Debapriyay Mukhopadhyay - cp's OEIS frontend

A354078 Squares k that are not divisible by 10, and whose reverse and digit sum are also squares, such that the digit sum divides both k and its reverse.

1, 4, 9, 144, 441, 10404, 12321, 40401, 69696, 1004004, 1022121, 1212201, 4004001, 4088484, 4848804, 100040004, 100220121, 102030201, 121022001, 400040001, 400880484, 404492544, 420578064, 445294404, 460875024, 484088004, 617323716, 10000400004, 10002200121

Offset: 1

Debapriyay Mukhopadhyay, May 17 2022

Palindromic terms include 12321, 69696, 102030201, 617323716.
144, 10404, 1004004, 100040004, 10000400004, 1000004000004, ... are all terms, so the sequence is infinite.
From Jon E. Schoenfield, May 20 2022: (Start)
Among the 3358 terms < 10^21, several classes of terms are rare or nonexistent:
- no term has an even number of digits
- no term begins or ends with a 5
- no term begins with 18 or 92
- no term that begins with 16 has any digit other than 9 as its third digit
- only one term (420578064) begins with 42, and only one (460875024) begins with 46
- only one term (9488660854689) begins with 94, and only one (9864580668849) begins with 98
- only two terms begin or end with a 6: 69696 and 617323716 (each of which is a palindrome)
- only three terms begin with a 9 and end with anything other than a 1: 9, 9488660854689, and 9864580668849
Do there exist any terms > 10^21 of any of these classes?
(End)

			10404 and its reverse, 40401 are terms because both are squares,
10404 = 102^2 and 40401 = 201^2, both have digit sum 9 and digit sum divides both 10404 and its reverse. 10404/9 = 1156, and 40401/9 = 4489.

Jon E. Schoenfield, Table of n, a(n) for n = 1..3358
Index entry for sequences related to reversing digits of squares

Cf. A001102, A028839.

Subsequence of A061457.

int get_digit_sum(int integer, int *reverse)
{
    int sum = 0;
    int rev_num = 0;
    int num = integer;
    int rem = 0;
    while (num != 0) {
        rem = num % 10;
        sum += rem;
        num = num / 10;
        rev_num = 10*rev_num + rem;
    }
    *reverse = rev_num;
    return sum;
}
int is_square(int integer)
{
    int mid = (int)(sqrt(integer));
    if ((mid*mid) == integer) {
        return mid;
    }
    else {
        return 0;
    }
}
int main(int argc, char *argv[])
{
   int reverse = 0;
   for (int j = 1; j <= 100011; j++) {
       if (j % 10 == 0) {
           continue;
       }
       int i = j*j;
       int digit_sum = get_digit_sum(i, &reverse);
       if ((i % digit_sum == 0) && (reverse % digit_sum == 0) &&
                (is_square(digit_sum) != 0) && (is_square(reverse) != 0)) {
           printf("%d, ", i);
       }
   }
   printf("\n");
   return 0;
}

J:=100011; a:=[]; for j in [1..J] do if j mod 10 ne 0 then k:=j^2; I:=Intseq(k); s:=&+I; if (k mod s eq 0) and IsSquare(s) then r:=Seqint(Reverse(I)); if (r mod s eq 0) and IsSquare(r) then a[#a+1]:=k; end if; end if; end if; end for; a; // Jon E. Schoenfield, May 19 2022

isok1(k) = if (k % 100, my(s=sumdigits(k), q=k/s); issquare(s) && issquare(q) && (denominator(q)==1));
isok(k) = isok1(k) && isok1(fromdigits(Vecrev(digits(k)))); \\ Michel Marcus, May 17 2022

A316312 Numbers k such that the sum of the digits of the numbers 1, 2, 3, ... up to (k - 1) is divisible by k.

Original entry on oeis.org

1, 3, 5, 7, 9, 12, 15, 20, 27, 40, 45, 60, 63, 80, 81, 100, 180, 181, 300, 360, 363, 500, 540, 545, 700, 720, 727, 900, 909, 912, 915, 1137, 1140, 1200, 1500, 1560, 1563, 2000, 2700, 2720, 2727, 4000, 4500, 4540, 4545, 6000, 6300, 6360, 6363, 8000, 8100, 8180

Offset: 1

Debapriyay Mukhopadhyay, Jun 29 2018

Numbers k such that A007953(A007908(k - 1)) is divisible by k. - Felix Fröhlich, Jun 29 2018
From Robert Israel, Jun 29 2018: (Start)
Numbers k such that A037123(k - 1) is divisible by k.
If m is even, then 10^m, 3 * 10^m, 5 * 10^m, 7 * 10^m and 9 * 10^m are included.
If m is odd, then 2 * 10^m, 4 * 10^m, 6 * 10^m, and 8 * 10^m are included. (End)
Is it true that if k is a term then 100 * k is a term?

