A330927 -id:A330927 - cp's OEIS frontend

A085775 Numbers k such that (k / sum of digits of k) and (k+1 / sum of digits of k+1) are both prime.

152, 803, 1016, 1853, 3031, 3032, 3438, 7361, 7542, 7587, 8226, 8337, 10095, 10278, 10307, 11354, 11646, 13116, 13117, 13881, 17153, 21434, 21906, 23412, 26221, 28824, 30254, 31112, 32166, 34218, 35513, 38322, 40335, 41058, 44373, 45380

Offset: 1

Jason Earls, Jul 23 2003

			152 is a term since 152/(1+5+2) = 19 and 153/(1+5+3) = 17 are both prime.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Subsequence of A001101 and A330927.

moranQ[n_] := PrimeQ[n / Plus @@ IntegerDigits[n]]; Select[Range[50000], moranQ[#] && moranQ[#+1] &] (* Amiram Eldar, Apr 25 2020 *)

Offset corrected by Amiram Eldar, Apr 25 2020

A364217 Numbers k such that k and k+1 are both Jacobsthal-Niven numbers (A364216).

Original entry on oeis.org

1, 2, 3, 8, 11, 14, 15, 27, 32, 42, 43, 44, 45, 51, 56, 75, 86, 87, 92, 95, 99, 104, 125, 128, 135, 144, 155, 171, 176, 182, 183, 195, 204, 264, 267, 275, 287, 305, 344, 363, 375, 387, 428, 444, 455, 474, 497, 512, 524, 535, 544, 545, 552, 555, 581, 605, 623, 639

Offset: 1

Amiram Eldar, Jul 14 2023

A001045(2*n+1) = A007583(n) = (2^(2*n+1) + 1)/3 is a term for n >= 0, since its representation is 2*n 1's, so A364215(A001045(2*n+1)) = 1 divides A001045(2*n+1), and the representation of A001045(2*n+1) + 1 = (2^(2*n+1) + 4)/3 is max(2*n-1, 0) 0's between 2 1's, so A364215(A001045(2*n+1) + 1) = 2 which divides (2^(2*n+1) + 4)/3.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A364215.
Subsequence of A364216.
Subsequences: A364218, A364219, A364220, A364221.
Similar sequences: A330927, A328205, A328209, A328213, A330931, A331086, A333427, A334309, A331820, A342427, A344342, A351715, A351720, A352090, A352108, A352321, A352343, A352509.

consecJacobsthalNiven[kmax_, len_] := Module[{m = 1, c = Table[False, {len}], s = {}}, Do[c = Join[Rest[c], {Divisible[k, DigitCount[m, 2, 1]]}]; While[m++; OddQ[IntegerExponent[m, 2]]]; If[And @@ c, AppendTo[s, k - len + 1]], {k, 1, kmax}]; s]; consecJacobsthalNiven[640, 2]

lista(kmax, len) = {my(m = 1, c = vector(len)); for(k = 1, kmax, c = concat(vecextract(c, "^1"), !(k % sumdigits(m, 2))); until(valuation(m, 2)%2 == 0, m++); if(vecsum(c) == len, print1(k-len+1, ", ")));}
lista(640, 2)

A364380 Numbers k such that k and k+1 are both greedy Jacobsthal-Niven numbers (A364379).

Original entry on oeis.org

1, 2, 3, 4, 5, 8, 9, 10, 11, 14, 15, 20, 21, 26, 27, 32, 42, 43, 44, 45, 51, 56, 68, 75, 84, 85, 86, 87, 92, 99, 104, 105, 111, 115, 116, 125, 128, 135, 144, 155, 170, 171, 176, 182, 183, 195, 204, 213, 219, 224, 260, 264, 267, 275, 304, 305, 324, 329, 341, 344

Offset: 1

Amiram Eldar, Jul 21 2023

The positive Jacobsthal numbers, A001045(n) for n >= 1, are terms since their representation in Jacobsthal greedy base is one 1 followed by n-1 0's, so A265745(A001045(n)) = 1 divides A001045(n), and the representation of A001045(n) + 1 is 2 if n <= 2 and otherwise n-3 0's between two 1's, so A265745(A001045(n) + 1) = 2 which divides A001045(n) + 1.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A265745, A265747.
Subsequence of A364379.
Subsequences: A364381, A364382, A364383.
Similar sequences: A330927, A328205, A328209, A328213, A330931, A331086, A333427, A334309, A331820, A342427, A344342, A351715, A351720, A352090, A352108, A352321, A352343, A352509, A364217.

