A123157 -id:A123157 - cp's OEIS frontend

A165492 Primes p where the digital sum of p^2 is equal to 13.

7, 29, 47, 61, 79, 151, 349, 389, 461, 601, 1051, 1249, 1429, 1451, 1789, 2239, 3499, 4799, 5011, 5101, 5501, 5749, 10501, 11149, 14249, 50101, 51001, 55001, 100501, 100649, 101149, 150001, 318751, 448999, 501001, 1009951, 5000011, 5000101, 5001001, 50000101

Offset: 1

Vincenzo Librandi, Sep 21 2009

			29 is in the sequence because 29^2 = 841 and 8+4+1 = 13.
1789 is in the sequence because 1789^2 = 3200521 and 3+2+0+0+5+2+1 = 13.

Vincenzo Librandi and Hiroaki Yamanouchi, Table of n, a(n) for n = 1..61 (terms a(1)-a(44) from Vincenzo Librandi)

Cf. A165502 - A165504.

[p: p in PrimesUpTo(5*10^7) | &+Intseq(p^2) eq 13]; // Vincenzo Librandi, Sep 26 2015

Select[Prime[Range[80000]],Total[IntegerDigits[#^2]]==13&]  (* Harvey P. Dale, Apr 08 2011 *)

{A000040(i) : A123157(i) = 13}. [R. J. Mathar, Sep 29 2009]

More terms from R. J. Mathar, Sep 29 2009

a(37)-a(40) from Hiroaki Yamanouchi, Sep 23 2014

A165493 Primes p where the digital sum of p^2 is equal to 19.

Original entry on oeis.org

17, 37, 53, 73, 89, 107, 109, 127, 181, 199, 269, 271, 379, 503, 521, 557, 701, 739, 1009, 1061, 1097, 1151, 1171, 1439, 1511, 1601, 1619, 1747, 1801, 1871, 2251, 3169, 3259, 3329, 3347, 3511, 3761, 3851, 3889, 4051, 4139, 4519, 4751, 4951, 5003, 5021, 5849

Offset: 1

Vincenzo Librandi, Sep 21 2009

			17 is in the sequence because 17^2=289 and 2+8+9=19.
1801 is in the sequence because 1801^2=3243601 and 3+2+4+3+6+0+1=19.

Vincenzo Librandi, Table of n, a(n) for n = 1..215

[p: p in PrimesUpTo(6000) | &+Intseq(p^2) eq 19]; // Bruno Berselli, Jun 24 2013

A007953 := proc(n) add(d,d=convert(n,base,10)) ; end:
A123157 := proc(n) A007953((ithprime(n))^2) ; end:
for n from 1 to 100000 do if A123157(n) = 19 then printf("%d,",ithprime(n)) ; fi; od: # R. J. Mathar, Sep 29 2009

Select[Prime[Range[800]], Total[IntegerDigits[#^2]]== 19&] (* Vincenzo Librandi, Jun 24 2013 *)

{A000040(i) : A123157(i) = 19} [R. J. Mathar, Sep 29 2009]

More terms from R. J. Mathar, Sep 29 2009

A165502 Primes p where the digital sum of p^2 is equal to 31.

Original entry on oeis.org

83, 137, 173, 223, 263, 277, 281, 367, 443, 457, 547, 587, 673, 677, 691, 727, 853, 857, 907, 911, 997, 1033, 1087, 1109, 1163, 1181, 1213, 1217, 1307, 1433, 1447, 1523, 1613, 1627, 1699, 1721, 1811, 2027, 2153, 2203, 2221, 2297, 2347, 2459, 2473, 2477, 2531, 2549

Offset: 1

Vincenzo Librandi, Sep 21 2009

			83 is in the sequence because 83^2=6889 and 6+8+8+9=31.
1721 is in the sequence because 1721^2=2961841 and 2+9+6+1+8+4+1=31.

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

[p: p in PrimesUpTo(2600) | &+Intseq(p^2) eq 31]; // Vincenzo Librandi, Sep 12 2013

A007953 := proc(n) add(d,d=convert(n,base,10)) ; end:
A123157 := proc(n) A007953((ithprime(n))^2) ; end:
for n from 1 to 10000 do if A123157(n) = 31 then printf("%d,",ithprime(n)) ; fi; od: # R. J. Mathar, Sep 29 2009

Select[Prime[Range[500]], Total[IntegerDigits[#^2]]== 31 &]  (* Harvey P. Dale, Apr 13 2011 *)

{A000040(i) : A123157(i) = 31} [R. J. Mathar, Sep 29 2009]

Edited by R. J. Mathar, Sep 29 2009

A165459 Primes p such that the sum of the digits of p^2 is 16.