			For n = 7, sum of the digits of the numbers 1 to 6 is 21, which is divisible by 7.
For n = 12, sum of the digits of the numbers 1 to 11 is 48, which is divisible by 12.
For n = 15, sum of the digits of the numbers 1 to 14 is 60, which is divisible by 15.
16 is not in the sequence because the sum of the digits of the numbers 1 to 15 is 66, which is not divisible by 16.

Henry Bottomley, Table of n, a(n) for n = 1..118

Cf. A007953, A007908, A037123, A110740, A114136.

t:= 0: Res:= NULL:
for n from 1 to 10000 do
  t:= t + convert(convert(n-1,base,10),`+`);
  if (t/n)::integer then Res:= Res, n fi
od:
Res; # Robert Israel, Jun 29 2018

s = 0; Reap[Do[If[Mod[s, n] == 0, Sow[n]]; s += Plus @@ IntegerDigits@n, {n, 10000}]][[2, 1]] (* Giovanni Resta, Jun 29 2018 *)

sumsod(n) = sum(i=1, n, sumdigits(i))
is(n) = sumsod(n-1)%n==0 \\ Felix Fröhlich, Jun 29 2018

upto(n) = my(s=0,res=List()); for(i=0, n, s += vecsum(digits(i)); if(s%(i+1)==0, listput(res, i+1))); res \\ David A. Corneth, Jun 29 2018

More terms from Felix Fröhlich, Jun 29 2018

A276509 Numbers k in base 10 such that the digits of 2 + k are the digits of 2k written in reverse order.

Original entry on oeis.org

2, 47, 497, 4997, 49997, 499997, 4999997, 49999997, 499999997, 4999999997, 49999999997, 499999999997, 4999999999997, 49999999999997, 499999999999997, 4999999999999997, 49999999999999997, 499999999999999997, 4999999999999999997, 49999999999999999997, 499999999999999999997

Offset: 1

Debapriyay Mukhopadhyay, Sep 06 2016

			47 is in the sequence because 47 + 2 = 49 and 47*2 = 94.
497 is in the sequence because 497 + 2 = 499 and 497*2 = 994.

Xander Faber and Jon Grantham, On Integers Whose Sum is the Reverse of their Product, arXiv:2108.13441 [math.NT], 2021.
Index entries for linear recurrences with constant coefficients, signature (11,-10).

[5*10^(n-1)-3: n in [1..25]]; // Vincenzo Librandi, Sep 09 2016

Select[Range[10^6], IntegerDigits[# + 2] == Reverse@ IntegerDigits[2 #] &] (* or *)
Table[5 (10^(n - 1)) - 3, {n, 22}] (* or *)
CoefficientList[Series[x(2 + 25 x)/(1 - 11 x + 10 x^2), {x, 0, 21}], x] (* or *)
{2}~Join~Table[FromDigits@ Join[{4}, ConstantArray[9, {n - 2}], {7}], {n, 2, 22}] (* Michael De Vlieger, Sep 06 2016 *)

isok(n) = digits(n+2) == Vecrev(digits(2*n)); \\ Michel Marcus, Sep 07 2016

a(n) = 5 * 10^(n - 1) - 3. - Peter Bala, Sep 06 2016
G.f.: x*(2 + 25*x)/(1 - 11*x + 10*x^2). - Michael De Vlieger, Sep 06 2016
E.g.f.: (exp(10*x) - 6*exp(x) + 5)/2. - Stefano Spezia, Mar 04 2023

A275475 Primes p such that p+2^3, p+2^5 and p+2^7 are all primes.

Original entry on oeis.org

11, 29, 71, 149, 491, 599, 701, 1439, 1451, 2339, 3761, 4211, 5399, 5651, 6269, 6701, 7541, 9059, 9311, 9689, 9941, 10859, 11831, 12569, 12791, 13679, 15299, 15551, 16979, 18089, 19301, 19469, 22031, 22541, 23549, 23879, 25229, 25841, 27329, 27791, 28541, 30809

Offset: 1

Debapriyay Mukhopadhyay, Jul 29 2016

nonn

			11 is in the sequence because 11+8 = 19, 11+32 = 43 and 11+128 = 139 are all primes.
29 is in the sequence because 29+8 = 37, 29+32 = 61 and 29+128 = 157 are all primes.

