A334311
Starts of runs of 4 consecutive base phi Niven numbers (A334308).
Original entry on oeis.org
285129, 1958893, 2501533, 6488440, 7069840, 8803023, 16514327, 23826399, 34031773, 52256248, 68198847, 72969138, 76779087, 77622950, 87430210, 87474672, 96485487, 114137958, 120197293, 136275022, 151444458, 173740578, 174878352, 183872325, 188385855, 196268415
Offset: 1
285129 is a term since 285129, 285130, 285131 and 285132 are all base phi Niven numbers.
-
phiDigSum[1] = 1; phiDigSum[n_] := Plus @@ RealDigits[n, GoldenRatio, 2*Ceiling[ Log[GoldenRatio, n] ]][[1]]; phiNivenQ[n_] := Divisible[n, phiDigSum[n]]; q1 = phiNivenQ[1]; q2 = phiNivenQ[2]; q3 = phiNivenQ[3]; seq = {}; Do[q4 = phiNivenQ[n]; If[q1 && q2 && q3 && q4, AppendTo[seq, n - 3]]; q1 = q2; q2 = q3; q3 = q4, {n, 4, 10^5}]; seq
A352092
Starts of runs of 4 consecutive tribonacci-Niven numbers (A352089).
Original entry on oeis.org
1602, 218349, 296469, 1213749, 1291869, 1896630, 1952070, 2153709, 2399550, 3149109, 3753870, 3809310, 3983229, 4226208, 4256790, 4449288, 4711482, 5707897, 5727708, 6141750, 6589230, 6969429, 7205757, 7229208, 7276143, 7292943, 7454710, 7752588, 7937109, 8877069
Offset: 1
1602 is a term since 1602, 1603, 1604 and 1605 are all divisible by the number of terms in their minimal tribonacci representation:
k A278038(k) A278043(k) k/A278043(k)
--------------------------------------------
1602 110100011010 6 267
1603 110100011011 7 229
1604 110100100000 4 401
1605 110100100001 5 321
-
t[1] = 1; t[2] = 2; t[3] = 4; t[n_] := t[n] = t[n - 1] + t[n - 2] + t[n - 3]; triboNivenQ[n_] := Module[{s = {}, m = n, k}, While[m > 0, k = 1; While[t[k] <= m, k++]; k--; AppendTo[s, k]; m -= t[k]; k = 1]; Divisible[n, DigitCount[Total[2^(s - 1)], 2, 1]]]; seq[count_, nConsec_] := Module[{tri = triboNivenQ /@ Range[nConsec], s = {}, c = 0, k = nConsec + 1}, While[c < count, If[And @@ tri, c++; AppendTo[s, k - nConsec]]; tri = Join[Rest[tri], {triboNivenQ[k]}]; k++]; s]; seq[6, 4]
A352110
Starts of runs of 4 consecutive lazy-tribonacci-Niven numbers (A352107).
Original entry on oeis.org
1081455, 1976895, 2894175, 5886255, 6906912, 15604110, 16588752, 19291479, 20387232, 25919439, 32394942, 34801557, 35654175, 36813582, 36907899, 39117219, 41407392, 43520832, 46181055, 47954499, 52145952, 54524319, 54815397, 56733639, 57775102, 58942959, 59292177
Offset: 1
1081455 is a term since 1081455, 1081456, 1081457 and 1081458 are all divisible by the number of terms in their maximal tribonacci representation:
k A352103(k) A352104(k) k/A352104(k)
------- ----------------------- ---------- ------------
1081455 10101011011110110011110 15 72097
1081456 10101011011110110011111 16 67591
1081457 10101011011110110100100 13 83189
1081458 10101011011110110100101 14 77247
A330929
Starts of runs of 6 consecutive Niven (or Harshad) numbers (A005349).
Original entry on oeis.org
1, 2, 3, 4, 5, 10000095, 10000096, 12751220, 14250624, 22314620, 22604423, 25502420, 28501224, 35521222, 41441420, 41441421, 51004820, 56511023, 57002424, 70131620, 71042422, 71253024, 97740760, 102009620, 111573020, 114004824, 121136420, 124324220, 124324221
Offset: 1
10000095 is a term since 10000095 is divisible by 1 + 0 + 0 + 0 + 0 + 0 + 9 + 5 = 15, 10000096 is divisible by 16, ..., and 10000100 is divisible by 2.
