Gary Croft - cp's OEIS frontend

A301628 Numbers not divisible by 2, 3 or 5 (A007775) with digital root 7.

7, 43, 61, 79, 97, 133, 151, 169, 187, 223, 241, 259, 277, 313, 331, 349, 367, 403, 421, 439, 457, 493, 511, 529, 547, 583, 601, 619, 637, 673, 691, 709, 727, 763, 781, 799, 817, 853, 871, 889, 907, 943, 961, 979, 997, 1033, 1051, 1069, 1087, 1123

Offset: 1

Gary Croft, Mar 24 2018

Numbers == {7, 43, 61, 79} mod 90 with additive sum sequence 7{+36+18+18+18} {repeat ...}. Includes all prime numbers > 5 with digital root 7.

			7+36=43; 43+18=61; 61+18=79; 79+18=97; 97+36=133.

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).

Intersection of A007775 and A017245.

Filtered(Filtered([1..1200],n->n mod 2 <> 0 and n mod 3 <> 0 and n mod 5 <> 0),i->i-9*Int((i-1)/9)=7); # Muniru A Asiru, Apr 22 2018

Vec(x*(7 + 36*x + 18*x^2 + 18*x^3 + 11*x^4) / ((1 - x)^2*(1 + x)*(1 + x^2)) + O(x^40)) \\ Colin Barker, Sep 21 2019

Numbers == {7, 43, 61, 79} mod 90.
From Colin Barker, Sep 21 2019: (Start)
G.f.: x*(7 + 36*x + 18*x^2 + 18*x^3 + 11*x^4) / ((1 - x)^2*(1 + x)*(1 + x^2)).
a(n) = a(n-1) + a(n-4) - a(n-5) for n>5.
(End)

A301623 Numbers not divisible by 2, 3 or 5 (A007775) with digital root 5.

Original entry on oeis.org

23, 41, 59, 77, 113, 131, 149, 167, 203, 221, 239, 257, 293, 311, 329, 347, 383, 401, 419, 437, 473, 491, 509, 527, 563, 581, 599, 617, 653, 671, 689, 707, 743, 761, 779, 797, 833, 851, 869, 887, 923, 941, 959, 977, 1013, 1031, 1049, 1067, 1103, 1121

Offset: 1

Gary Croft, Mar 24 2018

Numbers == {23, 41, 59, 77} mod 90 with additive sum sequence 23{+18+18+18+36} {repeat ...}. Includes all primes number > 5 with digital root 5.

			23+18=41; 41+18=59; 59+18=77; 77+36=113; 113+18=131.

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).

Intersection of A007775 and A017221.

Filtered(Filtered([1..1200],n->n mod 2 <> 0 and n mod 3 <> 0 and n mod 5 <> 0),i->i-9*Int((i-1)/9)=5); # Muniru A Asiru, Apr 22 2018

LinearRecurrence[{1,0,0,1,-1},{23,41,59,77,113},50] (* Harvey P. Dale, Jul 28 2018 *)

Vec(x*(23 + 18*x + 18*x^2 + 18*x^3 + 13*x^4) / ((1 - x)^2*(1 + x)*(1 + x^2)) + O(x^60)) \\ Colin Barker, Mar 25 2018

Numbers == {23, 41, 59, 77} mod 90.
From Colin Barker, Mar 25 2018: (Start)
G.f.: x*(23 + 18*x + 18*x^2 + 18*x^3 + 13*x^4) / ((1 - x)^2*(1 + x)*(1 + x^2)).
a(n) = a(n-1) + a(n-4) - a(n-5) for n>5.
(End)

A301622 Numbers not divisible by 2, 3 or 5 (A007775) with digital root 4.