Dana Jacobsen, Table of n, a(n) for n = 1..10000

Cf. A269257, A269258, A269259, A269859, A270203.

Cf. A275485 (a subsequence).

Select[Prime@ Range@ 3450, Function[k, Times @@ Boole@ PrimeQ@ Map[k + 2^# &, {3, 5, 7}] == 1]] (* Michael De Vlieger, Aug 10 2016 *)
Select[Prime[Range[4000]],AllTrue[#+{8,32,128},PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Apr 26 2018 *)

use ntheory ":all"; say for sieve_prime_cluster(2, 1e6, 2**3, 2**5, 2**7); # Dana Jacobsen, Sep 29 2016

A271568 Squarefree semiprimes n such that phi(n) - 1 is prime.

Original entry on oeis.org

10, 14, 15, 21, 26, 33, 35, 38, 39, 51, 62, 65, 69, 77, 86, 91, 93, 95, 111, 122, 123, 129, 133, 146, 159, 161, 201, 203, 206, 209, 213, 215, 217, 218, 221, 249, 278, 287, 291, 299, 301, 302, 303, 305, 321, 335, 339, 362, 371, 381, 386, 395, 398, 403

Offset: 1

Debapriyay Mukhopadhyay, Jul 13 2016

nonn

Equals (A001358 intersection A078892) - A001248.

Appears to be equal to A088710 without the 9. - R. J. Mathar, Jun 21 2025

			15 is in the sequence, because 15 = 3*5 is a semiprime with omega(15) = 2 and phi(15) - 1 = 2*4 - 1 = 7 is a prime.
21 is in the sequence, because 21 = 3*7 is a semiprime with omega(21) = 2 and phi(21) - 1 = 2*6 - 1 = 11 is a prime.

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

Cf. A001358, A078892, A001248.

[n: n in [1..500] |(EulerPhi(n)+DivisorSigma(1,n)) eq 2*(n+1) and IsPrime(EulerPhi(n)-1)]; // Vincenzo Librandi, Jul 29 2016

with(numtheory):
is_A271568 := n -> issqrfree(n) and bigomega(n) = 2 and isprime(phi(n)-1):
select(is_A271568, [$1..403]); # Peter Luschny, Jul 21 2016

A271568Q = SquareFreeQ[#] && PrimeNu[#] == 2 && PrimeQ[EulerPhi[#] - 1] &; Select[Range[500], A271568Q] (* JungHwan Min, Jul 29 2016 *)

is_a001358(n) = bigomega(n)==2
is_a005117(n) = issquarefree(n)
is_a078892(n) = ispseudoprime(eulerphi(n)-1)
is(n) = is_a001358(n) && is_a005117(n) && is_a078892(n) \\ Felix Fröhlich, Jul 21 2016

is(n)=my(f=factor(n)); f[,2]==[1,1]~ && isprime((f[1,1]-1)*(f[2,1]-1)-1) \\ Charles R Greathouse IV, Jul 21 2016

list(lim)=my(v=List()); forprime(p=2, sqrt(lim), forprime(q=p+1, lim\p, if(isprime((p-1)*(q-1)-1), listput(v, p*q)))); Set(v) \\ Charles R Greathouse IV, Aug 29 2016

New name from Charles R Greathouse IV, Jul 29 2016

A271550 Numbers n such that n is a squarefree semiprime (i.e., omega(n) = 2 = Omega(n)) and phi(n) + 1 is a prime.

Original entry on oeis.org

6, 10, 14, 21, 22, 26, 34, 38, 46, 55, 57, 58, 62, 74, 77, 82, 86, 91, 93, 94, 95, 106, 111, 115, 118, 119, 122, 133, 134, 142, 145, 146, 158, 166, 178, 194, 202, 206, 209, 214, 217, 218, 221, 226, 237, 254, 262, 274, 278, 287, 291, 295, 298, 302, 305, 314, 319, 326, 329

Offset: 1

Debapriyay Mukhopadhyay, Jul 13 2016

nonn

Equals (A001358 intersection A039698) - A001248.