- Jean-Marie De Koninck, Those Fascinating Numbers, American Mathematical Society, 2009, p. 36, entry 110.
- Amiram Eldar, Table of n, a(n) for n = 1..4000
- Curtis Cooper and Robert E. Kennedy, On consecutive Niven numbers, Fibonacci Quarterly, Vol. 21, No. 2 (1993), pp. 146-151.
- Helen G. Grundman, Sequences of consecutive Niven numbers, Fibonacci Quarterly, Vol. 32, No. 2 (1994), pp. 174-175.
- Wikipedia, Harshad number.
- Brad Wilson, Construction of 2n consecutive n-Niven numbers, Fibonacci Quarterly, Vol. 35, No. 2 (1997), pp. 122-128.
-
f:=func; a:=[]; for k in [1..30000000] do if forall{m:m in [0..5]|f(k+m)} then Append(~a,k); end if; end for; a; // Marius A. Burtea, Jan 03 2020
-
nivenQ[n_] := Divisible[n, Total @ IntegerDigits[n]]; niv = nivenQ /@ Range[6]; seq = {}; Do[niv = Join[Rest[niv], {nivenQ[k]}]; If[And @@ niv, AppendTo[seq, k - 5]], {k, 6, 10^7}]; seq
A330930
Starts of runs of 7 consecutive Niven (or Harshad) numbers (A005349).
Original entry on oeis.org
1, 2, 3, 4, 10000095, 41441420, 124324220, 124324221, 124324222, 207207020, 233735070, 331531220, 350602590, 409036350, 414414020, 467470110, 621621020, 621621021, 621621022, 1030302012, 1036035020, 1051807710, 1201800620, 1243242020, 1243242021, 1243242022
Offset: 1
10000095 is a term since 10000095 is divisible by 1 + 0 + 0 + 0 + 0 + 0 + 9 + 5 = 15, 10000096 is divisible by 16, ..., and 10000101 is divisible by 3.
- Jean-Marie De Koninck, Those Fascinating Numbers, American Mathematical Society, 2009, p. 36, entry 110.
- Amiram Eldar, Table of n, a(n) for n = 1..400
- Curtis Cooper and Robert E. Kennedy, On consecutive Niven numbers, Fibonacci Quarterly, Vol. 21, No. 2 (1993), pp. 146-151.
- Helen G. Grundman, Sequences of consecutive Niven numbers, Fibonacci Quarterly, Vol. 32, No. 2 (1994), pp. 174-175.
- Wikipedia, Harshad number.
- Brad Wilson, Construction of 2n consecutive n-Niven numbers, Fibonacci Quarterly, Vol. 35, No. 2 (1997), pp. 122-128.
-
nivenQ[n_] := Divisible[n, Total @ IntegerDigits[n]]; niv = nivenQ /@ Range[7]; seq = {}; Do[niv = Join[Rest[niv], {nivenQ[k]}]; If[And @@ niv, AppendTo[seq, k - 6]], {k, 7, 10^7}]; seq
A352345
Starts of runs of 4 consecutive lazy-Pell-Niven numbers (A352342).
Original entry on oeis.org
750139, 41765247, 54831951, 56423275, 136038447, 151175724, 223956843, 227483124, 293913170, 362557214, 382572475, 457616575, 502106253, 562407324, 586380624, 637133390, 724382239, 771849439, 774421478, 859463253, 926398647, 953750523, 1043787390, 1193063550
Offset: 1
750139 is a term since 750139, 750140, 750141 and 750142 are all divisible by the sum of the digits in their maximal Pell representation:
k A352339(k) A352340(k) k/A352340(k)
------ ---------------- --------- -----------
750139 1102022021112220 19 39481
750140 1102022021112221 20 37507
750141 1102022021112222 21 35721
750142 1102022021120210 17 44126
Similar sequences:
A141769,
A328211,
A328207,
A328215,
A330933,
A331824,
A334311,
A342429,
A344344,
A352092,
A352110.
A352511
Starts of runs of 4 consecutive Catalan-Niven numbers (A352508).