consecGreedyJN[kmax_, len_] := Module[{m = 1, c = Table[False, {len}], s = {}}, Do[c = Join[Rest[c], {greedyJacobNivenQ[k]}]; If[And @@ c, AppendTo[s, k - len + 1]], {k, 1, kmax}]; s]; consecGreedyJN[350, 2] (* using the function greedyJacobNivenQ[n] from A364379 *)

lista(kmax, len) = {my(c = vector(len)); for(k = 1, kmax, c = concat(vecextract(c, "^1"), isA364379(k)); if(vecsum(c) == len, print1(k-len+1, ", ")));} \\ using the function isA364379(n) from A364379
lista(350, 2)

A337076 Niven numbers (A005349) with a record gap to the next Niven number.

Original entry on oeis.org

1, 10, 12, 63, 72, 90, 288, 378, 558, 2889, 3784, 6480, 19872, 28971, 38772, 297864, 478764, 589860, 989867, 2879865, 9898956, 49989744, 88996914, 689988915, 879987906, 989888823, 2998895823, 6998899824, 8889999624, 8988988866, 9879997824, 18879988824, 286889989806

Offset: 1

Amiram Eldar, Aug 14 2020

The corresponding record gaps are 1, 2, 6, 7, 8, 10, 12, 14, 18, 23, 32, 36, 44, 45, 54, 60, 66, 72, 88, 90, 99, 108, 126, 135, 144, 150, 153, 192, 201, 234, 258, 276, 294, ...
Kennedy and Cooper (1984) proved that the asymptotic density of the Niven numbers is 0. Therefore, this sequence is infinite.
De Koninck and Doyon proved that for sufficiently large k the least number m such that the interval[m, m+k-1] does not contain any Niven numbers is < (100*(k+2))^(k+3).

			10 is a term since it is a Niven number, and the next Niven number is 12, with a gap 12 - 10 = 2, which is a record, since all the numbers below 10 are also Niven numbers.

Jean-Marie De Koninck and Nicolas Doyon, Large and Small Gaps Between Consecutive Niven Numbers, J. Integer Seqs., Vol. 6, 2003, Article 03.2.5.
R. E. Kennedy and C. N. Cooper, On the natural density of the Niven numbers, The College Mathematics Journal, Vol. 15, No. 4 (1984), pp. 309-312.

Cf. A005349, A330927, A337077.

nivenQ[n_] := Divisible[n, Plus @@ IntegerDigits[n]]; gapmax = 0; n1 = 1; s = {}; Do[If[nivenQ[n], gap = n - n1; If[gap > gapmax, gapmax = gap; AppendTo[s, n1]]; n1 = n], {n, 2, 10^6}]; s

A381582 Numbers k such that k and k+1 are both terms in A381581.

Original entry on oeis.org

1, 2, 3, 20, 21, 27, 44, 55, 56, 57, 75, 95, 110, 111, 115, 152, 175, 207, 264, 287, 291, 304, 305, 344, 364, 365, 377, 380, 395, 398, 399, 404, 425, 435, 455, 534, 584, 605, 815, 846, 847, 864, 888, 930, 987, 992, 1004, 1011, 1024, 1025, 1064, 1084, 1085, 1145, 1182

Offset: 1

Amiram Eldar, Feb 28 2025

If k is not divisible by 3 (A001651), then A001906(k) = Fibonacci(2*k) is a term.

			1 is a term since A291711(1) = 1 divides 1 and A291711(2) = 2 divides 2.
20 is a term since A291711(20) = 4 divides 20 and A291711(21) = 1 divides 21.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000045, A001906, A001651, A291711.
Subsequence of A381581.
Subsequences: A381583, A381584, A381585.
Similar sequences: A328209, A330927, A330931, A351720.

f[n_] := f[n] = Fibonacci[2*n]; q[n_] := q[n] = Module[{s = 0, m = n, k}, While[m > 0, k = 1; While[m > f[k], k++]; If[m < f[k], k--]; If[m >= 2*f[k], s += 2; m -= 2*f[k], s++; m -= f[k]]]; Divisible[n, s]]; Select[Range[1200], q[#] && q[#+1] &]

mx = 20; fvec = vector(mx, i, fibonacci(2*i)); f(n) = if(n <= mx, fvec[n], fibonacci(2*n));
is1(n) = {my(s = 0, m = n, k); while(m > 0, k = 1; while(m > f(k), k++); if(m < f(k), k--); if(m >= 2*f(k), s += 2; m -= 2*f(k), s++; m -= f(k))); !(n % s);}
list(lim) = {my(q1 = is1(1), q2); for(k = 2, lim, q2 = is1(k); if(q1 && q2, print1(k-1, ", ")); q1 = q2);}

A331089 Positive numbers k such that -k and -(k + 1) are both negative negaFibonacci-Niven numbers (A331088).