			21 is in the sequence, because 21 = 3*7 is a semiprime with omega(21) = 2 and phi(21) + 1 = 2*6 + 1 = 13 is a prime.
55 is in the sequence, because 55 = 5*11 is a semiprime with omega(55) = 2 and phi(55) + 1 = 4*10 + 1 = 41 is a prime.

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

Cf. A001358, A039698, A001248.

Select[Range[400],SquareFreeQ[#]&&PrimeOmega[#]==2&&PrimeQ[EulerPhi[ #]+ 1]&] (* Harvey P. Dale, Aug 08 2020 *)

is(n)=my(f=factor(n)); f[, 2]==[1, 1]~ && isprime((f[1, 1]-1)*(f[2, 1]-1)+1) \\ Charles R Greathouse IV, Jul 21 2016

A270203 Primes p such that p+2^4, p+2^6, p+2^8, p+2^10, p+2^12, p+2^14 and p + 2^16 are all primes.

Original entry on oeis.org

163, 15667, 234067, 607093, 671353, 1447153, 1457857, 2162323, 5772097, 7717873, 9139453, 9549373, 11170933, 12039883, 13243063, 16442407, 16836163, 17784253, 18116473, 19433863, 21960577, 28209703, 29175283, 32380177, 33890803, 34613287, 34682113

Offset: 1

Debapriyay Mukhopadhyay, Jul 12 2016

nonn

			The prime 163 is in the sequence, since 163 + 16 = 179, 163 + 64 = 227, 163 + 256 = 419, 163 + 1024 = 1187, 163 + 4096 = 4259, 163 + 16384 = 16547 and 163 + 65536 = 65699 are all primes.

Dana Jacobsen, Table of n, a(n) for n = 1..10957

Subsequence of A269859.

Cf. A269257, A269258, A269259.

[p: p in PrimesInInterval(2,40000000) | forall{i: i in [16,64,256,1024,4096,16384,65536] | IsPrime(p+i)}]; // Vincenzo Librandi, Jul 16 2016

m = {2^4, 2^6, 2^8, 2^10, 2^12, 2^14, 2^16}; Select[Prime@ Range[3*10^6], Times @@ Boole@ PrimeQ[# + m] == 1 &] (* Michael De Vlieger, Jul 13 2016 *)
Select[Prime[Range[22*10^5]],AllTrue[#+2^Range[4,16,2],PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Dec 12 2018 *)

use ntheory ":all"; say for sieve_prime_cluster(2,1e8, 16,64,256,1024,4096,16384,65536); # Dana Jacobsen, Jul 13 2016

A269859 Primes p such that p+2^4, p+2^6, p+2^8, p+2^10, p+2^12 and p+2^14 are all primes.

Original entry on oeis.org

37, 163, 15667, 47287, 120607, 142543, 234067, 263047, 263803, 444607, 607093, 671353, 1447153, 1457857, 1562983, 2162323, 2694157, 2841337, 2979043, 3362143, 3567337, 4890307, 5037433, 5353987, 5772097, 6404773, 6776023, 7717873, 9139453, 9549373, 10550467

Offset: 1

Debapriyay Mukhopadhyay, Jul 12 2016

nonn

			The prime 37 is in the sequence, since 37 + 16 = 53, 37 + 64 = 101, 37 + 256 = 293, 37 + 1024 = 1061, 37 + 4096 = 4133 and 37 + 16384 = 16421 are all primes.
The prime 163 is in the sequence, since 163 + 16 = 179, 163 + 64 = 227, 163 + 256 = 419, 163 + 1024 = 1187, 163 + 4096 = 4259 and 163 + 16384 = 16547 are all primes.

Dana Jacobsen, Table of n, a(n) for n = 1..10801

Subsequence of A269259.

Cf. A269257, A269258.

[p: p in PrimesInInterval(2,12000000) | forall{i: i in [16,64,256,1024,4096,16384] | IsPrime(p+i)}]; // Vincenzo Librandi, Jul 16 2016

m = Map[2^# &, 2 Range[2, 7]]; Select[Prime@ Range[10^6], Times @@ Boole@ PrimeQ[# + m] == 1 &] (* Michael De Vlieger, Jul 13 2016 *)

use ntheory ":all"; say for sieve_prime_cluster(2,1e6, 16,64,256,1024,4096,16384); # Dana Jacobsen, Jul 13 2016

A269259 Primes p such that p+2^4, p+2^6, p+2^8, p+2^10 and p+2^12 are all primes.