Original entry on oeis.org

13, 31, 49, 67, 103, 121, 139, 157, 193, 211, 229, 247, 283, 301, 319, 337, 373, 391, 409, 427, 463, 481, 499, 517, 553, 571, 589, 607, 643, 661, 679, 697, 733, 751, 769, 787, 823, 841, 859, 877, 913, 931, 949, 967, 1003, 1021, 1039, 1057, 1093, 1111

Offset: 1

Gary Croft, Mar 24 2018

Numbers == {13, 31, 49, 67} mod 90 with additive sum sequence 13{+18+18+18+36} {repeat ...}. Includes all prime numbers > 5 with digital root 4.

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).

Intersection of A007775 and A017209.

Filtered(Filtered([1..1200],n->n mod 2 <> 0 and n mod 3 <> 0 and n mod 5 <> 0),i->i-9*Int((i-1)/9)=4); # Muniru A Asiru, Apr 22 2018

Rest@ CoefficientList[Series[x (13 + 18 x + 18 x^2 + 18 x^3 + 23 x^4)/((1 - x)^2*(1 + x) (1 + x^2)), {x, 0, 50}], x] (* Michael De Vlieger, Apr 21 2018 *)
LinearRecurrence[{1,0,0,1,-1},{13,31,49,67,103},50] (* Harvey P. Dale, May 11 2019 *)

Vec(x*(13 + 18*x + 18*x^2 + 18*x^3 + 23*x^4) / ((1 - x)^2*(1 + x)*(1 + x^2)) + O(x^60)) \\ Colin Barker, Mar 25 2018

Numbers == {13, 31, 49, 67} mod 90.
From Colin Barker, Mar 25 2018: (Start)
G.f.: x*(13 + 18*x + 18*x^2 + 18*x^3 + 23*x^4) / ((1 - x)^2*(1 + x)*(1 + x^2)).
a(n) = a(n-1) + a(n-4) - a(n-5) for n>5.
(End)

Last term corrected by Colin Barker, Mar 25 2018

A301621 Numbers not divisible by 2, 3 or 5 (A007775) with digital root 2.

Original entry on oeis.org

11, 29, 47, 83, 101, 119, 137, 173, 191, 209, 227, 263, 281, 299, 317, 353, 371, 389, 407, 443, 461, 479, 497, 533, 551, 569, 587, 623, 641, 659, 677, 713, 731, 749, 767, 803, 821, 839, 857, 893, 911, 929, 947, 983, 1001, 1019, 1037, 1073, 1091, 1109

Offset: 1

Gary Croft, Mar 24 2018

Numbers congruent to 11, 29, 47, or 83 mod 90 with additive sum sequence 11 { + 18 + 18 + 36 + 18} {repeat ...}. Includes all prime numbers greater than 5 with digital root 2.

			11+18=29; 29+18=47; 47+36=83; 83+18=101; 101+18=119.

Index entries for linear recurrences with constant coefficients, signature (1, 0, 0, 1, -1).

Intersection of A007775 and A017185.

Filtered(Filtered([1..1200],n->n mod 2 <> 0 and n mod 3 <> 0 and n mod 5 <> 0),i->i-9*Int((i-1)/9)=2); # Muniru A Asiru, Apr 22 2018

Flatten[Table[90n - {79, 61, 43, 7}, {n, 30}]] (* Alonso del Arte, Mar 29 2018 *)

Vec(x*(11 + 18*x + 18*x^2 + 36*x^3 + 7*x^4) / ((1 - x)^2*(1 + x)*(1 + x^2)) + O(x^60)) \\ Colin Barker, Mar 26 2018

n == {11, 29, 47, 83} mod 90.
From Colin Barker, Mar 26 2018: (Start)
G.f.: x*(11 + 18*x + 18*x^2 + 36*x^3 + 7*x^4) / ((1 - x)^2*(1 + x)*(1 + x^2)).
a(n) = a(n-1) + a(n-4) - a(n-5) for n>5.
(End)

A295869 Numbers not divisible by 2, 3 or 5 (A007775) with digital root 8.