Original entry on oeis.org

1, 2, 3, 15, 20, 21, 44, 50, 54, 55, 56, 57, 75, 104, 110, 111, 115, 128, 141, 152, 175, 207, 264, 291, 304, 308, 335, 351, 363, 376, 377, 380, 392, 398, 399, 435, 452, 455, 534, 584, 594, 605, 623, 654, 735, 740, 744, 753, 795, 804, 875, 884, 897, 924, 964, 968

Offset: 1

Amiram Eldar, Jan 08 2020

The Fibonacci numbers F(6*k + 2) and F(6*k + 4) are terms.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A328209, A328213, A330927, A330931, A331086, A331088.

ind[n_] := Floor[Log[Abs[n]*Sqrt[5] + 1/2]/Log[GoldenRatio]];
f[1] = 1; f[n_] := If[n > 0, i = ind[n - 1]; If[EvenQ[i], i++]; i, i = ind[-n]; If[OddQ[i], i++]; i];
negaFibTermsNum[n_] := Module[{k = n, s = 0}, While[k != 0, i = f[k]; s += 1; k -= Fibonacci[-i]]; s];
negFibQ[n_] := Divisible[n, negaFibTermsNum[-n]];
nConsec = 2; neg = negFibQ /@ Range[nConsec]; seq = {}; c = 0;
k = nConsec+1; While[c < 55, If[And @@ neg, c++; AppendTo[seq, k - nConsec]];neg = Join[Rest[neg], {negFibQ[k]}]; k++]; seq

A364007 Numbers k such that k and k+1 are both Wythoff-Niven numbers (A364006).

Original entry on oeis.org

3, 6, 7, 20, 39, 51, 54, 55, 90, 135, 143, 294, 305, 321, 356, 365, 369, 374, 375, 376, 784, 800, 924, 978, 979, 980, 986, 1904, 1945, 1970, 2043, 2199, 2232, 2289, 2394, 2424, 2439, 2499, 2525, 2562, 2580, 2583, 4185, 4598, 4707, 4774, 4790, 4796, 4879, 5004

Offset: 1

Amiram Eldar, Jul 01 2023

A035508(n) = Fibonacci(2*n+2) - 1 is a term for n >= 2 since A135818(Fibonacci(2*n+2) - 1) = A135818(Fibonacci(2*n+2)) = 1.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A035508, A135818.
Subsequence of A364006.
Subsequences: A364008, A364009.
Similar sequences: A330927, A328205, A328209, A328213, A330931, A331086, A333427, A334309, A331820, A342427, A344342, A351715, A351720, A352090, A352108, A352321, A352509.

seq[count_, nConsec_] := Module[{cn = wnQ /@ Range[nConsec], s = {}, c = 0, k = nConsec + 1}, While[c < count, If[And @@ cn, c++; AppendTo[s, k - nConsec]]; cn = Join[Rest[cn], {wnQ[k]}]; k++]; s]; seq[50, 2] (* using the function wnQ[n] from A364006 *)

A364124 Numbers k such that k and k+1 are both Stolarsky-Niven numbers (A364123).

Original entry on oeis.org

8, 56, 84, 159, 195, 224, 384, 399, 405, 995, 1140, 1224, 1245, 1295, 1309, 1419, 1420, 1455, 1474, 1507, 2585, 2597, 2600, 2680, 2681, 2727, 2744, 2750, 2799, 2855, 3122, 3311, 3339, 3345, 3618, 3707, 3795, 4004, 6770, 6774, 6984, 6985, 7014, 7074, 7154, 7405

Offset: 1

Amiram Eldar, Jul 07 2023

Amiram Eldar, Table of n, a(n) for n = 1..10000

Subsequence of A364123.
Subsequences: A364125, A364126.
Similar sequences: A330927, A328205, A328209, A328213, A330931, A331086, A333427, A334309, A331820, A342427, A344342, A351715, A351720, A352090, A352108, A352321, A352343, A352509.