Original entry on oeis.org
144, 15630, 164862, 202761, 373788, 450189, 753183, 1403961, 1779105, 2588415, 2673774, 2814229, 2850880, 3009174, 3013722, 3045870, 3091023, 3702390, 3942519, 4042950, 4432128, 4725432, 4938348, 5718942, 5907312, 6268248, 6519615, 6592752, 6791379, 7095492, 8567802
Offset: 1
144 is a term since 144, 145, 146 and 147 are all divisible by the sum of the digits in their Catalan representation:
k A014418(k) A014420(k) k/A014420(k)
--- ---------- ---------- ------------
144 100210 4 36
145 100211 5 29
146 101000 2 73
147 101001 3 49
Similar sequences:
A141769,
A328211,
A328207,
A328215,
A330933,
A331824,
A334311,
A342429,
A344344,
A352092,
A352110,
A352345.
-
c[n_] := c[n] = CatalanNumber[n]; catNivQ[n_] := Module[{s = {}, m = n, i}, While[m > 0, i = 1; While[c[i] <= m, i++]; i--; m -= c[i]; AppendTo[s, i]]; Divisible[n, Plus @@ IntegerDigits[Total[4^(s - 1)], 4]]]; seq[count_, nConsec_] := Module[{cn = catNivQ /@ Range[nConsec], s = {}, c = 0, k = nConsec + 1}, While[c < count, If[And @@ cn, c++; AppendTo[s, k - nConsec]]; cn = Join[Rest[cn], {catNivQ[k]}]; k++]; s]; seq[5, 4]
A364219
Starts of runs of 4 consecutive integers that are Jacobsthal-Niven numbers (A364216).
Original entry on oeis.org
1, 42, 43, 2731, 11605, 13024, 14229, 25983, 39390, 45727, 46624, 47529, 60073, 96039, 111390, 131103, 132010, 133984, 134430, 140767, 148180, 148181, 148509, 174762, 174763, 187744, 197790, 237609, 247114, 266453, 275229, 287988, 312190, 330847, 354429, 370269
Offset: 1
Similar sequences:
A141769,
A328211,
A328207,
A328215,
A330933,
A331824,
A334311,
A342429,
A344344,
A352092,
A352110,
A352345,
A352511.
-
consecJacobsthalNiven[4*10^5, 4] (* using the function from A364217 *)
-
lista(4*10^5, 4) \\ using the function from A364217
A364382
Starts of runs of 4 consecutive integers that are greedy Jacobsthal-Niven numbers (A364379).
Original entry on oeis.org
1, 2, 3, 8, 9, 42, 43, 84, 85, 2730, 2731, 5460, 5461, 21864, 21865, 59477, 60073, 66303, 75048, 112509, 156607, 174762, 174763, 283327, 312190, 320768, 349524, 349525, 351570, 354429, 374589, 384039, 479037, 504510, 527103, 624040, 625470, 656829, 688830, 711423
Offset: 1
Similar sequences:
A141769,
A328211,
A328207,
A328215,
A330933,
A331824,
A334311,
A342429,
A344344,
A352092,
A352110,
A352345,
A352511,
A364219.
-
consecGreedyJN[72000, 4] (* using the function consecGreedyJN from A364380 *)
-
lista(10^5, 4) \\ using the function lista from A364380
A381584
Starts of runs of 4 consecutive integers that are all terms in A381581.
Original entry on oeis.org
1, 55, 2104, 5222, 24784, 63510, 64264, 69487, 95463, 121393, 184327, 327303, 374589, 463110, 468168, 561069, 572550, 596868, 671407, 740310, 759030, 819948, 902670, 956680, 1023009, 1036230, 1065030, 1259817, 1274910, 1359552, 1683154, 1714470, 1731750, 2182023
Offset: 1
1 is a term since A291711(1) = 1 divides 1, A291711(2) = 2 divides 2, A291711(3) = 1 divides 3, and A291711(4) = 2 divides 4.
55 is a term since A291711(55) = 1 divides 55, A291711(56) = 2 divides 56, A291711(57) = 3 divides 57, and A291711(58) = 2 divides 58.
-
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]]; seq[count_, nConsec_] := Module[{cn = q /@ Range[nConsec], s = {}, c = 0, k = nConsec + 1}, While[c < count, If[And @@ cn, c++; AppendTo[s, k - nConsec]]; cn = Join[Rest[cn], {q[k]}]; k++]; s]; seq[12, 4]
-
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 = is1(2), q3 = is1(3), q4); for(k = 4, lim, q4 = is1(k); if(q1 && q2 && q3 && q4, print1(k-3, ", ")); q1 = q2; q2 = q3; q3 = q4);}
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