Original entry on oeis.org

37, 163, 15667, 22093, 40177, 47287, 53593, 114577, 120607, 142543, 234067, 242377, 255907, 263047, 263803, 305407, 388117, 444607, 460387, 503287, 527143, 607093, 671353, 784897, 904663, 938947, 1063903, 1086493, 1172803, 1216807, 1233523, 1288543

Offset: 1

Debapriyay Mukhopadhyay, Jul 12 2016

nonn

			The prime 37 is in the sequence, since 37 + 16 = 53, 37 + 64 = 101, 37 + 256 = 293, 37 + 1024 = 1061 and 37 + 4096 = 4133 are all primes.
The prime 163 is in the sequence, since 163 + 16 = 179, 163 + 64 = 227, 163 + 256 = 419, 163 + 1024 = 1187 and 163 + 4096 = 4259 are all primes.

Dana Jacobsen, Table of n, a(n) for n = 1..10476

Subsequence of A269258.

Cf. A269257.

[p: p in PrimesInInterval(2,1600000) | forall{i: i in [16,64,256,1024,4096] | IsPrime(p+i)}]; // Vincenzo Librandi, Jul 16 2016

m = {2^4, 2^6, 2^8, 2^10, 2^12}; Select[Prime@ Range[2*10^5], Times @@ Boole@ PrimeQ[# + m] == 1 &] (* Michael De Vlieger, Jul 13 2016 *)

is(n) = for(k=2, 6, if(!ispseudoprime(2^(2*k)+n), return(0))); return(1)
forprime(p=1, 16e5, if(is(p), print1(p, ", "))) \\ Felix Fröhlich, Jul 12 2016

use ntheory ":all"; say for sieve_prime_cluster(2,1e6, 16,64,256,1024,4096); # Dana Jacobsen, Jul 13 2016

A269258 Primes p such that p+2^4, p+2^6, p+2^8 and p+2^10 are all primes.

Original entry on oeis.org

7, 37, 163, 337, 2647, 5023, 9157, 9277, 15667, 22093, 24907, 40177, 43597, 47287, 53593, 56893, 59077, 59497, 66553, 78877, 83407, 84793, 92737, 93307, 102043, 111577, 114577, 116953, 120607, 135193, 137383, 141397, 142543, 150067, 165463, 173713, 180007, 181903, 183943

Offset: 1

Debapriyay Mukhopadhyay, Jul 12 2016

nonn

			The prime 7 is in the sequence because 7+16 = 23, 7+64 = 71, 7+256 = 263 and 7+1024 = 1031 are all primes.
The prime 37 is in the sequence because 37+16 = 53, 37+64 = 101, 37+256 = 293 and 37+1024 = 1061 are all primes.

Dana Jacobsen, Table of n, a(n) for n = 1..10727
Debapriyay Mukhopadhyay, C program to generate the terms of the sequences A269257, A269258, A269259, A269859 and A270203 up to 10^8

Subsequence of A269257.

[p: p in PrimesInInterval(2,200000) | forall{i: i in [16,64,256,1024] | IsPrime(p+i)}]; // Vincenzo Librandi, Jul 16 2016

Select[Prime@ Range[10^5], Times @@ Boole@ PrimeQ[# + 2^{4, 6, 8, 10}] == 1 &] (* Michael De Vlieger, Jul 13 2016 *)

use ntheory ":all"; say for sieve_prime_cluster(2,1e5, 16,64,256,1024); # Dana Jacobsen, Jul 13 2016

A269257 INTERSECT A361485. - R. J. Mathar, Mar 26 2024

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User: Debapriyay Mukhopadhyay

A354078 Squares k that are not divisible by 10, and whose reverse and digit sum are also squares, such that the digit sum divides both k and its reverse.

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A316312 Numbers k such that the sum of the digits of the numbers 1, 2, 3, ... up to (k - 1) is divisible by k.

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A276509 Numbers k in base 10 such that the digits of 2 + k are the digits of 2k written in reverse order.

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A275475 Primes p such that p+2^3, p+2^5 and p+2^7 are all primes.

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A271568 Squarefree semiprimes n such that phi(n) - 1 is prime.

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A271550 Numbers n such that n is a squarefree semiprime (i.e., omega(n) = 2 = Omega(n)) and phi(n) + 1 is a prime.

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A270203 Primes p such that p+2^4, p+2^6, p+2^8, p+2^10, p+2^12, p+2^14 and p + 2^16 are all primes.

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