Original entry on oeis.org

17, 53, 71, 89, 107, 143, 161, 179, 197, 233, 251, 269, 287, 323, 341, 359, 377, 413, 431, 449, 467, 503, 521, 539, 557, 593, 611, 629, 647, 683, 701, 719, 737, 773, 791, 809, 827, 863, 881, 899, 917, 953, 971, 989, 1007, 1043, 1061, 1079, 1097, 1133

Offset: 1

Gary Croft, Mar 24 2018

Numbers == {17, 53, 71, 89} mod 90 with additive sum sequence 17{+36+18+18+18} {repeat ...}. Includes all prime numbers >5 with digital root 8.

			17+36=53; 53+18=71; 71+18=89; 89+18=107; 107+36=143.

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).

Intersection of A007775 and A017257.

Filtered([1..1200],n->n mod 2<>0 and n mod 3 <>0 and n mod 5<>0 and n-9*Int((n-1)/9)=8); # Muniru A Asiru, May 30 2018

select(n->modp(n,2)<>0 and modp(n,3)<>0 and modp(n,5)<>0 and n-9*floor((n-1)/9)=8,[$1..1200]); # Muniru A Asiru, May 30 2018

Vec(x*(17 + 36*x + 18*x^2 + 18*x^3 + x^4) / ((1 - x)^2*(1 + x)*(1 + x^2)) + O(x^60)) \\ Colin Barker, Mar 26 2018

Numbers == {17, 53, 71, 89} mod 90.
From Colin Barker, Mar 26 2018: (Start)
G.f.: x*(17 + 36*x + 18*x^2 + 18*x^3 + x^4) / ((1 - x)^2*(1 + x)*(1 + x^2)).
a(n) = (5 + 9*(-1)^n - (9+9*i)*(-i)^n - (9-9*i)*i^n + 90*n) / 4, where i=sqrt(-1).
a(n) = a(n-1) + a(n-4) - a(n-5) for n>5.
(End)

A301617 Numbers not divisible by 2, 3 or 5 (A007775) with digital root 1.

Original entry on oeis.org

1, 19, 37, 73, 91, 109, 127, 163, 181, 199, 217, 253, 271, 289, 307, 343, 361, 379, 397, 433, 451, 469, 487, 523, 541, 559, 577, 613, 631, 649, 667, 703, 721, 739, 757, 793, 811, 829, 847, 883, 901, 919, 937, 973, 991, 1009, 1027, 1063, 1081, 1099

Offset: 1

Gary Croft, Mar 24 2018

Numbers == {1, 19, 37, 73} mod 90 with additive sum sequence 1{+18+18+36+18} {repeat ...}. Includes all prime numbers > 7 with digital root 1.

			1+18=19; 19+18=37; 37+36=73; 73+18=91; 91+18=109.

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).

Cf. A177704, A045572.

Intersection of A007775 and A017173.

seq(seq(i+90*j,i=[1,19,37,73]),j=0..30); # Robert Israel, Mar 25 2018

LinearRecurrence[{1,0,0,1,-1},{1,19,37,73,91},50] (* Harvey P. Dale, Dec 14 2019 *)

a(n) = 1 + 18 * (n - 1 + n\4) \\ David A. Corneth, Mar 24 2018

Vec(x*(1 + 18*x + 18*x^2 + 36*x^3 + 17*x^4) / ((1 - x)^2*(1 + x)*(1 + x^2)) + O(x^60)) \\ Colin Barker, Mar 24 2018

n == {1, 19, 37, 73} mod 90.
a(n + 1) = a(n) + 18 * A177704(n + 1). - David A. Corneth, Mar 24 2018
From Colin Barker, Mar 24 2018: (Start)
G.f.: x*(1 + 18*x + 18*x^2 + 36*x^3 + 17*x^4) / ((1 - x)^2*(1 + x)*(1 + x^2)).
a(n) = a(n-1) + a(n-4) - a(n-5) for n>5.
(End)

The missing term 1081 added to the sequence by Colin Barker, Mar 24 2018

A261371 Prime numbers that sum to a prime when added to their digital root.

