.

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