Original entry on oeis.org

13, 23, 31, 41, 59, 103, 131, 139, 211, 229, 239, 347, 401, 491, 499, 571, 751, 1021, 1201, 1229, 1453, 1489, 1499, 1741, 2003, 2011, 3001, 3821, 4001, 4639, 4649, 5701, 7079, 8951, 10111, 10247, 10301, 10499, 14251, 14639, 16249, 17321, 19751, 20011

Offset: 1

Vincenzo Librandi, Sep 20 2009

			31 is in the sequence because 31^2=961 and 9+6+1=16;
1489 is in the sequence because 1489^2=2217121 and 2+2+1+7+1+2+1=16;
3001 is in the sequence because 3001^2=9006001 and 9+0+0+6+0+0+1=16.

Vincenzo Librandi, Table of n, a(n) for n = 1..80

[p: p in PrimesUpTo(3*10^4) | &+Intseq(p^2) eq 16]; // Bruno Berselli, Jun 24 2013

A007953 := proc(n) local d ; add(d, d=convert(n,base,10)) ; end proc: A165459 := proc(n) local a ; if n = 1 then 13; else a := nextprime( procname(n-1)) ; while A007953(a^2) <> 16 do a := nextprime(a) ; end do ; return a ; end if; end proc: seq(A165459(n),n=1..50) ; # R. J. Mathar, Nov 09 2009

Select[Prime[Range[3000]], Total[IntegerDigits[#^2]]==16 &] (* Vincenzo Librandi, Jun 24 2013 *)

{A000040(i) : A123157(i) = 16}. [R. J. Mathar, Nov 09 2009]

Edited by N. J. A. Sloane, Sep 25 2009

347 inserted, more terms added by R. J. Mathar, Nov 09 2009

A165504 Primes p with a digits sum of p^2 equal to 43.

Original entry on oeis.org

887, 1697, 1723, 1867, 1913, 2083, 2137, 2417, 2543, 2633, 2687, 2767, 2803, 2957, 3083, 3109, 3137, 3433, 3793, 3847, 3947, 4073, 4217, 4423, 4567, 4657, 4783, 4793, 4937, 5099, 5233, 5279, 5333, 5387, 5431, 5647, 5683, 5827, 6043, 6053, 6133, 6143

Offset: 1

Vincenzo Librandi, Sep 21 2009

			887 is in the sequence because 887^2=786769 and 7+8+6+7+6+9=43.
1723 is in the sequence because 1723^2=2968729 and 2+9+6+8+7+2+9=43.

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

Cf. primes p where the digital sum of p^2 is equal to: A226803 (7), A165492 (13), A165493 (19), A165502 (31), A165503 (37), this sequence (43).

[p: p in PrimesUpTo(6150) | &+Intseq(p^2) eq 43]; // Vincenzo Librandi, Sep 12 2013

Select[Prime[Range[300]], Total[IntegerDigits[#^2]] == 43&] (* Vincenzo Librandi, Sep 12 2013 *)

{A000040(i) : A123157(i) = 43} [R. J. Mathar, Sep 29 2009]

More terms from R. J. Mathar, Sep 29 2009

A235398 Sum of digits of the cubes of prime numbers.

Original entry on oeis.org

8, 9, 8, 10, 8, 19, 17, 28, 17, 26, 28, 19, 26, 28, 17, 35, 26, 28, 19, 26, 28, 28, 35, 35, 28, 8, 28, 17, 28, 35, 28, 26, 26, 37, 35, 28, 46, 28, 35, 35, 35, 37, 44, 37, 35, 46, 37, 37, 35, 37, 35, 35, 37, 26, 44, 35, 35, 28, 28, 26, 37, 35, 37, 17, 37, 26

Offset: 1

Antonio Graciá Llorente, Jan 09 2014

Cf. A004164, A007953, A123157.

read("transforms") :
A235398 := proc(n)
    digsum(ithprime(n)^3) ;
end proc:
seq(A235398(n),n=1..40) ; # R. J. Mathar, Jul 19 2024

a(n) = sumdigits(prime(n)^3); \\ Michel Marcus, Jan 09 2014

a(n) = A007953(A030078(n)). - R. J. Mathar, Jul 19 2024

A165503 Primes p with a digits sum of p^2 equal to 37.