Original entry on oeis.org

11, 13, 29, 53, 67, 71, 89, 101, 103, 137, 191, 193, 227, 229, 233, 269, 281, 359, 431, 449, 461, 463, 499, 569, 593, 641, 643, 659, 683, 701, 719, 769, 821, 823, 857, 859, 877, 967, 1019, 1061, 1091, 1093, 1223, 1289, 1439, 1451, 1487, 1489, 1579, 1597, 1601, 1619

Offset: 1

Gary Croft, Sep 11 2015

			11 (prime) + 2 (its digital root) = 13 (prime);
89 (prime) + 8 (its digital root) = 97 (prime).

Cf. A000040, A010888, A038194.

[p: p in PrimesUpTo(2000) | IsPrime(p+(p mod 9))]; // Vincenzo Librandi, Sep 29 2015

Select[Prime[Range[300]], PrimeQ[# + Mod[#, 9]] &] (* Vincenzo Librandi, Sep 29 2015 *)

lista(nn) = forprime(p=2, nn, if (isprime(p + p%9), print1(p, ", "))); \\ Michel Marcus, Sep 12 2015

(a(n) mod 9) mod 2 = 0. - Altug Alkan, Sep 28 2015

A246508 Digital root of numbers congruent to {1,7,11,13,17,19,23,29} modulo 30.

Original entry on oeis.org

1, 7, 2, 4, 8, 1, 5, 2, 4, 1, 5, 7, 2, 4, 8, 5, 7, 4, 8, 1, 5, 7, 2, 8, 1, 7, 2, 4, 8, 1, 5, 2, 4, 1, 5, 7, 2, 4, 8, 5, 7, 4, 8, 1, 5, 7, 2, 8, 1, 7, 2, 4, 8, 1, 5, 2, 4, 1, 5, 7, 2, 4, 8, 5, 7, 4, 8, 1, 5, 7, 2, 8, 1, 7, 2, 4, 8, 1, 5, 2, 4, 1, 5, 7, 2, 4, 8, 5, 7, 4

Offset: 1

Gary Croft, Nov 14 2014

Period 24 repeating sequence, the digital root squares of which produce period 24 palindromic sequence A240924.

Index entries for linear recurrences with constant coefficients, signature (0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1).

Cf. A007775 (numbers not divisible by 2, 3 or 5), A240924 (digital root of this sequence squared).

a(n) = A010888(A007775(n)). - Michel Marcus, Nov 25 2014

G.f.: ( -x*(1 +7*x +2*x^2 +4*x^3 +8*x^4 +x^5 +5*x^6 +2*x^7 +4*x^8 +x^9 +5*x^10 +7*x^11 +2*x^12 +4*x^13 +8*x^14 +5*x^15 +7*x^16 +4*x^17 +8*x^18 +x^19 +5*x^20 +7*x^21 +2*x^22 +8*x^23) ) / ( (x-1) *(1+x+x^2) *(1+x) *(1-x+x^2) *(1+x^2) *(x^4-x^2+1) *(1+x^4) *(x^8-x^4+1) ). - R. J. Mathar, Sep 22 2016

A240924 Digital root of squares of numbers not divisible by 2, 3 or 5.

Original entry on oeis.org

1, 4, 4, 7, 1, 1, 7, 4, 7, 1, 7, 4, 4, 7, 1, 7, 4, 7, 1, 1, 7, 4, 4, 1, 1, 4, 4, 7, 1, 1, 7, 4, 7, 1, 7, 4, 4, 7, 1, 7, 4, 7, 1, 1, 7, 4, 4, 1, 1, 4, 4, 7, 1, 1, 7, 4, 7, 1, 7, 4, 4, 7, 1, 7, 4, 7, 1, 1, 7, 4, 4, 1

Offset: 1

Gary Croft, Aug 15 2014

This period 24 repeating sequence is palindromic.

			The first 8 numbers not divisible by 2, 3 or 5 are 1,7,11,13,17,19,23,29; with squares 1,49,121,169,289,361,529,841 and digital root sequence of 1,4,4,7,1,1,7,4.

