A337238 -id:A337238 - cp's OEIS frontend

A337258 Primes p such that p and the prime next to p are both digitally balanced numbers in base 2 (A031443).

37, 139, 557, 563, 613, 647, 653, 659, 2389, 2467, 2699, 2851, 8311, 8423, 8627, 8677, 8681, 8807, 8819, 9011, 9043, 9049, 9157, 9319, 9323, 9419, 9613, 9803, 9811, 9817, 9829, 9923, 10331, 10343, 10453, 10597, 11279, 11317, 11353, 11399, 11587, 11783, 11789

Offset: 1

Amiram Eldar, Nov 21 2020

Prime p such that p and the prime next to p are both terms of A066196.

			37 is a term since it is a prime number, and both 37 and the next prime, 41, are digitally balanced in base 2: the binary representation of 37 is 100101, the binary representation of 41 is 101001, and both contain 3 0's and 3 1's.

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

Cf. A031443, A066196, A095018, A337238.

digBalQ[n_] := Module[{d = IntegerDigits[n, 2], m}, EvenQ@(m = Length@d) && Count[d, 1] == m/2]; p = Select[Range[3*10^4], PrimeQ]; p[[Position[Partition[digBalQ /@ p, 2, 1], {True, True}] // Flatten]]

A373460 Numbers k such that k and k+1 both have an equal number of even and odd digits.

Original entry on oeis.org

29, 49, 69, 89, 1009, 1029, 1049, 1069, 1089, 1209, 1229, 1249, 1269, 1289, 1409, 1429, 1449, 1469, 1489, 1609, 1629, 1649, 1669, 1689, 1809, 1829, 1849, 1869, 1889, 2109, 2129, 2149, 2169, 2189, 2309, 2329, 2349, 2369, 2389, 2509, 2529, 2549, 2569, 2589, 2709

Offset: 1

Amiram Eldar, Jun 07 2024

The terms are of the form 100*m + j, where m is either 0 or a term of A227870 and j is in {29, 49, 69, 89} if m = 0 or in {9, 29, 49, 69, 89} if m > 0.

			29 is a term since it has one even digit (2) and one odd digit (9), and 29+1 = 30 also has one even digit (0) and one odd digit (3).

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

Subsequence of A017377 and A227870.

Cf. A337238 (binary analog), A373505.

q[n_] := Module[{d = Differences[Tally[Mod[IntegerDigits[n], 2]]]}, d != {} && d[[1, 2]] == 0]; Select[Range[3000], q[#] && q[# + 1] &]

iseq(n) = {my(o = 0, e = 0); while(n > 0, if((n%10) % 2 == 0, e++, o++); n \= 10); e == o;}
lista(kmax) = {my(q1 = 0, q2); for(k = 1, kmax, q2 = iseq(k); if(q1 && q2, print1(k-1, ", ")); q1 = q2);}

a(n) = 100 * A227870(floor(n/5)) + 20 * (n mod 5) + 9, for n > 4.

A373505 Numbers k such that k and k+1 both have an equal number of odd and even digits in their factorial-base representations.

Original entry on oeis.org

25, 29, 37, 41, 55, 67, 73, 77, 85, 89, 103, 115, 727, 739, 745, 749, 757, 761, 775, 787, 793, 797, 805, 809, 823, 835, 841, 845, 853, 857, 889, 893, 901, 905, 937, 941, 949, 953, 967, 979, 985, 989, 997, 1001, 1015, 1027, 1033, 1037, 1045, 1049, 1063, 1075, 1081

Offset: 1

Amiram Eldar, Jun 07 2024

If m is the sum of the first k odd-indexed factorial numbers (A000142), for k >= 2, then m-1 is a term, since the factorial-base representation of m is 1010...10, with the block "10" repeated k times, and the factorial-base representation of m-1 is the 1010...1001, with the block "10" repeated k-1 times and followed by "01" (these numbers are 25, 745, 41065, 3669865, 482671465, ...).

			25 is a term since the factorial-base representations of 25 and 26 are 1001 and 1010, respectively, and both have 2 odd digits and 2 even digits.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Wikipedia, Factorial number system.
Index entries for sequences related to factorial base representation..

Subsequence of A351895.
Cf. A000142, A007623.
Similar sequences: A337238, A373460.

With[{max = 7}, fctBaseDigits[n_] := IntegerDigits[n, MixedRadix[Range[max, 2, -1]]]; s = Select[Range[1, max!], EvenQ[Length[(d = fctBaseDigits[#])]] && Count[d, _?EvenQ] == Length[d]/2 &]; ind = Position[Differences[s], 1] // Flatten; s[[ind]]]

iseq(n) = {my(p = 2, o = 0, e = 0); while(n > 0, if((n%p) %2  == 0, e++, o++); n \= p; p++); e == o;}
lista(kmax) = {my(q1 = 0, q2); for(k = 1, kmax, q2 = iseq(k); if(q1 && q2, print1(k-1, ", ")); q1 = q2);}

A351896 Numbers k such that k and k+2 both have an equal number of odd and even digits in their factorial-base representations.

Original entry on oeis.org

71, 743, 791, 839, 862, 910, 983, 1031, 1079, 1102, 1150, 1223, 1271, 1319, 1342, 1390, 1583, 1631, 1823, 1871, 2063, 2111, 2183, 2231, 2279, 2302, 2350, 2423, 2471, 2519, 2542, 2590, 2663, 2711, 2759, 2782, 2830, 3023, 3071, 3263, 3311, 3503, 3551, 3623, 3671, 3719

Offset: 1

Amiram Eldar, Feb 24 2022

			71 is a term since the factorial-base representations of 71 and 73 are 2321 and 3001, respectively, and both have 2 odd digits and 2 even digits.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Wikipedia, Factorial number system.
Index entries for sequences related to factorial base representation.

Subsequence of A351895.
Cf. A007623, A351893, A351894.
Similar sequence: A337238.

max = 7; fctBaseDigits[n_] := IntegerDigits[n, MixedRadix[Range[max, 2, -1]]]; s = Select[Range[1, max!], EvenQ[Length[(d = fctBaseDigits[#])]] && Count[d, _?EvenQ] == Length[d]/2 &]; ind = Position[Differences[s], 2] // Flatten; s[[ind]]

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A337258 Primes p such that p and the prime next to p are both digitally balanced numbers in base 2 (A031443).

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A373460 Numbers k such that k and k+1 both have an equal number of even and odd digits.

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A373505 Numbers k such that k and k+1 both have an equal number of odd and even digits in their factorial-base representations.

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Author

Keywords

Comments

Examples

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Programs

Mathematica

PARI

A351896 Numbers k such that k and k+2 both have an equal number of odd and even digits in their factorial-base representations.

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Author

Keywords

Examples

Links

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Programs

Mathematica