Original entry on oeis.org

433, 613, 683, 773, 827, 863, 1063, 1117, 1187, 1223, 1567, 1583, 1657, 1693, 1783, 1907, 1997, 2017, 2087, 2141, 2143, 2161, 2267, 2357, 2393, 2467, 2557, 2593, 2609, 2663, 2719, 2753, 2789, 2843, 2879, 2897, 2969, 2971, 3023, 3041, 3061, 3167, 3187

Offset: 1

Vincenzo Librandi, Sep 21 2009

			433 is in the sequence because 433^2=187489 and 1+8+7+4+8+9=37.
1783 is in the sequence because 1783^2=3179089 and 3+1+7+9+0+8+9=37.

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

Cf. primes p where the digital sum of p^2 is equal to: A226803 (7), A165492 (13), A165493 (19), A165502 (31), this sequence (37), A165504 (43).

[p: p in PrimesUpTo(6150) | &+Intseq(p^2) eq 37]; // Vincenzo Librandi, Sep 12 2013

Select[Prime[Range[500]], Total[IntegerDigits[#^2]]== 37&] (* Vincenzo Librandi, Sep 12 2013 *)

{A000040(i) : A123157(i) = 37} [R. J. Mathar, Sep 29 2009]

More terms from R. J. Mathar, Sep 29 2009

A226802 Primes p where the digital sum of p^2 is equal to 10.

Original entry on oeis.org

19, 71, 179, 251, 449, 20249, 24499, 100549

Offset: 1

Vincenzo Librandi, Jun 24 2013

The next term is > 24154957 (if it exists). - R. J. Mathar, Jul 05 2013
No more terms below 10^12. - Hiroaki Yamanouchi, Sep 23 2014.
No additional terms < 10^15. - Chai Wah Wu, Nov 15 2015
No other terms below 10^50. The sequence is likely finite and complete. - Max Alekseyev, Jun 13 2025

			19 is in the sequence because 19^2=361 and 3+6+1=10.
71 is in the sequence because 71^2=5041 and 5+0+4+1=10.

Cf. A123157, A165492, A165459, A165493. Subset of A135027.

[p: p in PrimesUpTo(5*10^6) | &+Intseq(p^2) eq 10];

select(p -> isprime(p) and convert(convert(p^2,base,10),`+`)=10, [seq(2*k+1,k=1..100000)]); # Robert Israel, Sep 23 2014

Select[Prime[Range[70000]], Total[IntegerDigits[#^2]]== 10&]

A290972 Primes p such that the sum of the squares of digits of p equals the sum of digits of p^2.

Original entry on oeis.org

2, 3, 3331, 3433, 11243, 13241, 21523, 22153, 22531, 31541, 32141, 32411, 33203, 34033, 34141, 34211, 35141, 41341, 41413, 42131, 43411, 44131, 51341, 51413, 52321, 54311, 102253, 102523, 104231, 104513, 110543, 111263, 111623, 112163

Offset: 1

K. D. Bajpai, Aug 16 2017

214007 is the smallest term that is in A017353 and 31111009 is the smallest term that is in A017377. - Altug Alkan, Aug 16 2017

			a(3) = 3331 is prime: [3^2 + 3^2 + 3^2 + 1^2 = 9 + 9 + 9 + 1] = 28; [3331^2 = 11095561, 1 + 1 + 0 + 9 + 5 + 5 + 1] = 28.
a(5) = 11243 is prime: [1^2 + 1^2 + 2^2 + 4^2 + 3^2 = 1 + 1 + 4 + 16 + 9] = 31: [11243^2 = 126405049;1 + 2 + 6 + 4 + 0 + 5 + 0 + 4 + 9] = 31.

Robert Israel, Table of n, a(n) for n = 1..4000

Intersection of A000040 and A165550.