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (2,-1,-1,2,-1,-1,2,-2,1,1,-2,1,1,-2,1).

Cf. A007775, A166923.

Vec(x*(1 + x)^2*(1 - 4*x^2 + 12*x^3 - 27*x^4 + 45*x^5 - 53*x^6 + 45*x^7 - 27*x^8 + 12*x^9 - 4*x^10 + x^12) / ((1 - x)*(1 - x + x^2)*(1 - x^2 + x^4)*(1 - x^4 + x^8)) + O(x^100)) \\ Colin Barker, Sep 21 2019

A240924 = [1 + (n*n-1) % 9 for n in range(1,10**3,2) if n % 3 and n % 5 ]
# Chai Wah Wu, Sep 03 2014

From Colin Barker, Sep 21 2019: (Start)
G.f.: x*(1 + x)^2*(1 - 4*x^2 + 12*x^3 - 27*x^4 + 45*x^5 - 53*x^6 + 45*x^7 - 27*x^8 + 12*x^9 - 4*x^10 + x^12) / ((1 - x)*(1 - x + x^2)*(1 - x^2 + x^4)*(1 - x^4 + x^8)).
a(n) = 2*a(n-1) - a(n-2) - a(n-3) + 2*a(n-4) - a(n-5) - a(n-6) + 2*a(n-7) - 2*a(n-8) + a(n-9) + a(n-10) - 2*a(n-11) + a(n-12) + a(n-13) - 2*a(n-14) + a(n-15) for n>15.
(End)

A230113 Digital root of summed Fibonacci and Lucas digital roots indexed by numbers not divisible by 2, 3 or 5.

Original entry on oeis.org

3, 4, 5, 6, 6, 5, 4, 3, 4, 6, 6, 5, 4, 3, 3, 5, 6, 5, 4, 3, 3, 4, 5, 6, 5, 3, 3, 4, 5, 6, 6, 4, 3, 4, 5, 6, 6, 5, 4, 3, 4, 6, 6, 5, 4, 3, 3, 5, 6, 5, 4, 3, 3, 4, 5, 6, 5, 3, 3, 4, 5, 6, 6, 4, 3, 4, 5, 6, 6, 5, 4, 3, 4, 6, 6, 5, 4, 3, 3, 5, 6, 5, 4, 3, 3, 4, 5, 6, 5, 3, 3, 4, 5, 6, 6, 4

Offset: 1

Gary Croft, Dec 20 2013

32-beat repeating sequence is periodically palindromic starting at Length(40), then at Lengths (72)...(104)...(136)...(168)...{+32 terms ... repeat ... n}.

			Referencing A227896 (Fibo) and A233766 (Lucas): 1st Fibo term (1) + 1st Lucas term (2) = 3 = digital root 3. Likewise, 2nd Fibo term (4) + 2nd Lucas term (9) = 13 = digital root 4.

Cf. A007775, A227896, A233766, A000032, A000033.

Conjectures from Colin Barker, Sep 22 2019: (Start)
G.f.: x*(3 + x + x^2 + x^3 - x^5 - x^6 - x^7 + x^8 + 2*x^9 - x^11 - x^12 - x^13 + 2*x^15 + 4*x^16) / ((1 - x)*(1 + x^16)).
a(n) = a(n-1) - a(n-16) + a(n-17) for n>17.
(End)

cp's OEIS Frontend

User: Gary Croft

A301628 Numbers not divisible by 2, 3 or 5 (A007775) with digital root 7.

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A301623 Numbers not divisible by 2, 3 or 5 (A007775) with digital root 5.

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A301622 Numbers not divisible by 2, 3 or 5 (A007775) with digital root 4.

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A301621 Numbers not divisible by 2, 3 or 5 (A007775) with digital root 2.

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A295869 Numbers not divisible by 2, 3 or 5 (A007775) with digital root 8.

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A301617 Numbers not divisible by 2, 3 or 5 (A007775) with digital root 1.

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