seq[count_, nConsec_] := Module[{cn = stolNivQ /@ Range[nConsec], s = {}, c = 0, k = nConsec + 1}, While[c < count, If[And @@ cn, c++; AppendTo[s, k - nConsec]]; cn = Join[Rest[cn], {stolNivQ[k]}]; k++]; s]; seq[50, 2] (* using the function stolNivQ[n] from A364123 *)

lista(count, nConsec) = {my(cn = vector(nConsec, i, isStolNivQ(i)), c = 0, k = nConsec + 1); while(c < count, if(vecsum(cn) == nConsec, c++; print1(k-nConsec, ", ")); cn = concat(vecextract(cn, "^1"), isStolNivQ(k)); k++);} \\ using the function isA364123(n) from A364123
lista(50, 2)

A338514 Numbers k such that k and k+1 are both divisible by the total binary weight of their divisors (A093653).

Original entry on oeis.org

1, 2, 54, 2119, 11100, 13727, 14382, 15799, 16399, 20159, 20950, 33421, 34617, 36328, 36396, 39400, 42198, 42438, 42650, 46253, 46873, 50370, 55368, 56600, 58793, 67013, 67320, 69023, 72325, 76057, 86393, 90781, 92906, 93216, 105909, 132088, 134028, 134823, 140466

Offset: 1

Amiram Eldar, Oct 31 2020

Numbers k such that k and k+1 are both in A093705, or, equivalently, k is divisible by A093653(k) and k+1 is divisible by A093653(k+1).

			1 is a term since 1 and 2 are both terms of A093705.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000120, A093653, A093705.

Similar sequences: A330927, A330931, A334345, A338452.

divQ[n_] := Divisible[n, DivisorSum[n, DigitCount[#, 2, 1] &]]; q1 = divQ[1]; Reap[Do[q2 = divQ[n]; If[q1 && q2, Sow[n - 1]]; q1 = q2, {n, 2, 10^5}]][[2, 1]]
SequencePosition[Table[If[Divisible[n,Total[DigitCount[Divisors[n],2,1]]],1,0],{n,150000}],{1,1}][[All,1]] (* Harvey P. Dale, Jun 14 2022 *)

A331821 Positive numbers k such that -k and -(k + 1) are both negabinary-Niven numbers (A331728).

Original entry on oeis.org

2, 3, 8, 9, 15, 24, 27, 32, 33, 39, 54, 55, 63, 77, 111, 114, 115, 123, 128, 129, 135, 144, 159, 174, 175, 203, 234, 235, 245, 255, 264, 294, 295, 329, 370, 371, 384, 413, 414, 415, 444, 447, 474, 475, 495, 504, 507, 512, 513, 519, 534, 535, 543, 580, 581, 624

Offset: 1

Amiram Eldar, Jan 27 2020

			8 is a term since both -8 and -(8 + 1) = -9 are negabinary-Niven numbers: A039724(-8) = 1000 and 1 + 0 + 0 + 0 = 1 is a divisor of 8, and A039724(-9) = 1011 and 1 + 0 + 1 + 1 = 3 is a divisor of 9.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A005352, A027615, A039724, A328209, A328213, A330927, A330931, A331086, A331089, A331728, A331819, A331820.

negaBinWt[n_] := negaBinWt[n] = If[n == 0, 0, negaBinWt[Quotient[n - 1, -2]] + Mod[n, 2]]; negaBinNivenQ[n_] := Divisible[n, negaBinWt[-n]]; c = 0; k = 1; s = {}; v = Table[-1, {2}]; While[c < 60, If[negaBinNivenQ[k], v = Join[Rest[v], {k}]; If[AllTrue[Differences[v], # == 1 &], c++; AppendTo[s, k - 1]]]; k++]; s

cp's OEIS Frontend

A085775 Numbers k such that (k / sum of digits of k) and (k+1 / sum of digits of k+1) are both prime.

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A364217 Numbers k such that k and k+1 are both Jacobsthal-Niven numbers (A364216).

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A364380 Numbers k such that k and k+1 are both greedy Jacobsthal-Niven numbers (A364379).

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A337076 Niven numbers (A005349) with a record gap to the next Niven number.

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A381582 Numbers k such that k and k+1 are both terms in A381581.

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A331089 Positive numbers k such that -k and -(k + 1) are both negative negaFibonacci-Niven numbers (A331088).

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A364007 Numbers k such that k and k+1 are both Wythoff-Niven numbers (A364006).

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A364124 Numbers k such that k and k+1 are both Stolarsky-Niven numbers (A364123).

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A338514 Numbers k such that k and k+1 are both divisible by the total binary weight of their divisors (A093653).