Cf. A123157.

filter:= t -> convert(map(`^`,convert(t,base,10),2),`+`) = convert(convert(t^2,base,10),`+`) and isprime(t):
select(filter, [2,seq(i,i=3..200000,2)]); # Robert Israel, Aug 16 2017

Select[Prime[Range[20000]], Plus @@ IntegerDigits[#^2] == Total[IntegerDigits[#]^2] &]

forprime(p=1, 30000, v=digits(p); if(sum(i=1, length(v), v[i]^2) == sumdigits(p^2), print1(p", ")));

A234429 Numbers which are the digital sum of the square of some prime.

Original entry on oeis.org

4, 7, 9, 10, 13, 16, 19, 22, 25, 28, 31, 34, 37, 40, 43, 46, 49, 52, 55, 58, 61, 64, 67, 70, 73, 76, 79, 82, 85, 88, 91, 94, 97, 100, 103, 106, 109, 112, 115, 118, 121, 124, 127, 130, 133, 136, 139, 142, 145, 148, 151, 154, 157, 160, 163, 166, 169, 172, 175, 178

Offset: 1

Antonio Graciá Llorente, Dec 26 2013

A123157 sorted and duplicates removed.

Cf. A067180, A067523, A123157, A056991.

Cf. A062397, A226803, A226802, A165492, A165459 A165493, A229058, A165502, A165503, A165504.

terms(nn) = {v = []; forprime (p = 1, nn, v = concat(v, sumdigits(p^2));); vecsort(v,,8);} \\ Michel Marcus, Jan 08 2014

From Robert G. Wilson v, Sep 28 2014: (Start)
Except for 3, all primes are congruent to +-1 (mod 3). Therefore, (3n +- 1)^2 = 9n^2 +- 6n + 1 which is congruent to 1 (mod 3).
2, 11, 101, ... (A062397);
5, 149, 1049, ... (A226803);
only 3;
19, 71, 179, 251, 449, 20249, 24499, 100549, ... (A226802);
7, 29, 47, 61, 79, 151, 349, 389, 461, 601, 1051, 1249, 1429, ... (A165492);
13, 23, 31, 41, 59, 103, 131, 139, 211, 229, 239, 347, 401, ... (A165459);
17, 37, 53, 73, 89, 107, 109, 127, 181, 199, 269, 271, 379, ... (A165493);
43, 97, 191, 227, 241, 317, 331, 353, 421, 439, 479, 569, 619, 641, ...;
67, 113, 157, 193, 257, 283, 311, 337, 373, 409, 419, 463, ... (A229058);
163, 197, 233, 307, 359, 397, 431, 467, 487, 523, 541, 577, 593, 631, ...;
83, 137, 173, 223, 263, 277, 281, 367, 443, 457, 547, 587, ... (A165502);
167, 293, 383, 563, 607, 617, 733, 787, 823, 859, 877, 941, 967, 977, ...;
433, 613, 683, 773, 827, 863, 1063, 1117, 1187, 1223, 1567, ... (A165503);
313, 947, 983, 1303, 1483, 1609, 1663, 1933, 1973, 1987, 2063, 2113, ...;
887, 1697, 1723, 1867, 1913, 2083, 2137, 2417, 2543, 2633, ... (A165504);
883, 937, 1367, 1637, 2213, 2447, 2683, 2791, 2917, 3313, 3583, 3833, ...;
1667, 2383, 2437, 2617, 2963, 4219, 4457, 5087, 5281, 6113, 6163, ...;
...  Also see A229058. (End)
Conjectures from Chai Wah Wu, Apr 16 2025: (Start)
a(n) = 2*a(n-1) - a(n-2) for n > 5.
G.f.: x*(2*x^4 - x^3 - x^2 - x + 4)/(x - 1)^2. (End)

a(36) from Michel Marcus, Jan 08 2014
a(37)-a(54) from Robert G. Wilson v, Sep 28 2014
More terms from Giovanni Resta, Aug 15 2019

cp's OEIS Frontend

A165492 Primes p where the digital sum of p^2 is equal to 13.

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A165493 Primes p where the digital sum of p^2 is equal to 19.

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A165502 Primes p where the digital sum of p^2 is equal to 31.

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A165459 Primes p such that the sum of the digits of p^2 is 16.

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A165504 Primes p with a digits sum of p^2 equal to 43.

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A235398 Sum of digits of the cubes of prime numbers.

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A165503 Primes p with a digits sum of p^2 equal to 37.

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